Atmos Chem Phys 7 3163ndash3193 2007wwwatmos-chem-physnet731632007copy Author(s) 2007 This work is licensedunder a Creative Commons License

AtmosphericChemistry

and Physics

Reformulating atmospheric aerosol thermodynamics andhygroscopic growth into fog haze and clouds

S Metzger and J Lelieveld

Max Planck Institute for Chemistry Mainz Germany

Received 30 November 2006 ndash Published in Atmos Chem Phys Discuss 18 January 2007Revised 29 May 2007 ndash Accepted 1 June 2007 ndash Published 20 June 2007

Abstract Modeling atmospheric aerosol and cloud micro-physics is rather complex even if chemical and thermody-namical equilibrium is assumed We show however thatthe thermodynamics can be considerably simplified by re-formulating equilibrium to consistently include water andtransform laboratory-based concepts to atmospheric condi-tions We generalize the thermodynamic principles that ex-plain hydration and osmosis ndash merely based on solute solu-bilities ndash to explicitly account for the water mass consumedby hydration As a result in chemical and thermodynamicalequilibrium the relative humidity (RH) suffices to determinethe saturation molality including solute and solvent activities(and activity coefficients) since the water content is fixed byRH for a given aerosol concentration and type As a conse-quence gasliquidsolid aerosol equilibrium partitioning canbe solved analytically and non-iteratively Our new conceptenables an efficient and accurate calculation of the aerosolwater mass and directly links the aerosol hygroscopic growthto fog haze and cloud formation

We apply our new concept in the 3rd Equilibrium Sim-plified Aerosol Model (EQSAM3) for use in regional andglobal chemistry-transport and climate models Its input islimited to the speciesrsquo solubilities from which a newly intro-duced stoichiometric coefficient for water is derived Analo-gously we introduce effective stoichiometric coefficients forthe solutes to account for complete or incomplete dissocia-tion We show that these coefficients can be assumed con-stant over the entire activity range and calculated for variousinorganic organic and non-electrolyte compounds includingalcohols sugars and dissolved gases EQSAM3 calculatesthe aerosol composition and gasliquidsolid partitioning ofmixed inorganicorganic multicomponent solutions and theassociated water uptake for almost 100 major compoundsIt explicitly accounts for particle hygroscopic growth by

Correspondence toS Metzger(metzgermpch-mainzmpgde)

computing aerosol properties such as single solute molali-ties molal based activities including activity coefficients forvolatile compounds efflorescence and deliquescence relativehumidities of single solute and mixed solutions Various ap-plications and a model inter-comparison indicate that a) theapplication is not limited to dilute binary solutions b) sen-sitive aerosol properties such as hygroscopic growth and thepH of binary and mixed inorganicorganic salt solutions upto saturation can be computed accurately and c) aerosol wa-ter is central in modeling atmospheric chemistry visibilityweather and climate

1 Introduction

It is widely acknowledged that atmospheric aerosol parti-cles affect human and ecosystem health clouds and climate(eg EPA 1996 Holgate et al 1999 Seinfeld and Pandis1998 IPCC 2007) However it is less well recognized thatgasliquidsolid partitioning of atmospheric particles and pre-cursor gases largely determine the composition and hygro-scopicity of the aerosols which in turn govern the size distri-bution the atmospheric lifetime of both the particles and theinteracting gases and the particle optical properties For in-stance sea salt particles can deliquesce at a very low relativehumidity (RH) ofsim32 since they contain a small amountof the very hygroscopic salt magnesium chloride (MgCl2)Therefore marine air is often much hazier than continentalair at the same ambient temperature (T ) and RH

Overall the most abundant aerosol species is water Fora givenT and RH aerosol water determines the phase parti-tioning between the gas-liquid-solid and ice phases and thecomposition of atmospheric aerosols due to changes in thevapor pressure above the particle surface (Pruppacher andKlett 1997) The hygroscopic growth of the aerosol particlesinfluences heterogeneous reactions light extinction and vis-ibility whereby aerosol water is most relevant for the direct

Published by Copernicus Publications on behalf of the European Geosciences Union

3164 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

radiative forcing of Earthrsquos climate (Pilinis et al 1995)However aerosol water depends besides the meteorologicalconditions on the ionic composition of the particles whichin turn depends on the aerosol water mass Consequentlygas-aerosol partitioning and aerosol water mass are difficultto measure or predict numerically (by established methods)even if the complex gas-aerosol system is simplified by as-suming thermodynamic gasliquidsolid aerosol equilibrium(Seinfeld and Pandis 1998 and references therein)

The underlying principles that govern the gas-aerosolequilibrium partitioning and hygroscopic growth have beenformulated toward the end of the nineteenth century by Gibbs(1839ndash1903) the architect of equilibrium thermodynamicsMost of our current understanding of equilibrium which fol-lows from the second law of thermodynamics derives fromGibbs (1876) Among the numerous publications that haveappeared since none has consistently transformed the ba-sic principles of equilibrium thermodynamics to atmosphericaerosol modeling applications

The problem is twofold (a) the amount of water mass con-sumed by hydration is not explicitly accounted for althoughhydration drives the hygroscopic growth of natural and man-made aerosols and (b) the water mass used to define theaerosol activity coefficients is kept constant which is rea-sonable for laboratory but not for atmospheric conditions ndashthus leading to a conceptual difficulty

Here we overcome these problems by introducing a newapproach for the calculation of the aerosol hygroscopicgrowth and the subsequent water uptake into fog haze andclouds which has several advantagesFirst the complexsystem of the gasliquidsolid aerosol equilibrium partition-ing can be solved analytically which limits computationalrequirementsSecond a large number of aerosol physical-chemical properties can be directly and explicitly computedThis includes aerosol activities (and activity coefficients) thewater activity (with or without the Kelvin term) single so-lute molalities of binary and mixed solutions of pure inor-ganic or mixed inorganicorganic compounds efflorescenceand deliquescence RHs of soluble salt compounds (binaryand mixed solutions) and related optical properties so thatsubsequently all relevant aerosol properties such as dry andambient radii mass and number distribution can be directlyderived at a given RH andT

T hird our new concept allows to consistently and effi-ciently link aerosol thermodynamics and cloud microphysicsthrough explicit computation of the aerosol water mass fromwhich the initial cloud waterice mass cloud dropleticenumber concentrations and the cloud cover can be calcu-lated Fourth the explicit account for the water mass con-sumed by hydration can be directly connected to aerosolchemical composition and emission sources being charac-terized by a certain mix of compounds that can undergo at-mospheric chemistry

Thus our method allows to explicitly link emissions toatmospheric conditions including visibility reduction and

climate forcing through anthropogenic activities This notonly helps to abandon the use of ambiguous terms such asldquomarinerdquo and ldquocontinentalrdquo aerosols it also allows to refinelumped categories such as mineral dust biomass burningsea salt organic and sulfate aerosols currently used in at-mospheric modeling Consequently our method is more ex-plicit than the traditional concept of cloud condensation nu-clei (CCN) which relies on activation thresholds that do notexplicitly relate the particle chemical composition to dropletformation

In Sect 2 we generalize the basic thermodynamic princi-ples of hydration and osmosis and translate them to atmo-spheric conditions In Sect 3 we introduce new formulationsrequired to consistently calculate the activity (including so-lute molalities and activity coefficients) and the water massof atmospheric aerosols and we demonstrate how the aerosoland cloud thermodynamics can be directly coupled by refor-mulating chemical equilibrium to consistently include waterIn Sect 4 we apply our new approach in a new version (3) ofthe EQuilibrium Simplified Aerosol Model (EQSAM3) andpresent results of a model inter-comparison The new aspectsand limitations of our new approach are discussed in Sect 5and we conclude with Sect 6 The appendix includes tablesof acronyms abbreviations and symbols used in this study

In an electronic supplement(httpwwwatmos-chem-physnet731632007acp-7-3163-2007-supplementzip) we provide (1) ad-ditional figures of single solute molalities and the associatedwater uptake for all compounds listed in Table 1 (com-plementing Fig 2a and b) including a list of compoundsand (2) the derivation of the standard definitions of theldquoclassicalrdquo equilibrium thermodynamics (eg Denbigh1981 Seinfeld and Pandis 1998 Wexler and Potukuchi1998)

2 Revisiting thermodynamic principles

We refer to ldquoclassicalrdquo equilibrium thermodynamics sincewater is generally omitted in equilibrium equations unlessexplicitly involved in the reaction based on the assumptionthat water is neither consumed nor produced However whenhydration is involved water is consumed and released whichcauses inconsistencies in the standard treatment

Furthermore equilibrium thermodynamics of atmosphericaerosols have ndash thus far ndash been defined for laboratory con-ditions and subsequently applied to atmospheric modelingThis introduces a conceptual difficulty in particular for labo-ratory based activity coefficients which are central in aerosolthermodynamics In the laboratory the water mass is usuallyheld constant to measure aerosol activity and activity coeffi-cients as a function of solute concentration

In contrast to the laboratory condensed water in the at-mosphere is not a constant At equilibrium the aerosol wa-ter is proportional to the solute concentration More soluble

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

3164 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

radiative forcing of Earthrsquos climate (Pilinis et al 1995)However aerosol water depends besides the meteorologicalconditions on the ionic composition of the particles whichin turn depends on the aerosol water mass Consequentlygas-aerosol partitioning and aerosol water mass are difficultto measure or predict numerically (by established methods)even if the complex gas-aerosol system is simplified by as-suming thermodynamic gasliquidsolid aerosol equilibrium(Seinfeld and Pandis 1998 and references therein)

The underlying principles that govern the gas-aerosolequilibrium partitioning and hygroscopic growth have beenformulated toward the end of the nineteenth century by Gibbs(1839ndash1903) the architect of equilibrium thermodynamicsMost of our current understanding of equilibrium which fol-lows from the second law of thermodynamics derives fromGibbs (1876) Among the numerous publications that haveappeared since none has consistently transformed the ba-sic principles of equilibrium thermodynamics to atmosphericaerosol modeling applications

The problem is twofold (a) the amount of water mass con-sumed by hydration is not explicitly accounted for althoughhydration drives the hygroscopic growth of natural and man-made aerosols and (b) the water mass used to define theaerosol activity coefficients is kept constant which is rea-sonable for laboratory but not for atmospheric conditions ndashthus leading to a conceptual difficulty

Here we overcome these problems by introducing a newapproach for the calculation of the aerosol hygroscopicgrowth and the subsequent water uptake into fog haze andclouds which has several advantagesFirst the complexsystem of the gasliquidsolid aerosol equilibrium partition-ing can be solved analytically which limits computationalrequirementsSecond a large number of aerosol physical-chemical properties can be directly and explicitly computedThis includes aerosol activities (and activity coefficients) thewater activity (with or without the Kelvin term) single so-lute molalities of binary and mixed solutions of pure inor-ganic or mixed inorganicorganic compounds efflorescenceand deliquescence RHs of soluble salt compounds (binaryand mixed solutions) and related optical properties so thatsubsequently all relevant aerosol properties such as dry andambient radii mass and number distribution can be directlyderived at a given RH andT

T hird our new concept allows to consistently and effi-ciently link aerosol thermodynamics and cloud microphysicsthrough explicit computation of the aerosol water mass fromwhich the initial cloud waterice mass cloud dropleticenumber concentrations and the cloud cover can be calcu-lated Fourth the explicit account for the water mass con-sumed by hydration can be directly connected to aerosolchemical composition and emission sources being charac-terized by a certain mix of compounds that can undergo at-mospheric chemistry

Thus our method allows to explicitly link emissions toatmospheric conditions including visibility reduction and

climate forcing through anthropogenic activities This notonly helps to abandon the use of ambiguous terms such asldquomarinerdquo and ldquocontinentalrdquo aerosols it also allows to refinelumped categories such as mineral dust biomass burningsea salt organic and sulfate aerosols currently used in at-mospheric modeling Consequently our method is more ex-plicit than the traditional concept of cloud condensation nu-clei (CCN) which relies on activation thresholds that do notexplicitly relate the particle chemical composition to dropletformation

In Sect 2 we generalize the basic thermodynamic princi-ples of hydration and osmosis and translate them to atmo-spheric conditions In Sect 3 we introduce new formulationsrequired to consistently calculate the activity (including so-lute molalities and activity coefficients) and the water massof atmospheric aerosols and we demonstrate how the aerosoland cloud thermodynamics can be directly coupled by refor-mulating chemical equilibrium to consistently include waterIn Sect 4 we apply our new approach in a new version (3) ofthe EQuilibrium Simplified Aerosol Model (EQSAM3) andpresent results of a model inter-comparison The new aspectsand limitations of our new approach are discussed in Sect 5and we conclude with Sect 6 The appendix includes tablesof acronyms abbreviations and symbols used in this study

In an electronic supplement(httpwwwatmos-chem-physnet731632007acp-7-3163-2007-supplementzip) we provide (1) ad-ditional figures of single solute molalities and the associatedwater uptake for all compounds listed in Table 1 (com-plementing Fig 2a and b) including a list of compoundsand (2) the derivation of the standard definitions of theldquoclassicalrdquo equilibrium thermodynamics (eg Denbigh1981 Seinfeld and Pandis 1998 Wexler and Potukuchi1998)

2 Revisiting thermodynamic principles

We refer to ldquoclassicalrdquo equilibrium thermodynamics sincewater is generally omitted in equilibrium equations unlessexplicitly involved in the reaction based on the assumptionthat water is neither consumed nor produced However whenhydration is involved water is consumed and released whichcauses inconsistencies in the standard treatment

Furthermore equilibrium thermodynamics of atmosphericaerosols have ndash thus far ndash been defined for laboratory con-ditions and subsequently applied to atmospheric modelingThis introduces a conceptual difficulty in particular for labo-ratory based activity coefficients which are central in aerosolthermodynamics In the laboratory the water mass is usuallyheld constant to measure aerosol activity and activity coeffi-cients as a function of solute concentration

In contrast to the laboratory condensed water in the at-mosphere is not a constant At equilibrium the aerosol wa-ter is proportional to the solute concentration More soluble

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics 3165

(hygroscopic) particles take up more water from the atmo-sphere for solute hydration since each particle solution willbecome saturated at equilibrium (per definition) An increasein solute concentration (eg due to condensation of volatilecompounds coagulation or chemical reactions) therefore ei-ther leads to additional water uptake or to solute precipi-tation (causing a solid phase to co-exists with the aqueousphase) while a decrease of the solute concentration (eg dueto evaporational loss or chemical reactions) would be asso-ciated with the evaporation of aerosol water so that finallywhen equilibrium is reached again the aerosol molality re-mains constant at a given RH andT

The available water mass for condensation depends pri-marily on the available water vapor(Pw [Pa]) and the tem-perature dependent saturation vapor pressure(Pwsat[Pa])the ratio defines the relative humidity (RH=PwPwsat)However the aerosol water mass depends for a given amountof solute also on the hygroscopicity of the solute hygroscop-icity is the ability to absorb (release) water vapor from (to)the surrounding atmosphere In particular the ability of saltsolutes to hydrate causes hygroscopic growth of aerosol par-ticles at subsaturated atmospheric conditions (RHlt1) wherethe equilibrium water uptake of atmospheric aerosols is gen-erally limited by the available water vapor mass (respectivePw)

Instead near saturation or at supersaturation the hygro-scopic growth of aerosol particles continues and yields clouddroplets whereby the water uptake at RHge1 is only lim-ited by the temperature dependent saturation water vapormass Excess water vapor directly condenses into clouddropletscrystals and with limited water vapor availablelarger particles ultimately grow mainly dynamically at theexpense of smaller particles since their ability to collect wa-ter is largest (either due to their larger cross-sections or fallvelocities)

In either case (RHlt1 and RHge1) the water uptake is de-termined by the amount and type of solutes The water as-sociated with a certain amount and type of solute can beobtained from water activity measurements in the labora-tory (eg Tang and Munkelwitz 1994) or directly calculatedfrom the vapor pressure reduction that occurs after dissolvinga salt solute in water ndash well-known as Raoultrsquos law (Raoult1888) ndash if solution non-idealities are taken into account (War-neck 1988 Pruppacher and Klett 1997) Note Raoultrsquos lawcharacterizes the solvent Henryrsquos law the solute

However in the atmosphere at equilibrium conditionswhere evaporation balances condensation the vapor pres-sure reduction is fully compensated by the associated wa-ter uptake so that the equilibrium growth of aerosol parti-cles can be directly approximated as a function of relativehumidity (Metzger 2000 Metzger et al 2002a) Nev-ertheless no general concept yet exists despite the vari-ous efforts to predict the hygroscopic growth of atmosphericaerosols (Kohler 1936 Zadanovskii 1948 Robinson andStokes 1965 Stokes and Robinson 1966 Low 1969 Hanel

1970 Winkler 1973 Fitzgerald 1975 Hanel 1976 Win-kler 1988 for a more general overview see eg the textbooksof Pruppacher and Klett 1997 Seinfeld and Pandis 1998and references therein)

We therefore introduce in the following a new approachthat describes (a) the water uptake not only as a function ofrelative humidity for various inorganic and organic salt com-pounds found in natural or man-made pollution aerosols butalso (b) explicitly accounts for the water mass consumed byhydration (Sect 22) upon (c) considering the required trans-formation to atmospheric conditions (Sect 23)

Our new approach generalizes hydration and osmosis ac-cording to standard chemical methods by accounting for astoichiometric constant for water (Sect 221) which can bederived for any compound based on the measured or esti-mated solubility (Sect 34) This new stoichiometric con-stant for water is introduced in analogy to the stoichiometricconstants of the solutes For the latter we use effective onesto account for incomplete dissociation of certain solutes (egweak electrolytes or non-electrolytes) New formulas willalso be derived in Sect 35 36 37 to analytically (non-iteratively) predict for a given RH andT the aerosol molalityefflorescence and deliquescence RHs and the aerosol watermass associated with major inorganic and organic salt com-pounds and their mixtures including various non-electrolytecompounds such as sugars alcohols or dissolved gases

21 Laboratory conditions

211 Osmosis

In general the nature of hygroscopic growth of solutes is bestunderstood by using an osmotic system represented by onesolution separated from another by a semi-permeable mem-brane as illustrated in Fig 1a Osmosis first investigated byPfeffer (1881) describes the net flow of water through themembrane driven by a difference in solute concentrationswhich results in an osmotic pressure (turgor) The size of themembrane pores is large enough to separate small molecules(eg water or small ions) from larger ones (eg hydratedsodium and chloride ions)

Osmosis produces a pressure on a membrane5 [Pa])which depends primarily on the concentration of the solutethough also on its nature Adding a salt solute (eg 1 mole ofsodium chloride (NaCl) to the left compartment of Fig 1a)develops an osmotic pressure due to the additional volumeoccupied by the hydrated solute Depending on the natureof the solute ndash in particular its chemical bond strengths ndash thesalt solute can dissociate thus occupying more volume Ifthe volume expands in height a hydrostatic pressure buildsup which can be regarded as an osmotic counter pressure(Fig 1a) The associated decrease of the partial vapor pres-sure of either solute or solvent above the solution allows tomeasure the osmotic pressure quantitatively from the result-ing pressure differences above the two compartments For

wwwatmos-chem-physnet731632007 Atmos Chem Phys 7 3163ndash3193 2007

3166 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

wo A

A

TV(g) = const

Osmosis - Pfeffer cell

G = 0

w

Pw

(g)V(g)= nw

(g)RT

Pwo(g)V(g)= nwo

(g)RT

Pw

(g) = -w lt 0

Pw

(g)

w o

Pwo(g)

nwo = 5551 [molkgw]

= w + s

w = $wnw($wnw+$ens)

w

= w

-wo

TV(aq) = const

ns [mol] NaCl

$e = $e

+ + $e

- $w = $w

++ $w

-

H3O+

OH-

EQ

EQ

H2O

2 H2O = 1 H3O++ 1 OH

-

Pw(g) =w

Letzte Aumlnderung Thu Jul 31 2003 160754

W Pfeffer

copy Peter v Sengbusch - ImpressumW Pfeffer 1881

nwo = V(aq)(aq)Mw

pH = - log c(H3O+)

Pwo(g) =wo

Pw

Pwo(aq)

w o o

= awo

+ -

= o +

aw = RH awo = RHo

mss = ns nw 5551

w

= aw

$e

+ Na

+$e

- Cl

-

$w

+$w

-

w o = o

= 1

o Vo

(aq) = Pwo(aq) Vo

(aq)+o Ao

= $wnwo RT

V(aq) = ( Pw(aq) + Ps

(aq)) V(aq)+ A

= ($wnw + $ens) RT

Figure 1a)

Fig 1a Laboratory conditions schematic of an osmotic system

instance the total change in partial pressures yields for water1P

(g)w =P

(g)w minus P

(g)wo This change is primarily a measure of

the solutersquos concentration but also measures the solutersquos hy-groscopicity if the amount of water consumed by hydrationper mole of solute is explicitly accounted for

In equilibrium evaporation and condensation of watermolecules above each surface balances (Fig 1a) so that1P

(g)w adjusts to a maximum equal to the total change of

the osmotic pressure of water though with opposite sign(1P

(g)w =minus15w

) Although the magnitude of a molar pres-

sure change is characteristic for the solute different solutesthat occupy the same volume cause the same osmotic pres-sure (change) Note that we consider changes since bothcompartments may contain solutes We further consider thatthe osmotic pressure change15 includes a change that re-sults from both the solute(s) and solvent (water) ie a)15s which accounts for the additional volume of the so-lute and its partial or complete dissociation b)15w whichaccounts for the volume increase due to the associated addi-tional amount of water that causes solute hydration and dis-sociation and the associated dilution of the solution

212 Gas-solution analogy

An important aspect of the osmotic pressure is that it di-rectly relates aqueous and gas phase properties For in-stance for a closed osmotic system at equilibrium and con-stantT (Fig 1a) where evaporation and condensation of wa-ter molecules above each surface balances the water vaporpressures above both compartments must equal the corre-sponding osmotic pressures ieP

(g)w =5w andP

(g)wo=5wo

respectively The total energy of the system is conserved ie

1E = P(g)w V (g)

minus P(g)woV

(g)+ 5V (aq)

minus 5oV(aq)

= 0 (1)

For the gas phase the energy terms (egP(g)woV

(g)[kJ]) canbe expressed with the gas law (withR [JmolK] andT [K])in terms of moles (egn(g)

wo[mol]) so that eg for pure water

P(g)woV

(g)= n

(g)woRT (2)

Since equilibrium requiresP (g)wo=5wo we obtain accord-

ingly for the aqueous phase

5woV(aq)

= nwoRT (3)

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics 3167

Note that this gas-solution analogy also applies to non-reference conditions whereby Eqs (2) and (3) remain thesame but without the index ldquoordquo and that the gas-solutionanalogy was noted and used by van rsquot Hoff and Ostwald abouthalf a decade after Pfefferrsquos investigations to interpret the os-motic pressure (van rsquot Hoff 1887)

The important aspect about this gas-solution analogy isthat it already allows to explain the principles of osmosiswhereby it is ndash together with Arrheniusrsquo theory of partial dis-sociation of electrolytes in solutions (Arrhenius 1887) ndash di-rectly applicable to any solute concentration provided thatthe water consumed by hydration is explicitly accounted for

Therefore we first generalize in the following the princi-ples that explain osmosis whereby we consistently accountfor the amount of water consumed by hydration (Sect 22)Subsequently we transform these equations to atmosphericapplications (Sect 23) and reformulate equilibrium thermo-dynamics to consistently account for water (Sect 3)

22 Generalizing thermodynamic principles

To consistently account for the amount of water consumedby hydration we build on van rsquot Hoffrsquos gas-solution analogyand Arrheniusrsquo theory of partial dissociation of electrolytesin solutions We introduce a stoichiometric coefficient forwater ndash analogously to the stoichiometric coefficient(s) forthe solute(s) ndash to explicitly account on a molar basis for theamount of water consumed by hydration

221 Hydration

The osmotic pressure (Fig 1a) is caused by the hydration ofns moles of solute The hydration ldquoconsumesrdquo water whichleads to a change in volume For an equilibrium system anychange needs to be compensated resulting in water uptakeThe volume changes because of a) the additional volume ofsolute by which the solute partly or completely dissociatesdue to hydration b) due to the volume of water that is ldquocon-sumedrdquo by the hydration and c) the chemical restructuringof the solute and water molecules For some hygroscopic so-lutes such as eg magnesium chloride (MgCl2) this restruc-turing can even lead to a volume depression whereby theentropy of the hydrated magnesium chloride ions is smallerthan that of the crystalline salt (the hydrated MgCl2 ions havea higher structural order occupying less volume)

Strong electrolytes such as NaCl or MgCl2 dissociate prac-tically completely due to hydration The chemical dissocia-tion of eg sodium chloride (NaCl) (see example given inFig 1a) involves water that is consumed by the hydrationprocesses for which we formulate the equilibrium reaction

1 times nNaCl(cr)s + 1 times n

H2O(aq)w hArr ν+

e times nNa+

(aq)s +

νminuse times n

Clminus(aq)

s + ν+w times n

H3O+

(aq)w + νminus

w times nOHminus

(aq)w (R1)

The subscript ldquoerdquo denotes the stoichiometric coefficientsthat account for effective solute dissociation

(νe=ν+

e +νminuse

)

For strong electrolytes such as NaCl they equal the stoichio-metric coefficients for complete dissociation(νs=ν+

s +νminuss )

The solute hydration is associated with the consumptionof a certain number of water moles We therefore intro-duce a stoichiometric coefficientνw=ν+

w+νminusw for the solvent

(water) to account for the number of moles of solvent (wa-ter) needed for solvation (hydration) and solute dissociationIn Fig 1a the solution (left compartment) contains ldquowater-bindingrdquo particles and its volume expands at the expense ofthe right compartment due to water consumption In case ofa closed system the total aqueous

(V (aq)

)and gaseous

(V (g)

)volumes remain constant The total change in energy (leftand right compartment) can thus be expressed in analogy toEqs (2) and (3) in terms of moles of water

15wV (aq)= (5w minus 5wo)V

(aq)= νwnw RT (4a)

Similar to Eq (4a) we can express the energy contained in ei-ther of the compartments in terms of the number of moles ofsolute and solvent so that for the solution (left compartment)

5 V (aq)= (5wo + 15w)V (aq)

= (νwnw + νens)RT (4b)

222 Generalized mole fraction

The ratio of Eqs (4a) and (4b) provides a very useful ex-pression as it explicitly relates measurable changes in theosmotic pressure with the actual solute concentration It maybe regarded as the generalized mole fractionχw

15w5 = νwnw

(νwnw + νens) = χw (5)

χw expresses the ratio of a change relative to the total os-motic pressure of the solution in terms of water needed forthe hydration and effective dissociation for any amount ofsolute (throughνwnw andνens)

Note that this generalized mole fraction of water (χw)is introduced to explicitly account for the amount of wa-ter consumed by hydration while the ldquoclassicalrdquo mole frac-tion of water is less explicit being merely defined asχw=nw (nw+ns) Therefore various correction factorssuch as the vanrsquot Hoff factor practical osmotic coefficient oractivity coefficients have been introduced in the past to cor-rect the calculated values to the measured ones (Low 1969)

In analogy to the ldquoclassicalrdquo solute mole fractionχs=ns (nw+ns) we further introduce the generalized molefraction of the soluteχs=νens (νwnw+νens) whereby thesum of the mole fractions is unityχs+χw=1

Equation (5) involves the following relations

ndash 15=15w+15s rArr 15w=15minus15s

ndash 15=5 minus 5o with 5=5w+5s

ndash 15s=5sminus5so with 5so=0

wwwatmos-chem-physnet731632007 Atmos Chem Phys 7 3163ndash3193 2007

3168 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

T RH = const

G = V(aq) = 0Pw

(g) =w

(aq)Pw

(g) =wo(aq)

III) RH 1 =gt nwnwsat

(g)

Pwsat(g)(T)V(g) = nwsat

(g)

RT

RH = Pw(g)Pwsat

(g)(T)

RH = nw

(g)nwsat

(g)

II) RH 1 =gt nwnw

(g)

I) =gt wnw $ ens

V(aq) = ( Pw(aq) + Ps

(aq)) V(aq) + A o Vo

(aq) = ( Pwo(aq) + Pso

(aq)) Vo(aq) +o Ao

V(aq)= ens RT + wnw RT

Pwo(aq)=wo

(aq) -wo Ao Vo(aq)

Pw

(aq) =w

(aq) -w A V(aq)

RH = awo = aw

ase = aw

-w

mss = [we 5551(1RH-1)]

we

= (wnwo + enso) RT= (wnw + ens) RT

Pw(g)

EQA

ns = 1 micromol NaCl

w

H3O+

OH-

nw (ns RH)

V(aq)Pw(aq)

e = e

+ + e

-

e

+ Na

+

w

-+

w = w

++ w

-

- w

+

e

- Cl

-

H2O

Pw(g)

EQo Ao

nso = 1 nmol NaCl

wo

H3O+

OH-

nwo (nso RH)

Vo(aq)Pwo

(aq)

e = e

+ + e

-

e

+ Na

+

w

-+

w = w

++ w

-

- w

+

e

- Cl

-

H2O

Pw(g)

EQA

ns = 1 micromol NaCl

w

H3O+

OH-

nw (ns RH)

V(aq)Pw(aq)

e = e

+ + e

-

e

+ Na

+

w

-+

w = w

++ w

-

- w

+

e

- Cl

-

H2O

Pw(g)

EQo Ao

nso = 1 nmol NaCl

wo

H3O+

OH-

nwo (nso RH)

Vo(aq)Pwo

(aq)

e = e

+ + e

-

e

+ Na

+

w

-+

w = w

++ w

-

- w

+

e

- Cl

-

H2O

Aerosol water uptake

Figure 1 b)

Fig 1b Atmospheric conditions schematic of aerosol water uptake

ndash 15w=5wminus5wo with 5wo=5o

ndash 15w5=5w5=minus5s5=1minusχs

Equation (5) expresses the fraction of water required forhydration as the system compensates the vapor pressure re-duction through a net flow of water from the right into the leftcompartment (Fig 1a) For an osmotic (closed) system thisresults in a change in water activity(1aw=awominusaw) whichwould equal a (local) change in relative humidity(1RH)since at equilibrium the osmotic pressure of the solute orsolvent (in solution) equals the corresponding partial vaporpressure of solute or solvent above the solution

Note that the osmotic pressure is independent of the curva-ture of the surface In contrast to the theoretical solvent par-tial pressure in solution the (measurable) osmotic pressureis an effective pressure and hence at equilibrium it implicitlyaccounts for any surface tension or non-ideality effects Sur-face tension (or curvature) affects the solute solubility andthrough the solubility the osmotic pressure We account forsurface tension or non-ideality effects by our method throughνw as it is a pure function of the solutersquos solubility andνe

(see Sect 34 Eq 19)For an open system without a membrane the equilibrium

water uptake is the same It only depends on the hygroscopic

nature of the solute since the water activity is maintainedconstant by RH Atmospheric aerosols provide an exampleof such an open system

23 Atmosphere

The principles of osmosis that explain the nature of hygro-scopic growth of solutes in the laboratory also apply to at-mospheric aerosols However in case of equilibrium the sit-uation differs in the atmosphere since the vapor pressurereduction associated with the hydration of a solute is en-tirely compensated by water uptake as schematically shownin Fig 1b The reason is that in contrast to controlled equilib-rium conditions in the laboratory (closed system at constantT ) the aerosol(s) surrounding RH remains ndash practically ndashconstant in the atmosphere (open system) provided that thewater vapor concentration does not change due to the rela-tively small amount of condensing water needed for hydra-tion ndash a requirement that holds for tropospheric subsaturatedconditions (RHlt1) ndash see also cloud Sect 432

Thus at constantT and RH the water activity of atmo-spheric aerosols is fixed at equilibrium by the available wa-ter vapor concentration and hence equals the fractional rel-ative humidity (aw=RH) Similar to the laboratory a solute

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics 3169

specific amount of water is required for hydration whereasin this case the water needs to condense from the gas phaseFurthermore sinceaw=RH=const for the time period con-sidered to reach equilibrium and no membrane separates so-lute and solvent no hydrostatic counter pressure can buildup At equilibrium the vapor pressure reduction is there-fore fully compensated by the associated water uptake of thewater-binding solute(s)

1P(g)w = 0 (6)

The equilibrium condition further requires that the totalchange in energy is zero

1E = 15 1V (aq)= 0 (7)

Changes in energy resulting from the hydration of the solutecan be expressed in terms of the effective numbers of molesof hydrated solute(s) and the total amount of water that driveshydration

15 1V (aq)= νe 1nsR T + νw 1nwR T (8)

Since1P(g)w =15=0 we can rewrite Eq (8)

νe1ns = minusνw1nw (9)

or analogously to Eq (5) if divided by the osmotic pressureenergy of the solution

νe1ns

(νwnw + νens) = minusνw1nw

(νwnw + νens) (10)

whereby the total change in the amount of solute(νe1ns)

again causes a change in water activity(1aw) ndash equal to achange in relative humidity(1RH) ndash but compensated by theassociated water uptake(minusνw1nw) so thataw and RH re-main unchanged Note that the rhs of Eq (10) equals Eq (5)with respect to the reference condition wherenso=nwo=0and1nw=nwminusnwo and1ns=nsminusnso and that the wateractivity is defined as the ratio of the fugacity (the real gasequivalent of an ideal gas partial pressure) of the water toits fugacity under reference conditions usually approximatedby the more easily determined ratio of partial pressures

3 Reformulating equilibrium thermodynamics

In this section we reformulate the ldquoclassicalrdquo equilibriumthermodynamics to consistently include water (see the elec-tronic supplementhttpwwwatmos-chem-physnet731632007acp-7-3163-2007-supplementzipfor a brief summaryof the relevant standard treatment)

One difficulty arising from solution non-idealities is usu-ally circumvented by applying measured activity coefficientsndash being central in both aerosol thermodynamics and atmo-spheric modeling ndash to atmospheric aerosols but without therequired transformation to atmospheric conditions

The problem is that activity coefficients are usually mea-sured as a function of solute concentration while in the at-mosphere the aerosol activity is not a function of the solute

concentration since the water activity is fixed by RH if equi-librium is assumed Although it is well-known thataw=RH(eg Wexler and Potukuchi 1998) it has been overlookedthat this condition fixes all activities including solute andsolvent activity coefficients Therefore for a given type ofsolute they are all a function of RH independent of their con-centration

Despite polynomial fits or mathematical approximationsare used in atmospheric modeling which have been derivedfor various solute activities from laboratory measurements asa function of solute concentrations (eg Clegg et al 19961998 Clegg and Seinfeld 2006) whereby their applicationin numerical schemes requires computationally demandingsolutions We will show in the following that we can over-come this if we (a) explicitly include in all equilibrium reac-tions the amount of water consumed by solute hydration and(b) consistently limit the amount of water that is available forcondensation as mentioned in Sect 2 and described in moredetail in Sect 432

According to Sect 23 the aerosol water mass is ndash besidesits dependency on RH ndash a function of the mass and type of so-lute In contrast the activity of atmospheric aerosols is onlya function of the water vapor mass available for condensationand of the type of solute but not of its mass

The amount of water needed for solute hydration isonly determined by the solute specific constantsνe andνw and RH being independent of the solute concentration(Sect 23) The reason is that only the solutesrsquo solubility de-termines the amount of solute that can exist in a saturatedsolution Any excess of a non-volatile solute can only pre-cipitate out of the solution so that a solid and aqueous phaseco-exist while (semi)-volatile compounds can additionallyevaporate (hence maintaining gasliquidsolid equilibrium)

This has important implications because for equilibriumconditions the solute specific constantsνe andνw enable theexplicit calculation of single solute molalities from which inturn the solute specific water uptake and derived propertiescan be calculated as a function of RHνe andνw

Note that this approach extends beyond binary solutions tomixed solutions under the additional and also quite realisticand widely applied assumption of molar volume additivity(Hanel 1976) which holds for most soluble salt compoundsas they can in principle dissolve completely (depending onlyon the saturation) Our new approach may also be appliedto laboratory conditions if the term RH used in the followingequations is substituted byaw

31 Solubility constants

Molality is a measure of solubility At equilibrium the solu-tion is saturated so that it contains the maximum concentra-tion of ions that can exist in equilibrium with its solid (crys-talline) phase The amount of solute that must be added toa given volume of solvent to form a saturated solution is itssolubility At equilibrium the ion product equals the solubil-

wwwatmos-chem-physnet731632007 Atmos Chem Phys 7 3163ndash3193 2007

3166 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

wo A

A

TV(g) = const

Osmosis - Pfeffer cell

G = 0

w

Pw

(g)V(g)= nw

(g)RT

Pwo(g)V(g)= nwo

(g)RT

Pw

(g) = -w lt 0

Pw

(g)

w o

Pwo(g)

nwo = 5551 [molkgw]

= w + s

w = $wnw($wnw+$ens)

w

= w

-wo

TV(aq) = const

ns [mol] NaCl

$e = $e

+ + $e

- $w = $w

++ $w

-

H3O+

OH-

EQ

EQ

H2O

2 H2O = 1 H3O++ 1 OH

-

Pw(g) =w

Letzte Aumlnderung Thu Jul 31 2003 160754

W Pfeffer

copy Peter v Sengbusch - ImpressumW Pfeffer 1881

nwo = V(aq)(aq)Mw

pH = - log c(H3O+)

Pwo(g) =wo

Pw

Pwo(aq)

w o o

= awo

+ -

= o +

aw = RH awo = RHo

mss = ns nw 5551

w

= aw

$e

+ Na

+$e

- Cl

-

$w

+$w

-

w o = o

= 1

o Vo

(aq) = Pwo(aq) Vo

(aq)+o Ao

= $wnwo RT

V(aq) = ( Pw(aq) + Ps

(aq)) V(aq)+ A

= ($wnw + $ens) RT

Figure 1a)

Fig 1a Laboratory conditions schematic of an osmotic system

instance the total change in partial pressures yields for water1P

(g)w =P

(g)w minus P

(g)wo This change is primarily a measure of

the solutersquos concentration but also measures the solutersquos hy-groscopicity if the amount of water consumed by hydrationper mole of solute is explicitly accounted for

In equilibrium evaporation and condensation of watermolecules above each surface balances (Fig 1a) so that1P

(g)w adjusts to a maximum equal to the total change of

the osmotic pressure of water though with opposite sign(1P

(g)w =minus15w

) Although the magnitude of a molar pres-

sure change is characteristic for the solute different solutesthat occupy the same volume cause the same osmotic pres-sure (change) Note that we consider changes since bothcompartments may contain solutes We further consider thatthe osmotic pressure change15 includes a change that re-sults from both the solute(s) and solvent (water) ie a)15s which accounts for the additional volume of the so-lute and its partial or complete dissociation b)15w whichaccounts for the volume increase due to the associated addi-tional amount of water that causes solute hydration and dis-sociation and the associated dilution of the solution

212 Gas-solution analogy

An important aspect of the osmotic pressure is that it di-rectly relates aqueous and gas phase properties For in-stance for a closed osmotic system at equilibrium and con-stantT (Fig 1a) where evaporation and condensation of wa-ter molecules above each surface balances the water vaporpressures above both compartments must equal the corre-sponding osmotic pressures ieP

(g)w =5w andP

(g)wo=5wo

respectively The total energy of the system is conserved ie

1E = P(g)w V (g)

minus P(g)woV

(g)+ 5V (aq)

minus 5oV(aq)

= 0 (1)

For the gas phase the energy terms (egP(g)woV

(g)[kJ]) canbe expressed with the gas law (withR [JmolK] andT [K])in terms of moles (egn(g)

wo[mol]) so that eg for pure water

P(g)woV

(g)= n

(g)woRT (2)

Since equilibrium requiresP (g)wo=5wo we obtain accord-

ingly for the aqueous phase

5woV(aq)

= nwoRT (3)

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics 3167

Note that this gas-solution analogy also applies to non-reference conditions whereby Eqs (2) and (3) remain thesame but without the index ldquoordquo and that the gas-solutionanalogy was noted and used by van rsquot Hoff and Ostwald abouthalf a decade after Pfefferrsquos investigations to interpret the os-motic pressure (van rsquot Hoff 1887)

The important aspect about this gas-solution analogy isthat it already allows to explain the principles of osmosiswhereby it is ndash together with Arrheniusrsquo theory of partial dis-sociation of electrolytes in solutions (Arrhenius 1887) ndash di-rectly applicable to any solute concentration provided thatthe water consumed by hydration is explicitly accounted for

Therefore we first generalize in the following the princi-ples that explain osmosis whereby we consistently accountfor the amount of water consumed by hydration (Sect 22)Subsequently we transform these equations to atmosphericapplications (Sect 23) and reformulate equilibrium thermo-dynamics to consistently account for water (Sect 3)

22 Generalizing thermodynamic principles

To consistently account for the amount of water consumedby hydration we build on van rsquot Hoffrsquos gas-solution analogyand Arrheniusrsquo theory of partial dissociation of electrolytesin solutions We introduce a stoichiometric coefficient forwater ndash analogously to the stoichiometric coefficient(s) forthe solute(s) ndash to explicitly account on a molar basis for theamount of water consumed by hydration

221 Hydration

The osmotic pressure (Fig 1a) is caused by the hydration ofns moles of solute The hydration ldquoconsumesrdquo water whichleads to a change in volume For an equilibrium system anychange needs to be compensated resulting in water uptakeThe volume changes because of a) the additional volume ofsolute by which the solute partly or completely dissociatesdue to hydration b) due to the volume of water that is ldquocon-sumedrdquo by the hydration and c) the chemical restructuringof the solute and water molecules For some hygroscopic so-lutes such as eg magnesium chloride (MgCl2) this restruc-turing can even lead to a volume depression whereby theentropy of the hydrated magnesium chloride ions is smallerthan that of the crystalline salt (the hydrated MgCl2 ions havea higher structural order occupying less volume)

Strong electrolytes such as NaCl or MgCl2 dissociate prac-tically completely due to hydration The chemical dissocia-tion of eg sodium chloride (NaCl) (see example given inFig 1a) involves water that is consumed by the hydrationprocesses for which we formulate the equilibrium reaction

1 times nNaCl(cr)s + 1 times n

H2O(aq)w hArr ν+

e times nNa+

(aq)s +

νminuse times n

Clminus(aq)

s + ν+w times n

H3O+

(aq)w + νminus

w times nOHminus

(aq)w (R1)

The subscript ldquoerdquo denotes the stoichiometric coefficientsthat account for effective solute dissociation

(νe=ν+

e +νminuse

)

For strong electrolytes such as NaCl they equal the stoichio-metric coefficients for complete dissociation(νs=ν+

s +νminuss )

The solute hydration is associated with the consumptionof a certain number of water moles We therefore intro-duce a stoichiometric coefficientνw=ν+

w+νminusw for the solvent

(water) to account for the number of moles of solvent (wa-ter) needed for solvation (hydration) and solute dissociationIn Fig 1a the solution (left compartment) contains ldquowater-bindingrdquo particles and its volume expands at the expense ofthe right compartment due to water consumption In case ofa closed system the total aqueous

(V (aq)

)and gaseous

(V (g)

)volumes remain constant The total change in energy (leftand right compartment) can thus be expressed in analogy toEqs (2) and (3) in terms of moles of water

15wV (aq)= (5w minus 5wo)V

(aq)= νwnw RT (4a)

Similar to Eq (4a) we can express the energy contained in ei-ther of the compartments in terms of the number of moles ofsolute and solvent so that for the solution (left compartment)

5 V (aq)= (5wo + 15w)V (aq)

= (νwnw + νens)RT (4b)

222 Generalized mole fraction

The ratio of Eqs (4a) and (4b) provides a very useful ex-pression as it explicitly relates measurable changes in theosmotic pressure with the actual solute concentration It maybe regarded as the generalized mole fractionχw

15w5 = νwnw

(νwnw + νens) = χw (5)

χw expresses the ratio of a change relative to the total os-motic pressure of the solution in terms of water needed forthe hydration and effective dissociation for any amount ofsolute (throughνwnw andνens)

Note that this generalized mole fraction of water (χw)is introduced to explicitly account for the amount of wa-ter consumed by hydration while the ldquoclassicalrdquo mole frac-tion of water is less explicit being merely defined asχw=nw (nw+ns) Therefore various correction factorssuch as the vanrsquot Hoff factor practical osmotic coefficient oractivity coefficients have been introduced in the past to cor-rect the calculated values to the measured ones (Low 1969)

In analogy to the ldquoclassicalrdquo solute mole fractionχs=ns (nw+ns) we further introduce the generalized molefraction of the soluteχs=νens (νwnw+νens) whereby thesum of the mole fractions is unityχs+χw=1

Equation (5) involves the following relations

ndash 15=15w+15s rArr 15w=15minus15s

ndash 15=5 minus 5o with 5=5w+5s

ndash 15s=5sminus5so with 5so=0

wwwatmos-chem-physnet731632007 Atmos Chem Phys 7 3163ndash3193 2007

3168 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

T RH = const

G = V(aq) = 0Pw

(g) =w

(aq)Pw

(g) =wo(aq)

III) RH 1 =gt nwnwsat

(g)

Pwsat(g)(T)V(g) = nwsat

(g)

RT

RH = Pw(g)Pwsat

(g)(T)

RH = nw

(g)nwsat

(g)

II) RH 1 =gt nwnw

(g)

I) =gt wnw $ ens

V(aq) = ( Pw(aq) + Ps

(aq)) V(aq) + A o Vo

(aq) = ( Pwo(aq) + Pso

(aq)) Vo(aq) +o Ao

V(aq)= ens RT + wnw RT

Pwo(aq)=wo

(aq) -wo Ao Vo(aq)

Pw

(aq) =w

(aq) -w A V(aq)

RH = awo = aw

ase = aw

-w

mss = [we 5551(1RH-1)]

we

= (wnwo + enso) RT= (wnw + ens) RT

Pw(g)

EQA

ns = 1 micromol NaCl

w

H3O+

OH-

nw (ns RH)

V(aq)Pw(aq)

e = e

+ + e

-

e

+ Na

+

w

-+

w = w

++ w

-

- w

+

e

- Cl

-

H2O

Pw(g)

EQo Ao

nso = 1 nmol NaCl

wo

H3O+

OH-

nwo (nso RH)

Vo(aq)Pwo

(aq)

e = e

+ + e

-

e

+ Na

+

w

-+

w = w

++ w

-

- w

+

e

- Cl

-

H2O

Pw(g)

EQA

ns = 1 micromol NaCl

w

H3O+

OH-

nw (ns RH)

V(aq)Pw(aq)

e = e

+ + e

-

e

+ Na

+

w

-+

w = w

++ w

-

- w

+

e

- Cl

-

H2O

Pw(g)

EQo Ao

nso = 1 nmol NaCl

wo

H3O+

OH-

nwo (nso RH)

Vo(aq)Pwo

(aq)

e = e

+ + e

-

e

+ Na

+

w

-+

w = w

++ w

-

- w

+

e

- Cl

-

H2O

Aerosol water uptake

Figure 1 b)

Fig 1b Atmospheric conditions schematic of aerosol water uptake

ndash 15w=5wminus5wo with 5wo=5o

ndash 15w5=5w5=minus5s5=1minusχs

Equation (5) expresses the fraction of water required forhydration as the system compensates the vapor pressure re-duction through a net flow of water from the right into the leftcompartment (Fig 1a) For an osmotic (closed) system thisresults in a change in water activity(1aw=awominusaw) whichwould equal a (local) change in relative humidity(1RH)since at equilibrium the osmotic pressure of the solute orsolvent (in solution) equals the corresponding partial vaporpressure of solute or solvent above the solution

Note that the osmotic pressure is independent of the curva-ture of the surface In contrast to the theoretical solvent par-tial pressure in solution the (measurable) osmotic pressureis an effective pressure and hence at equilibrium it implicitlyaccounts for any surface tension or non-ideality effects Sur-face tension (or curvature) affects the solute solubility andthrough the solubility the osmotic pressure We account forsurface tension or non-ideality effects by our method throughνw as it is a pure function of the solutersquos solubility andνe

(see Sect 34 Eq 19)For an open system without a membrane the equilibrium

water uptake is the same It only depends on the hygroscopic

nature of the solute since the water activity is maintainedconstant by RH Atmospheric aerosols provide an exampleof such an open system

23 Atmosphere

The principles of osmosis that explain the nature of hygro-scopic growth of solutes in the laboratory also apply to at-mospheric aerosols However in case of equilibrium the sit-uation differs in the atmosphere since the vapor pressurereduction associated with the hydration of a solute is en-tirely compensated by water uptake as schematically shownin Fig 1b The reason is that in contrast to controlled equilib-rium conditions in the laboratory (closed system at constantT ) the aerosol(s) surrounding RH remains ndash practically ndashconstant in the atmosphere (open system) provided that thewater vapor concentration does not change due to the rela-tively small amount of condensing water needed for hydra-tion ndash a requirement that holds for tropospheric subsaturatedconditions (RHlt1) ndash see also cloud Sect 432

Thus at constantT and RH the water activity of atmo-spheric aerosols is fixed at equilibrium by the available wa-ter vapor concentration and hence equals the fractional rel-ative humidity (aw=RH) Similar to the laboratory a solute

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics 3169

specific amount of water is required for hydration whereasin this case the water needs to condense from the gas phaseFurthermore sinceaw=RH=const for the time period con-sidered to reach equilibrium and no membrane separates so-lute and solvent no hydrostatic counter pressure can buildup At equilibrium the vapor pressure reduction is there-fore fully compensated by the associated water uptake of thewater-binding solute(s)

1P(g)w = 0 (6)

The equilibrium condition further requires that the totalchange in energy is zero

1E = 15 1V (aq)= 0 (7)

Changes in energy resulting from the hydration of the solutecan be expressed in terms of the effective numbers of molesof hydrated solute(s) and the total amount of water that driveshydration

15 1V (aq)= νe 1nsR T + νw 1nwR T (8)

Since1P(g)w =15=0 we can rewrite Eq (8)

νe1ns = minusνw1nw (9)

or analogously to Eq (5) if divided by the osmotic pressureenergy of the solution

νe1ns

(νwnw + νens) = minusνw1nw

(νwnw + νens) (10)

whereby the total change in the amount of solute(νe1ns)

again causes a change in water activity(1aw) ndash equal to achange in relative humidity(1RH) ndash but compensated by theassociated water uptake(minusνw1nw) so thataw and RH re-main unchanged Note that the rhs of Eq (10) equals Eq (5)with respect to the reference condition wherenso=nwo=0and1nw=nwminusnwo and1ns=nsminusnso and that the wateractivity is defined as the ratio of the fugacity (the real gasequivalent of an ideal gas partial pressure) of the water toits fugacity under reference conditions usually approximatedby the more easily determined ratio of partial pressures

3 Reformulating equilibrium thermodynamics

In this section we reformulate the ldquoclassicalrdquo equilibriumthermodynamics to consistently include water (see the elec-tronic supplementhttpwwwatmos-chem-physnet731632007acp-7-3163-2007-supplementzipfor a brief summaryof the relevant standard treatment)

One difficulty arising from solution non-idealities is usu-ally circumvented by applying measured activity coefficientsndash being central in both aerosol thermodynamics and atmo-spheric modeling ndash to atmospheric aerosols but without therequired transformation to atmospheric conditions

The problem is that activity coefficients are usually mea-sured as a function of solute concentration while in the at-mosphere the aerosol activity is not a function of the solute

concentration since the water activity is fixed by RH if equi-librium is assumed Although it is well-known thataw=RH(eg Wexler and Potukuchi 1998) it has been overlookedthat this condition fixes all activities including solute andsolvent activity coefficients Therefore for a given type ofsolute they are all a function of RH independent of their con-centration

Despite polynomial fits or mathematical approximationsare used in atmospheric modeling which have been derivedfor various solute activities from laboratory measurements asa function of solute concentrations (eg Clegg et al 19961998 Clegg and Seinfeld 2006) whereby their applicationin numerical schemes requires computationally demandingsolutions We will show in the following that we can over-come this if we (a) explicitly include in all equilibrium reac-tions the amount of water consumed by solute hydration and(b) consistently limit the amount of water that is available forcondensation as mentioned in Sect 2 and described in moredetail in Sect 432

According to Sect 23 the aerosol water mass is ndash besidesits dependency on RH ndash a function of the mass and type of so-lute In contrast the activity of atmospheric aerosols is onlya function of the water vapor mass available for condensationand of the type of solute but not of its mass

The amount of water needed for solute hydration isonly determined by the solute specific constantsνe andνw and RH being independent of the solute concentration(Sect 23) The reason is that only the solutesrsquo solubility de-termines the amount of solute that can exist in a saturatedsolution Any excess of a non-volatile solute can only pre-cipitate out of the solution so that a solid and aqueous phaseco-exist while (semi)-volatile compounds can additionallyevaporate (hence maintaining gasliquidsolid equilibrium)

This has important implications because for equilibriumconditions the solute specific constantsνe andνw enable theexplicit calculation of single solute molalities from which inturn the solute specific water uptake and derived propertiescan be calculated as a function of RHνe andνw

Note that this approach extends beyond binary solutions tomixed solutions under the additional and also quite realisticand widely applied assumption of molar volume additivity(Hanel 1976) which holds for most soluble salt compoundsas they can in principle dissolve completely (depending onlyon the saturation) Our new approach may also be appliedto laboratory conditions if the term RH used in the followingequations is substituted byaw

31 Solubility constants

Molality is a measure of solubility At equilibrium the solu-tion is saturated so that it contains the maximum concentra-tion of ions that can exist in equilibrium with its solid (crys-talline) phase The amount of solute that must be added toa given volume of solvent to form a saturated solution is itssolubility At equilibrium the ion product equals the solubil-

wwwatmos-chem-physnet731632007 Atmos Chem Phys 7 3163ndash3193 2007

S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics 3167

Note that this gas-solution analogy also applies to non-reference conditions whereby Eqs (2) and (3) remain thesame but without the index ldquoordquo and that the gas-solutionanalogy was noted and used by van rsquot Hoff and Ostwald abouthalf a decade after Pfefferrsquos investigations to interpret the os-motic pressure (van rsquot Hoff 1887)

The important aspect about this gas-solution analogy isthat it already allows to explain the principles of osmosiswhereby it is ndash together with Arrheniusrsquo theory of partial dis-sociation of electrolytes in solutions (Arrhenius 1887) ndash di-rectly applicable to any solute concentration provided thatthe water consumed by hydration is explicitly accounted for

Therefore we first generalize in the following the princi-ples that explain osmosis whereby we consistently accountfor the amount of water consumed by hydration (Sect 22)Subsequently we transform these equations to atmosphericapplications (Sect 23) and reformulate equilibrium thermo-dynamics to consistently account for water (Sect 3)

22 Generalizing thermodynamic principles

To consistently account for the amount of water consumedby hydration we build on van rsquot Hoffrsquos gas-solution analogyand Arrheniusrsquo theory of partial dissociation of electrolytesin solutions We introduce a stoichiometric coefficient forwater ndash analogously to the stoichiometric coefficient(s) forthe solute(s) ndash to explicitly account on a molar basis for theamount of water consumed by hydration

221 Hydration

The osmotic pressure (Fig 1a) is caused by the hydration ofns moles of solute The hydration ldquoconsumesrdquo water whichleads to a change in volume For an equilibrium system anychange needs to be compensated resulting in water uptakeThe volume changes because of a) the additional volume ofsolute by which the solute partly or completely dissociatesdue to hydration b) due to the volume of water that is ldquocon-sumedrdquo by the hydration and c) the chemical restructuringof the solute and water molecules For some hygroscopic so-lutes such as eg magnesium chloride (MgCl2) this restruc-turing can even lead to a volume depression whereby theentropy of the hydrated magnesium chloride ions is smallerthan that of the crystalline salt (the hydrated MgCl2 ions havea higher structural order occupying less volume)

Strong electrolytes such as NaCl or MgCl2 dissociate prac-tically completely due to hydration The chemical dissocia-tion of eg sodium chloride (NaCl) (see example given inFig 1a) involves water that is consumed by the hydrationprocesses for which we formulate the equilibrium reaction

1 times nNaCl(cr)s + 1 times n

H2O(aq)w hArr ν+

e times nNa+

(aq)s +

νminuse times n

Clminus(aq)

s + ν+w times n

H3O+

(aq)w + νminus

w times nOHminus

(aq)w (R1)

The subscript ldquoerdquo denotes the stoichiometric coefficientsthat account for effective solute dissociation

(νe=ν+

e +νminuse

)

For strong electrolytes such as NaCl they equal the stoichio-metric coefficients for complete dissociation(νs=ν+

s +νminuss )

The solute hydration is associated with the consumptionof a certain number of water moles We therefore intro-duce a stoichiometric coefficientνw=ν+

w+νminusw for the solvent

(water) to account for the number of moles of solvent (wa-ter) needed for solvation (hydration) and solute dissociationIn Fig 1a the solution (left compartment) contains ldquowater-bindingrdquo particles and its volume expands at the expense ofthe right compartment due to water consumption In case ofa closed system the total aqueous

(V (aq)

)and gaseous

(V (g)

)volumes remain constant The total change in energy (leftand right compartment) can thus be expressed in analogy toEqs (2) and (3) in terms of moles of water

15wV (aq)= (5w minus 5wo)V

(aq)= νwnw RT (4a)

Similar to Eq (4a) we can express the energy contained in ei-ther of the compartments in terms of the number of moles ofsolute and solvent so that for the solution (left compartment)

5 V (aq)= (5wo + 15w)V (aq)

= (νwnw + νens)RT (4b)

222 Generalized mole fraction

The ratio of Eqs (4a) and (4b) provides a very useful ex-pression as it explicitly relates measurable changes in theosmotic pressure with the actual solute concentration It maybe regarded as the generalized mole fractionχw

15w5 = νwnw

(νwnw + νens) = χw (5)

χw expresses the ratio of a change relative to the total os-motic pressure of the solution in terms of water needed forthe hydration and effective dissociation for any amount ofsolute (throughνwnw andνens)

Note that this generalized mole fraction of water (χw)is introduced to explicitly account for the amount of wa-ter consumed by hydration while the ldquoclassicalrdquo mole frac-tion of water is less explicit being merely defined asχw=nw (nw+ns) Therefore various correction factorssuch as the vanrsquot Hoff factor practical osmotic coefficient oractivity coefficients have been introduced in the past to cor-rect the calculated values to the measured ones (Low 1969)

In analogy to the ldquoclassicalrdquo solute mole fractionχs=ns (nw+ns) we further introduce the generalized molefraction of the soluteχs=νens (νwnw+νens) whereby thesum of the mole fractions is unityχs+χw=1

Equation (5) involves the following relations

ndash 15=15w+15s rArr 15w=15minus15s

ndash 15=5 minus 5o with 5=5w+5s

ndash 15s=5sminus5so with 5so=0

wwwatmos-chem-physnet731632007 Atmos Chem Phys 7 3163ndash3193 2007

3168 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

T RH = const

G = V(aq) = 0Pw

(g) =w

(aq)Pw

(g) =wo(aq)

III) RH 1 =gt nwnwsat

(g)

Pwsat(g)(T)V(g) = nwsat

(g)

RT

RH = Pw(g)Pwsat

(g)(T)

RH = nw

(g)nwsat

(g)

II) RH 1 =gt nwnw

(g)

I) =gt wnw $ ens

V(aq) = ( Pw(aq) + Ps

(aq)) V(aq) + A o Vo

(aq) = ( Pwo(aq) + Pso

(aq)) Vo(aq) +o Ao

V(aq)= ens RT + wnw RT

Pwo(aq)=wo

(aq) -wo Ao Vo(aq)

Pw

(aq) =w

(aq) -w A V(aq)

RH = awo = aw

ase = aw

-w

mss = [we 5551(1RH-1)]

we

= (wnwo + enso) RT= (wnw + ens) RT

Pw(g)

EQA

ns = 1 micromol NaCl

w

H3O+

OH-

nw (ns RH)

V(aq)Pw(aq)

e = e

+ + e

-

e

+ Na

+

w

-+

w = w

++ w

-

- w

+

e

- Cl

-

H2O

Pw(g)

EQo Ao

nso = 1 nmol NaCl

wo

H3O+

OH-

nwo (nso RH)

Vo(aq)Pwo

(aq)

e = e

+ + e

-

e

+ Na

+

w

-+

w = w

++ w

-

- w

+

e

- Cl

-

H2O

Pw(g)

EQA

ns = 1 micromol NaCl

w

H3O+

OH-

nw (ns RH)

V(aq)Pw(aq)

e = e

+ + e

-

e

+ Na

+

w

-+

w = w

++ w

-

- w

+

e

- Cl

-

H2O

Pw(g)

EQo Ao

nso = 1 nmol NaCl

wo

H3O+

OH-

nwo (nso RH)

Vo(aq)Pwo

(aq)

e = e

+ + e

-

e

+ Na

+

w

-+

w = w

++ w

-

- w

+

e

- Cl

-

H2O

Aerosol water uptake

Figure 1 b)

Fig 1b Atmospheric conditions schematic of aerosol water uptake

ndash 15w=5wminus5wo with 5wo=5o

ndash 15w5=5w5=minus5s5=1minusχs

Equation (5) expresses the fraction of water required forhydration as the system compensates the vapor pressure re-duction through a net flow of water from the right into the leftcompartment (Fig 1a) For an osmotic (closed) system thisresults in a change in water activity(1aw=awominusaw) whichwould equal a (local) change in relative humidity(1RH)since at equilibrium the osmotic pressure of the solute orsolvent (in solution) equals the corresponding partial vaporpressure of solute or solvent above the solution

Note that the osmotic pressure is independent of the curva-ture of the surface In contrast to the theoretical solvent par-tial pressure in solution the (measurable) osmotic pressureis an effective pressure and hence at equilibrium it implicitlyaccounts for any surface tension or non-ideality effects Sur-face tension (or curvature) affects the solute solubility andthrough the solubility the osmotic pressure We account forsurface tension or non-ideality effects by our method throughνw as it is a pure function of the solutersquos solubility andνe

(see Sect 34 Eq 19)For an open system without a membrane the equilibrium

water uptake is the same It only depends on the hygroscopic

nature of the solute since the water activity is maintainedconstant by RH Atmospheric aerosols provide an exampleof such an open system

23 Atmosphere

The principles of osmosis that explain the nature of hygro-scopic growth of solutes in the laboratory also apply to at-mospheric aerosols However in case of equilibrium the sit-uation differs in the atmosphere since the vapor pressurereduction associated with the hydration of a solute is en-tirely compensated by water uptake as schematically shownin Fig 1b The reason is that in contrast to controlled equilib-rium conditions in the laboratory (closed system at constantT ) the aerosol(s) surrounding RH remains ndash practically ndashconstant in the atmosphere (open system) provided that thewater vapor concentration does not change due to the rela-tively small amount of condensing water needed for hydra-tion ndash a requirement that holds for tropospheric subsaturatedconditions (RHlt1) ndash see also cloud Sect 432

Thus at constantT and RH the water activity of atmo-spheric aerosols is fixed at equilibrium by the available wa-ter vapor concentration and hence equals the fractional rel-ative humidity (aw=RH) Similar to the laboratory a solute

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics 3169

specific amount of water is required for hydration whereasin this case the water needs to condense from the gas phaseFurthermore sinceaw=RH=const for the time period con-sidered to reach equilibrium and no membrane separates so-lute and solvent no hydrostatic counter pressure can buildup At equilibrium the vapor pressure reduction is there-fore fully compensated by the associated water uptake of thewater-binding solute(s)

1P(g)w = 0 (6)

The equilibrium condition further requires that the totalchange in energy is zero

1E = 15 1V (aq)= 0 (7)

Changes in energy resulting from the hydration of the solutecan be expressed in terms of the effective numbers of molesof hydrated solute(s) and the total amount of water that driveshydration

15 1V (aq)= νe 1nsR T + νw 1nwR T (8)

Since1P(g)w =15=0 we can rewrite Eq (8)

νe1ns = minusνw1nw (9)

or analogously to Eq (5) if divided by the osmotic pressureenergy of the solution

νe1ns

(νwnw + νens) = minusνw1nw

(νwnw + νens) (10)

whereby the total change in the amount of solute(νe1ns)

again causes a change in water activity(1aw) ndash equal to achange in relative humidity(1RH) ndash but compensated by theassociated water uptake(minusνw1nw) so thataw and RH re-main unchanged Note that the rhs of Eq (10) equals Eq (5)with respect to the reference condition wherenso=nwo=0and1nw=nwminusnwo and1ns=nsminusnso and that the wateractivity is defined as the ratio of the fugacity (the real gasequivalent of an ideal gas partial pressure) of the water toits fugacity under reference conditions usually approximatedby the more easily determined ratio of partial pressures

3 Reformulating equilibrium thermodynamics

In this section we reformulate the ldquoclassicalrdquo equilibriumthermodynamics to consistently include water (see the elec-tronic supplementhttpwwwatmos-chem-physnet731632007acp-7-3163-2007-supplementzipfor a brief summaryof the relevant standard treatment)

One difficulty arising from solution non-idealities is usu-ally circumvented by applying measured activity coefficientsndash being central in both aerosol thermodynamics and atmo-spheric modeling ndash to atmospheric aerosols but without therequired transformation to atmospheric conditions

The problem is that activity coefficients are usually mea-sured as a function of solute concentration while in the at-mosphere the aerosol activity is not a function of the solute

concentration since the water activity is fixed by RH if equi-librium is assumed Although it is well-known thataw=RH(eg Wexler and Potukuchi 1998) it has been overlookedthat this condition fixes all activities including solute andsolvent activity coefficients Therefore for a given type ofsolute they are all a function of RH independent of their con-centration

Despite polynomial fits or mathematical approximationsare used in atmospheric modeling which have been derivedfor various solute activities from laboratory measurements asa function of solute concentrations (eg Clegg et al 19961998 Clegg and Seinfeld 2006) whereby their applicationin numerical schemes requires computationally demandingsolutions We will show in the following that we can over-come this if we (a) explicitly include in all equilibrium reac-tions the amount of water consumed by solute hydration and(b) consistently limit the amount of water that is available forcondensation as mentioned in Sect 2 and described in moredetail in Sect 432

According to Sect 23 the aerosol water mass is ndash besidesits dependency on RH ndash a function of the mass and type of so-lute In contrast the activity of atmospheric aerosols is onlya function of the water vapor mass available for condensationand of the type of solute but not of its mass

The amount of water needed for solute hydration isonly determined by the solute specific constantsνe andνw and RH being independent of the solute concentration(Sect 23) The reason is that only the solutesrsquo solubility de-termines the amount of solute that can exist in a saturatedsolution Any excess of a non-volatile solute can only pre-cipitate out of the solution so that a solid and aqueous phaseco-exist while (semi)-volatile compounds can additionallyevaporate (hence maintaining gasliquidsolid equilibrium)

This has important implications because for equilibriumconditions the solute specific constantsνe andνw enable theexplicit calculation of single solute molalities from which inturn the solute specific water uptake and derived propertiescan be calculated as a function of RHνe andνw

Note that this approach extends beyond binary solutions tomixed solutions under the additional and also quite realisticand widely applied assumption of molar volume additivity(Hanel 1976) which holds for most soluble salt compoundsas they can in principle dissolve completely (depending onlyon the saturation) Our new approach may also be appliedto laboratory conditions if the term RH used in the followingequations is substituted byaw

31 Solubility constants

Molality is a measure of solubility At equilibrium the solu-tion is saturated so that it contains the maximum concentra-tion of ions that can exist in equilibrium with its solid (crys-talline) phase The amount of solute that must be added toa given volume of solvent to form a saturated solution is itssolubility At equilibrium the ion product equals the solubil-

wwwatmos-chem-physnet731632007 Atmos Chem Phys 7 3163ndash3193 2007

3168 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

T RH = const

G = V(aq) = 0Pw

(g) =w

(aq)Pw

(g) =wo(aq)

III) RH 1 =gt nwnwsat

(g)

Pwsat(g)(T)V(g) = nwsat

(g)

RT

RH = Pw(g)Pwsat

(g)(T)

RH = nw

(g)nwsat

(g)

II) RH 1 =gt nwnw

(g)

I) =gt wnw $ ens

V(aq) = ( Pw(aq) + Ps

(aq)) V(aq) + A o Vo

(aq) = ( Pwo(aq) + Pso

(aq)) Vo(aq) +o Ao

V(aq)= ens RT + wnw RT

Pwo(aq)=wo

(aq) -wo Ao Vo(aq)

Pw

(aq) =w

(aq) -w A V(aq)

RH = awo = aw

ase = aw

-w

mss = [we 5551(1RH-1)]

we

= (wnwo + enso) RT= (wnw + ens) RT

Pw(g)

EQA

ns = 1 micromol NaCl

w

H3O+

OH-

nw (ns RH)

V(aq)Pw(aq)

e = e

+ + e

-

e

+ Na

+

w

-+

w = w

++ w

-

- w

+

e

- Cl

-

H2O

Pw(g)

EQo Ao

nso = 1 nmol NaCl

wo

H3O+

OH-

nwo (nso RH)

Vo(aq)Pwo

(aq)

e = e

+ + e

-

e

+ Na

+

w

-+

w = w

++ w

-

- w

+

e

- Cl

-

H2O

Pw(g)

EQA

ns = 1 micromol NaCl

w

H3O+

OH-

nw (ns RH)

V(aq)Pw(aq)

e = e

+ + e

-

e

+ Na

+

w

-+

w = w

++ w

-

- w

+

e

- Cl

-

H2O

Pw(g)

EQo Ao

nso = 1 nmol NaCl

wo

H3O+

OH-

nwo (nso RH)

Vo(aq)Pwo

(aq)

e = e

+ + e

-

e

+ Na

+

w

-+

w = w

++ w

-

- w

+

e

- Cl

-

H2O

Aerosol water uptake

Figure 1 b)

Fig 1b Atmospheric conditions schematic of aerosol water uptake

ndash 15w=5wminus5wo with 5wo=5o

ndash 15w5=5w5=minus5s5=1minusχs

Equation (5) expresses the fraction of water required forhydration as the system compensates the vapor pressure re-duction through a net flow of water from the right into the leftcompartment (Fig 1a) For an osmotic (closed) system thisresults in a change in water activity(1aw=awominusaw) whichwould equal a (local) change in relative humidity(1RH)since at equilibrium the osmotic pressure of the solute orsolvent (in solution) equals the corresponding partial vaporpressure of solute or solvent above the solution

Note that the osmotic pressure is independent of the curva-ture of the surface In contrast to the theoretical solvent par-tial pressure in solution the (measurable) osmotic pressureis an effective pressure and hence at equilibrium it implicitlyaccounts for any surface tension or non-ideality effects Sur-face tension (or curvature) affects the solute solubility andthrough the solubility the osmotic pressure We account forsurface tension or non-ideality effects by our method throughνw as it is a pure function of the solutersquos solubility andνe

(see Sect 34 Eq 19)For an open system without a membrane the equilibrium

water uptake is the same It only depends on the hygroscopic

nature of the solute since the water activity is maintainedconstant by RH Atmospheric aerosols provide an exampleof such an open system

23 Atmosphere

The principles of osmosis that explain the nature of hygro-scopic growth of solutes in the laboratory also apply to at-mospheric aerosols However in case of equilibrium the sit-uation differs in the atmosphere since the vapor pressurereduction associated with the hydration of a solute is en-tirely compensated by water uptake as schematically shownin Fig 1b The reason is that in contrast to controlled equilib-rium conditions in the laboratory (closed system at constantT ) the aerosol(s) surrounding RH remains ndash practically ndashconstant in the atmosphere (open system) provided that thewater vapor concentration does not change due to the rela-tively small amount of condensing water needed for hydra-tion ndash a requirement that holds for tropospheric subsaturatedconditions (RHlt1) ndash see also cloud Sect 432

Thus at constantT and RH the water activity of atmo-spheric aerosols is fixed at equilibrium by the available wa-ter vapor concentration and hence equals the fractional rel-ative humidity (aw=RH) Similar to the laboratory a solute

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics 3169

specific amount of water is required for hydration whereasin this case the water needs to condense from the gas phaseFurthermore sinceaw=RH=const for the time period con-sidered to reach equilibrium and no membrane separates so-lute and solvent no hydrostatic counter pressure can buildup At equilibrium the vapor pressure reduction is there-fore fully compensated by the associated water uptake of thewater-binding solute(s)

1P(g)w = 0 (6)

The equilibrium condition further requires that the totalchange in energy is zero

1E = 15 1V (aq)= 0 (7)

Changes in energy resulting from the hydration of the solutecan be expressed in terms of the effective numbers of molesof hydrated solute(s) and the total amount of water that driveshydration

15 1V (aq)= νe 1nsR T + νw 1nwR T (8)

Since1P(g)w =15=0 we can rewrite Eq (8)

νe1ns = minusνw1nw (9)

or analogously to Eq (5) if divided by the osmotic pressureenergy of the solution

νe1ns

(νwnw + νens) = minusνw1nw

(νwnw + νens) (10)

whereby the total change in the amount of solute(νe1ns)

again causes a change in water activity(1aw) ndash equal to achange in relative humidity(1RH) ndash but compensated by theassociated water uptake(minusνw1nw) so thataw and RH re-main unchanged Note that the rhs of Eq (10) equals Eq (5)with respect to the reference condition wherenso=nwo=0and1nw=nwminusnwo and1ns=nsminusnso and that the wateractivity is defined as the ratio of the fugacity (the real gasequivalent of an ideal gas partial pressure) of the water toits fugacity under reference conditions usually approximatedby the more easily determined ratio of partial pressures

3 Reformulating equilibrium thermodynamics

In this section we reformulate the ldquoclassicalrdquo equilibriumthermodynamics to consistently include water (see the elec-tronic supplementhttpwwwatmos-chem-physnet731632007acp-7-3163-2007-supplementzipfor a brief summaryof the relevant standard treatment)

One difficulty arising from solution non-idealities is usu-ally circumvented by applying measured activity coefficientsndash being central in both aerosol thermodynamics and atmo-spheric modeling ndash to atmospheric aerosols but without therequired transformation to atmospheric conditions

The problem is that activity coefficients are usually mea-sured as a function of solute concentration while in the at-mosphere the aerosol activity is not a function of the solute

concentration since the water activity is fixed by RH if equi-librium is assumed Although it is well-known thataw=RH(eg Wexler and Potukuchi 1998) it has been overlookedthat this condition fixes all activities including solute andsolvent activity coefficients Therefore for a given type ofsolute they are all a function of RH independent of their con-centration

Despite polynomial fits or mathematical approximationsare used in atmospheric modeling which have been derivedfor various solute activities from laboratory measurements asa function of solute concentrations (eg Clegg et al 19961998 Clegg and Seinfeld 2006) whereby their applicationin numerical schemes requires computationally demandingsolutions We will show in the following that we can over-come this if we (a) explicitly include in all equilibrium reac-tions the amount of water consumed by solute hydration and(b) consistently limit the amount of water that is available forcondensation as mentioned in Sect 2 and described in moredetail in Sect 432

According to Sect 23 the aerosol water mass is ndash besidesits dependency on RH ndash a function of the mass and type of so-lute In contrast the activity of atmospheric aerosols is onlya function of the water vapor mass available for condensationand of the type of solute but not of its mass

The amount of water needed for solute hydration isonly determined by the solute specific constantsνe andνw and RH being independent of the solute concentration(Sect 23) The reason is that only the solutesrsquo solubility de-termines the amount of solute that can exist in a saturatedsolution Any excess of a non-volatile solute can only pre-cipitate out of the solution so that a solid and aqueous phaseco-exist while (semi)-volatile compounds can additionallyevaporate (hence maintaining gasliquidsolid equilibrium)

This has important implications because for equilibriumconditions the solute specific constantsνe andνw enable theexplicit calculation of single solute molalities from which inturn the solute specific water uptake and derived propertiescan be calculated as a function of RHνe andνw

Note that this approach extends beyond binary solutions tomixed solutions under the additional and also quite realisticand widely applied assumption of molar volume additivity(Hanel 1976) which holds for most soluble salt compoundsas they can in principle dissolve completely (depending onlyon the saturation) Our new approach may also be appliedto laboratory conditions if the term RH used in the followingequations is substituted byaw

31 Solubility constants

Molality is a measure of solubility At equilibrium the solu-tion is saturated so that it contains the maximum concentra-tion of ions that can exist in equilibrium with its solid (crys-talline) phase The amount of solute that must be added toa given volume of solvent to form a saturated solution is itssolubility At equilibrium the ion product equals the solubil-

wwwatmos-chem-physnet731632007 Atmos Chem Phys 7 3163ndash3193 2007

S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics 3169

specific amount of water is required for hydration whereasin this case the water needs to condense from the gas phaseFurthermore sinceaw=RH=const for the time period con-sidered to reach equilibrium and no membrane separates so-lute and solvent no hydrostatic counter pressure can buildup At equilibrium the vapor pressure reduction is there-fore fully compensated by the associated water uptake of thewater-binding solute(s)

1P(g)w = 0 (6)

The equilibrium condition further requires that the totalchange in energy is zero

1E = 15 1V (aq)= 0 (7)

Changes in energy resulting from the hydration of the solutecan be expressed in terms of the effective numbers of molesof hydrated solute(s) and the total amount of water that driveshydration

15 1V (aq)= νe 1nsR T + νw 1nwR T (8)

Since1P(g)w =15=0 we can rewrite Eq (8)

νe1ns = minusνw1nw (9)

or analogously to Eq (5) if divided by the osmotic pressureenergy of the solution

νe1ns

(νwnw + νens) = minusνw1nw

(νwnw + νens) (10)

whereby the total change in the amount of solute(νe1ns)

again causes a change in water activity(1aw) ndash equal to achange in relative humidity(1RH) ndash but compensated by theassociated water uptake(minusνw1nw) so thataw and RH re-main unchanged Note that the rhs of Eq (10) equals Eq (5)with respect to the reference condition wherenso=nwo=0and1nw=nwminusnwo and1ns=nsminusnso and that the wateractivity is defined as the ratio of the fugacity (the real gasequivalent of an ideal gas partial pressure) of the water toits fugacity under reference conditions usually approximatedby the more easily determined ratio of partial pressures

3 Reformulating equilibrium thermodynamics

In this section we reformulate the ldquoclassicalrdquo equilibriumthermodynamics to consistently include water (see the elec-tronic supplementhttpwwwatmos-chem-physnet731632007acp-7-3163-2007-supplementzipfor a brief summaryof the relevant standard treatment)

One difficulty arising from solution non-idealities is usu-ally circumvented by applying measured activity coefficientsndash being central in both aerosol thermodynamics and atmo-spheric modeling ndash to atmospheric aerosols but without therequired transformation to atmospheric conditions

The problem is that activity coefficients are usually mea-sured as a function of solute concentration while in the at-mosphere the aerosol activity is not a function of the solute

concentration since the water activity is fixed by RH if equi-librium is assumed Although it is well-known thataw=RH(eg Wexler and Potukuchi 1998) it has been overlookedthat this condition fixes all activities including solute andsolvent activity coefficients Therefore for a given type ofsolute they are all a function of RH independent of their con-centration

Despite polynomial fits or mathematical approximationsare used in atmospheric modeling which have been derivedfor various solute activities from laboratory measurements asa function of solute concentrations (eg Clegg et al 19961998 Clegg and Seinfeld 2006) whereby their applicationin numerical schemes requires computationally demandingsolutions We will show in the following that we can over-come this if we (a) explicitly include in all equilibrium reac-tions the amount of water consumed by solute hydration and(b) consistently limit the amount of water that is available forcondensation as mentioned in Sect 2 and described in moredetail in Sect 432

According to Sect 23 the aerosol water mass is ndash besidesits dependency on RH ndash a function of the mass and type of so-lute In contrast the activity of atmospheric aerosols is onlya function of the water vapor mass available for condensationand of the type of solute but not of its mass

The amount of water needed for solute hydration isonly determined by the solute specific constantsνe andνw and RH being independent of the solute concentration(Sect 23) The reason is that only the solutesrsquo solubility de-termines the amount of solute that can exist in a saturatedsolution Any excess of a non-volatile solute can only pre-cipitate out of the solution so that a solid and aqueous phaseco-exist while (semi)-volatile compounds can additionallyevaporate (hence maintaining gasliquidsolid equilibrium)

This has important implications because for equilibriumconditions the solute specific constantsνe andνw enable theexplicit calculation of single solute molalities from which inturn the solute specific water uptake and derived propertiescan be calculated as a function of RHνe andνw

Note that this approach extends beyond binary solutions tomixed solutions under the additional and also quite realisticand widely applied assumption of molar volume additivity(Hanel 1976) which holds for most soluble salt compoundsas they can in principle dissolve completely (depending onlyon the saturation) Our new approach may also be appliedto laboratory conditions if the term RH used in the followingequations is substituted byaw

31 Solubility constants

Molality is a measure of solubility At equilibrium the solu-tion is saturated so that it contains the maximum concentra-tion of ions that can exist in equilibrium with its solid (crys-talline) phase The amount of solute that must be added toa given volume of solvent to form a saturated solution is itssolubility At equilibrium the ion product equals the solubil-

wwwatmos-chem-physnet731632007 Atmos Chem Phys 7 3163ndash3193 2007

3170 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

ity product constant(Ksp) for the solute For instance usingthe dissociation equilibrium constant for Reaction (R1) andexplicitly accounting for water

KNaCl(cr) times KH2O(aq)

= ν+e [Na+

(aq)] + νminus

e [Clminus(aq)

]

+ ν+w [H3O+

(aq)] + νminus

w [OHminus

(aq)] (K1)

where KH2O(aq)

denotes the dissociation equilibrium constant

for the associated water mass consumed by hydration ieKH2O(aq)=ν+

w [H3O+

(aq)] + νminusw [OHminus

(aq)]

We further express (K1) in terms of activities

KNaCl(cr) times KH2O(aq)

= aνe

s times aνw

w

= aν+e

Na+ times aνminuse

Clminustimes a

ν+w

H3O+ times aνminusw

OHminus (K2)

whereby the subscript ldquosrdquo denotes the solute activity ofthe plusmn-ion pair being a product of the cation and an-ion activity ie for the solute sodium chloride (NaCl)

aνeplusmn

NaClplusmn(aq)

=aνes =a

ν+e

Na+ times aνminuse

Clminus Similarly the subscript ldquowrdquo

denotes the activity of water ie the water activity

aνplusmnw

H2Oplusmn(aq)

=aνww =a

ν+w

H3O+ times aνminusw

OHminus

Considering that in the atmosphere the water activity isfixed by RH and that the equilibrium condition requires thatthe total energy change is zero we can directly derive arelationship between the solute and water activity that al-lows to considerably simplify the numerical solution of (K2)Since both requirements (a) the equilibrium condition and (b)aw=RH must also hold for the reference condition with re-spect to temperature and pressure (ie the standard state) the

summation over the partial Gibbs free energies(ksum

i=1νij go

ij )

must be zero if extended to include water

νe gos + νw go

w = 0 (11)

The equilibrium condition is fulfilled for a certain relationbetween the stoichiometric constants ieνw of water caus-ing the hydration of theνe moles of solute satisfying Eq (11)

νw = minusνegos g

ow (12)

This relation betweenνw and νe requires that the prod-uct of activities and subsequently of the equilibrium con-stants is unity when water is included (see eg the elec-tronic supplementhttpwwwatmos-chem-physnet731632007acp-7-3163-2007-supplementzipfor a definition ofequilibrium constants)

Thus the general formulation for thej th-chemical reac-tion (eg according to K2) yields

Kspj = exp(minus1RT

ksumi=1

νij goij ) = a

νej

sj times aνwj

wj = 1 (13)

being consistent with Eqs (9) and (10) By dropping theindexj ie

aνe

s = aminusνw

w (14)

or in terms of equilibrium constants (according to K2)KNaCl(cr)= Kminus1

H2O(aq)

Thus at equilibrium the energy gain associated with a neu-tralization reaction must be compensated for by the energyconsumed by the hydration process so thatdG=0 This alsofollows directly from the fact that for charged species forwhich the electrical forces must be considered the poten-tial for an electrochemical reaction is zero at equilibrium incase of electro-neutrality (Nernst 1889) ndash a condition thatis generally fulfilled for neutralization reactions which in-cludes the hydration of salt solutes

32 Aerosol activities

In general an aerosol system is not in equilibrium if the ionproduct deviates from the solubility product constant of thesolute But it can rapidly adjust according to Le Chatelierrsquosprinciple by which the reaction re-equilibrates after excessions precipitate or dissolve until the ion product deficit iscompensated The solubility product constant for a satu-rated binary solution (one solute and solvent) requires thatν+e cations are released forνminus

e anions so that at equilibrium

aν+e

s+ = aνminuse

sminus = aminusν

+w

w+ = aminusν

minusw

wminus (15)

aν+e

s+ andaνminuse

sminus denote the solutersquos(s) cation(+) and anion(minus)

activity aminusν

+w

w+ and aminusν

minusw

wminus denote the water activities On amolal scale (moles of solute per kilogram solvent) the soluteactivity is defined in terms of molality and corrected by themolal activity coefficient

aνe

s = aν+e

s+ times aνminuse

sminus = (γs+ times ms+)ν+e

times (γsminus times msminus)νminuse

= γνe

splusmn times mν+e

s+ times mνminuse

sminus = (γsplusmn times msplusmn)νe

(16)

mν+e

s+ mνminuse

sminus andmsplusmn denote the cation anion and the ion-

pair molality γν+e

s+ γνminuse

sminus and γsplusmn=(γν+e

s+ times γνminuse

sminus )1νe are themolal activity coefficients of the cation anion and the meanion-pair activity coefficient of the solute respectivelyν+

e

andνminuse are their stoichiometric constants ie the effective

number of moles of cations and anions per mole dissociatingsolute(s) The aqueous single solute(ss) molality is de-fined asmss=5551timesnsnw with ns andnw the number ofmoles [mol] of solute and solvent (water) 5551=1000Mw

[molkg H2O] is the molal concentration of water withMw=18015 [gmol] the molar mass of water

33 Solubility

For a saturated binary solutionns andnw can be directlydetermined from the solute solubility The solubility can beexpressed in terms of the saturation molality of the singlesolute or as mass of solute per 100 gram of water or mass

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics 3171

Table 1aThermodynamic data

Table 1 Thermodynamic data

Anions

Cations

Phosphate

PO43-

Sulfate

SO42-

Hydrogen Sulfate

HSO4-

Nitrate

NO3-

Chloride

Cl-Bromide

Br-

Iodide

I -

Hydrogen

H+

H3PO

4H

2SO

4 HNO

3 HCl HBr HI

97995 8457rsquo - 9808 70 183 - - - 6301 25 1513 3646 15 149 8091 25 3307 1279 30 5228

4 4 1626 3 3 1669 - - - 2 2 1398 2 2 1176 2 2 1398 2 2 1477

Ammonium

NH4+

(NH4)3PO

4$3H

2O (NH

4)2SO

4NH

4HSO

4NH

4NO

3NH

4Cl NH

4Br NH

4I

2031 20 - 13214 4331 177 11511 76 178 8004 6805 172 5349 2834 1519 9794 4392 2429 1449 6403 2514

4 4 1 3 215 1605 2 23 1820 2 197 1839 2 19 1475 2 2 1643 2 2 1806

Sodium

Na+

Na3PO

4$12H

2O Na

2SO

4NaHSO

4NaNO

3 NaCl NaBr NaI

38012 1259 162 14204 2194 27 1201 2218 243 85 477 226 5844 2647 217 1029 4861 32 1499 6479 367

4 4 1 3 19 1364 2 2 138 2 197 1685 2 2 1423 2 2 1687 2 2 1812

Potassium

K+

K3PO

4K

2SO

4KHSO

4KNO

3 KCl KBr KI

2123 5146 2564 1743 1071 266 1362 336 232 1011 2769 211 7455 2623 1988 119 4041 274 166 5968 312

4 4 141 3 3 1 2 2 1526 2 162 1534 2 19 1441 2 2 1607 2 2 1776

Calcium

Ca2+

Ca3(PO

4)2

CaSO4

Ca(NO3)2

CaCl2

CaBr2

CaI2

3102 0001rsquo 314 1361 0205 296 - - - 1641 5902 25 111 4484 215 1999 6094 338 2939 6825 396

3 3 1 2 18 1 - - - 3 271 1639 3 3 1476 3 3 1609 3 3 1658

Magnesium

Mg2+

Mg3(PO

4)2$5H

2O MgSO

4 Mg(NO

3)2

MgCl2

MgBr2

MgI2

3529 0001rsquo - 1204 2631 266 - - - 1483 4159 23 9521 359 2325 1841 505 372 2781 5935 443

3 3 1 2 2 142 - - - 3 251 152 3 3 138 3 3 1527 3 3 1597

Iron(IIIII)

Fe2+ Fe3+

FePO4$2H

2O Fe

2(SO

4)3

Fe(NO3)3

FeCl3

FeBr3

FeI2

1869 - 287 3999 8148rsquo 31 - - - 2419 4521rsquo - 1622 477 29 2956 8198 45 3097 - -

2 2 1 5 5 1513 - - - 4 4 1354 4 4 1378 4 4 1613 3 - -

Formula

Ms Ws s

amps ampe ampw

Ms = Solute molar mass [gmol]

Ws = Solute solubility mass percent []

s = Solute density [gcm3]

amps = Solute stoichiometric constant complete disscociation [-]

ampe = Solute stoichiometric constant effective disscociation [-]

ampw = Solvent stoichiometric constant effective disscociation [-]

Ms Ws s data from CRC-Handbook of Chemistry and Physics 85th Edition 2004-2005

Note Solubility measurements (Ws) correspond to aqueous solutions at T=25oC except noted ldquo= 15oC rsquo= 20oC

Ms Ws account for the dry mass excluding any mass of hydration Solvent (water) molar mass Mw = 18015 [gmol] density of the solution [gmL]

For a compound with low solubility (~Wslt 1) the error from approximating the density is generally less than the uncertainty in the experimental solubility measurement

Solute mass fraction [-] ws = ms(mw + ms)

Mass of solute per liter of solution [gL H2O] Rs = 1000 ws

Mass of solute per 100g of H2O [g100gH2O] rs = 100 (1ws-1)

Molarity [molL] Cs = 1000 ws Ms

Molality [molkg] cs = 1000 Ms (1ws-1)

Mole fraction [-] xs = (ws Ms) [(ws Ms) + (1-ws)Mw]

Relation of Ws = 100 ws between other common measures of solubility

percentWs [] ie mass of solute per total mass of solution(solute and solvent) For the latter

Ws = 100times ws = 100times ms(ms + mw) (17)

wherews denotes the solute mass fractionms=nsMs andmw=nwMw the mass[g] of solute and solvent (water) re-spectivelyMs andMw are the corresponding molar massesof solute and solvent [gmol]

At equilibrium a solution is saturated ie it contains themaximum number of moles of solute that can be dissolvedIf the solubility is known this number can be directly calcu-lated from the total mass of solution (fixed to 1000 g) from

ns = 1000Ms times ws (18a)

The associated number of (free) moles of water of the solu-tion can be obtained from

nw = 1000Mw times (1 minus ws) (18b)

The saturation molality is then related to the solute mass frac-tion (solubility) by

msssat = 1000Ms times 1(1ws minus 1) (18c)

and the mass of hydrated solute can be expressed in terms ofwater by

nwMw = 1000times ws (18d)

The relation between the moles of water is given bynw=nwominusnw with nwo=5551 [mol]

34 Stoichiometric constant of water

The stoichiometric constant of water(νw) that hydrates onemole of solute intoνe moles is related toνe according toEq (9) byνe1ns=minus νw1nw The term on the rhs expressesthe amount of water required for the hydration of1ns molesof solute (where1nwlt0 since water is consumed) in equi-librium giving rise to an effective dissociation intoνe moles

wwwatmos-chem-physnet731632007 Atmos Chem Phys 7 3163ndash3193 2007

3172 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

Table 1b ContinuedTable 1 Thermodynamic data (continued)

Anions

Cations

Carbonate

CO3

2-

Hydrogen Carbonate

HCO3

-

Hydroxide

OH-

Formate

CHO2

-

Acetate

C2H

3O

2-

Oxalate

C2O

42-

Citrate

C6H

5O

73-

Hydrogen

H+

H2CO

3 H

2O CH

2O

2C

2H

4O

2C

2H

2O

4C

6H

8O

7

62025 - - - - - 18015 100 0997 4603 68 122 6005 23 1045 9004 869rsquo 19 1921 59rsquo 1665

3 - - - - - 2 2 - 2 2 1833 2 2 1362 3 3 1000 4 4 1470

Ammonium

NH4

+

(NH4)2CO

3NH

4HCO

3NH

4OH NH

4CHO

2NH

4C

2H

3O

2(NH

4)2C

2O

4(NH

4)2HC

6H

5O

7

96086 5000rdquo - 7906 1987 1586 3505 100 - 6306 5885 127 7708 5968 1073 1241 494 15 2262 - 148

3 3 1523 2 2 1298 2 2 2 2 2 177 2 2 1776 3 3 1 4 - -

Sodium

Na+

Na2CO

3NaHCO

3 NaOH NaCHO2

NaC2H

3O

2Na

2C

2O

4Na

3C

6H

5O

7

10599 2349 254 8401 934 22 400 50 213 6801 4869 192 8203 3351 1528 134 348 361 2581 - -

3 19 1393 2 2 1 2 2 17 2 2 1687 2 2 1525 3 3 1 4 - -

Potassium

K+

K2CO

3KHCO

3 KOH KCHO2

KC2H

3O

2K

2C

2O

4$1H

2O K

3C

6H

5O

7

13821 5261 229 1001 2578 217 5611 5475 2044 8412 7680rsquo 191 9814 729 157 1842 2668 213 3064 - -

3 3 1545 2 2 1411 2 2 1738 2 2 1885 2 2 1863 3 3 125 4 - -

Calcium

Ca2+

CaCO3

Ca(OH)2

Ca(CHO2)2

Ca(C2H

3O

2)2

CaC2O

4Ca

3(C

6H

5O

7)2

10009 00007 283 - - - 7409 016rsquo 22 1301 1424rsquo 202 1582 - 15 1281 0006rsquo 22 4984 - -

2 2 1 - - - 3 3 1 3 3 1 3 - - 2 2 1 5 - -

Magnesium

Mg2+

MgCO3

Mg(OH)2

Mg(CHO2)2$2H

2O Mg(C

2H

3O

2)2

MgC2O

4Mg

3(C

6H

5O

7)2

84314 0018rsquo 305 - - - 5832 0007rsquo 237 1504 - - 1424 3961 15 1123 - - 451 - -

2 2 1 - - - 3 3 1 3 - - 3 3 1421 2 - - 5 - -

Iron(IIIII)

Fe2+ Fe3+

FeCO3

Fe(OH)3

Fe(CHO2)3

FeOH(C2H

3O

2)2

Fe2(C

2O

4)3

FeC6H

5O

7$5H

2O

11585 00006 39 - - - 1069 - 312 1909 4521rsquo - 1909 - - 3758 - - 335 - -

2 2 1 - - - 4 - - 4 4 1354 4 - - 5 - - 2 - -

Ammonia Acetone Methanol Ethanol D-Fructose D-Mannitol Sucrose

NH3

(CH3)2CO CH

3OH CH

3CH

2OH C

6H

12O

6C

6H

14O

6C

12H

22O

11

17031 30 0696 5808 10 0785 3204 100 0791 4607 100 0789 1802 48 16 1822 15 1489 3423 80 1581

1 1 1778 3 3 1 2 2 2 3 3 1824 6 6 1204 11 11 1 11 11 1163

Recalling that at equilibrium a binary solution is satu-rated for which the solubility product constant requires thatν+e cations are released forνminus

e anions with the total ofνe=ν+

e +νminuse ions of solute and for which electroneutrality

requires thatν+w moles of H3O+

(aq) andνminusw moles of OHminus

(aq)must be involved in the hydration of each mole of solutewhereby two moles of water are consumed for each mole ofH3O+

(aq) and OHminus

(aq) produced we can express the stoichio-metric constant of water as

νw = νwo + log(2 νe times 1000ws) (19)

with 1000ws=nwMw according to Eq (18d)Table 1 lists the stoichiometric constants of water together

with the required thermodynamic data for nearly 100 com-pounds

Note thatνw is determined by the solubility For near-100 solubilityνw converges to 2 (theoretically to 2301)while for less soluble compounds(Wsle10) νw approachesunity For pure waterνw is not defined and not neededνwo=minus1 and indicates that each mole of hydrated soluteldquoconsumesrdquo log(2 νe times 1000ws) moles of water

35 Single solute molality

The water mass consumed by solute hydration is for aclosed system (Fig 1a) the same as for an open system

(Fig 1b) According to Eq (9) the water uptake is pro-portional to the amount of solute whereby the relativeamount of water is given by the mole fraction of wateror by the molality mss=5551nsnw [molkg H2O] Themolality and the mole fraction of water are related byxw=nw(nw+ns)=1(1+nsnw)=1(1+mss5551)

According to Eq (5) the osmotic pressure change for wa-ter15w equals the osmotic pressure of water5w so that atequilibrium the ratio of5w to the total osmotic pressure ofthe solution equals the relative humidity ie5w5 =RHSince5w5=χw=1(1+νensνwnw) we can directly de-rive mss from RH if νe andνw are known

mss = [νwνe 5551 (1RH minus 1)]νwνe (20)

Note the transformation ofnsnw into molality (multiplica-tion of both sides with 1000Mw=5551 and consideringthatmss rarr m

νeνwss )

Figure 2a shows single solute molalities as a functionof RH for four selected compounds from Table 1 Notethat the full set of figures is presented in the electronicsupplement (httpwwwatmos-chem-physnet731632007acp-7-3163-2007-supplementzip) and that RH equals thewater activity(aw) The single solute molalities include

1 Measurements used in various thermodynamic equi-librium models (EQMs) (black line) Water activ-ity data of NaNO3 Na2SO4 NaHSO4 (NH4)2SO4

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

3172 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

Table 1b ContinuedTable 1 Thermodynamic data (continued)

Anions

Cations

Carbonate

CO3

2-

Hydrogen Carbonate

HCO3

-

Hydroxide

OH-

Formate

CHO2

-

Acetate

C2H

3O

2-

Oxalate

C2O

42-

Citrate

C6H

5O

73-

Hydrogen

H+

H2CO

3 H

2O CH

2O

2C

2H

4O

2C

2H

2O

4C

6H

8O

7

62025 - - - - - 18015 100 0997 4603 68 122 6005 23 1045 9004 869rsquo 19 1921 59rsquo 1665

3 - - - - - 2 2 - 2 2 1833 2 2 1362 3 3 1000 4 4 1470

Ammonium

NH4

+

(NH4)2CO

3NH

4HCO

3NH

4OH NH

4CHO

2NH

4C

2H

3O

2(NH

4)2C

2O

4(NH

4)2HC

6H

5O

7

96086 5000rdquo - 7906 1987 1586 3505 100 - 6306 5885 127 7708 5968 1073 1241 494 15 2262 - 148

3 3 1523 2 2 1298 2 2 2 2 2 177 2 2 1776 3 3 1 4 - -

Sodium

Na+

Na2CO

3NaHCO

3 NaOH NaCHO2

NaC2H

3O

2Na

2C

2O

4Na

3C

6H

5O

7

10599 2349 254 8401 934 22 400 50 213 6801 4869 192 8203 3351 1528 134 348 361 2581 - -

3 19 1393 2 2 1 2 2 17 2 2 1687 2 2 1525 3 3 1 4 - -

Potassium

K+

K2CO

3KHCO

3 KOH KCHO2

KC2H

3O

2K

2C

2O

4$1H

2O K

3C

6H

5O

7

13821 5261 229 1001 2578 217 5611 5475 2044 8412 7680rsquo 191 9814 729 157 1842 2668 213 3064 - -

3 3 1545 2 2 1411 2 2 1738 2 2 1885 2 2 1863 3 3 125 4 - -

Calcium

Ca2+

CaCO3

Ca(OH)2

Ca(CHO2)2

Ca(C2H

3O

2)2

CaC2O

4Ca

3(C

6H

5O

7)2

10009 00007 283 - - - 7409 016rsquo 22 1301 1424rsquo 202 1582 - 15 1281 0006rsquo 22 4984 - -

2 2 1 - - - 3 3 1 3 3 1 3 - - 2 2 1 5 - -

Magnesium

Mg2+

MgCO3

Mg(OH)2

Mg(CHO2)2$2H

2O Mg(C

2H

3O

2)2

MgC2O

4Mg

3(C

6H

5O

7)2

84314 0018rsquo 305 - - - 5832 0007rsquo 237 1504 - - 1424 3961 15 1123 - - 451 - -

2 2 1 - - - 3 3 1 3 - - 3 3 1421 2 - - 5 - -

Iron(IIIII)

Fe2+ Fe3+

FeCO3

Fe(OH)3

Fe(CHO2)3

FeOH(C2H

3O

2)2

Fe2(C

2O

4)3

FeC6H

5O

7$5H

2O

11585 00006 39 - - - 1069 - 312 1909 4521rsquo - 1909 - - 3758 - - 335 - -

2 2 1 - - - 4 - - 4 4 1354 4 - - 5 - - 2 - -

Ammonia Acetone Methanol Ethanol D-Fructose D-Mannitol Sucrose

NH3

(CH3)2CO CH

3OH CH

3CH

2OH C

6H

12O

6C

6H

14O

6C

12H

22O

11

17031 30 0696 5808 10 0785 3204 100 0791 4607 100 0789 1802 48 16 1822 15 1489 3423 80 1581

1 1 1778 3 3 1 2 2 2 3 3 1824 6 6 1204 11 11 1 11 11 1163

Recalling that at equilibrium a binary solution is satu-rated for which the solubility product constant requires thatν+e cations are released forνminus

e anions with the total ofνe=ν+

e +νminuse ions of solute and for which electroneutrality

requires thatν+w moles of H3O+

(aq) andνminusw moles of OHminus

(aq)must be involved in the hydration of each mole of solutewhereby two moles of water are consumed for each mole ofH3O+

(aq) and OHminus

(aq) produced we can express the stoichio-metric constant of water as

νw = νwo + log(2 νe times 1000ws) (19)

with 1000ws=nwMw according to Eq (18d)Table 1 lists the stoichiometric constants of water together

with the required thermodynamic data for nearly 100 com-pounds

Note thatνw is determined by the solubility For near-100 solubilityνw converges to 2 (theoretically to 2301)while for less soluble compounds(Wsle10) νw approachesunity For pure waterνw is not defined and not neededνwo=minus1 and indicates that each mole of hydrated soluteldquoconsumesrdquo log(2 νe times 1000ws) moles of water

35 Single solute molality

The water mass consumed by solute hydration is for aclosed system (Fig 1a) the same as for an open system

(Fig 1b) According to Eq (9) the water uptake is pro-portional to the amount of solute whereby the relativeamount of water is given by the mole fraction of wateror by the molality mss=5551nsnw [molkg H2O] Themolality and the mole fraction of water are related byxw=nw(nw+ns)=1(1+nsnw)=1(1+mss5551)

According to Eq (5) the osmotic pressure change for wa-ter15w equals the osmotic pressure of water5w so that atequilibrium the ratio of5w to the total osmotic pressure ofthe solution equals the relative humidity ie5w5 =RHSince5w5=χw=1(1+νensνwnw) we can directly de-rive mss from RH if νe andνw are known

mss = [νwνe 5551 (1RH minus 1)]νwνe (20)

Note the transformation ofnsnw into molality (multiplica-tion of both sides with 1000Mw=5551 and consideringthatmss rarr m

νeνwss )

Figure 2a shows single solute molalities as a functionof RH for four selected compounds from Table 1 Notethat the full set of figures is presented in the electronicsupplement (httpwwwatmos-chem-physnet731632007acp-7-3163-2007-supplementzip) and that RH equals thewater activity(aw) The single solute molalities include

1 Measurements used in various thermodynamic equi-librium models (EQMs) (black line) Water activ-ity data of NaNO3 Na2SO4 NaHSO4 (NH4)2SO4

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics 3173

Figure 2 a)

0

5

10

15

20

25

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Phosphoric acid - H3PO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

10

20

30

40

50

60

70

80

90

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Sulfuric acid - H2SO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

00002

00003

00004

00005

00006

00007

00008

00009

0001

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Hydrogen - dummy 03

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Nitric acid - HNO3

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Hydrogen chloride - HCl

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Hydrogen bromide - HBr

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Hydrogen iodide - HI

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

5

10

15

20

25

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Ammonium phosphate - (NH4)3PO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

1

2

3

4

5

6

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Ammonium sulfate - (NH4)2SO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

10

20

30

40

50

60

70

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Ammonium hydrogen sulfate - NH4HSO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

5

10

15

20

25

30

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Ammonium nitrate - NH4NO3

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

1

2

3

4

5

6

7

8

9

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Ammonium chloride - NH4Cl

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Ammonium bromide - NH4Br

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Ammonium iodide - NH4I

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

5

10

15

20

25

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Sodium phosphate - Na3PO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

05

1

15

2

25

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Sodium sulfate - Na2SO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

5

10

15

20

25

30

35

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Sodium hydrogen sulfate - NaHSO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

2

4

6

8

10

12

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Sodium nitrate - NaNO3

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

1

2

3

4

5

6

7

8

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Sodium chloride - NaCl

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Sodium bromide - NaBr

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Sodium iodide - NaI

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

5

10

15

20

25

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Potassium phosphate - K3PO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

01

02

03

04

05

06

07

08

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Potassium sulfate - K2SO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Potassium hydrogen sulfate - KHSO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

05

1

15

2

25

3

35

4

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Potassium nitrate - KNO3

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

1

2

3

4

5

6

7

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Potassium chloride - KCl

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Potassium bromide - KBr

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Potassium iodide - KI

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Calcium phosphate - Ca3(PO4)2

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

02

04

06

08

1

12

14

16

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Calcium sulfate - CaSO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

00002

00003

00004

00005

00006

00007

00008

00009

0001

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Calcium - dummy 03

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

1

2

3

4

5

6

7

8

9

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Calcium nitrate - Ca(NO3)2

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

2

4

6

8

10

12

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Calcium chloride - CaCl2

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

10

20

30

40

50

60

70

80

90

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Calcium bromide - CaBr2

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]Relative humidity (RH) []

Calcium iodide - CaI2

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Magnesium phosphate - Mg3(PO4)2

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

1

2

3

4

5

6

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Magnesium sulfate - MgSO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

00002

00003

00004

00005

00006

00007

00008

00009

0001

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Magnesium - dummy 03

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Magnesium nitrate - Mg(NO3)2

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

1

2

3

4

5

6

7

8

9

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Magnesium chloride - MgCl2

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

10

20

30

40

50

60

70

80

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Magnesium bromide - MgBr2

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

10

20

30

40

50

60

70

80

90

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Magnesium iodide - MgI2

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Iron(III) phosphate - FePO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

1

2

3

4

5

6

7

8

9

10

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Iron(III) sulfate - Fe2(SO4)3

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

00002

00003

00004

00005

00006

00007

00008

00009

0001

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Iron(III) - dummy 03

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

5

10

15

20

25

30

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Iron(III) nitrate - Fe(NO3)3

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

2

4

6

8

10

12

14

16

18

20

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Iron(III) chloride - FeCl3

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

5

10

15

20

25

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Iron(III) bromide - FeBr3

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

Mola

lity [m

olkg w

ate

r]

Relative humidity (RH) []

Iron(III) iodide - FeI3

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

Figure 2 Single solute molalities for all compounds of Table 1

Fig 2a (a)Single solute molalities(b) associated water mass (aerosol water uptake) Shown is a selection of figures presented in theelectronic supplement (httpwwwatmos-chem-physnet731632007acp-7-3163-2007-supplementzip Fig A1)

(NH4)HSO4 from Tang and Munkelwitz (1994)NH4NO3 from Chan et al (1992) KCl from Cohenet al (1987) All other sources are given in Kim etal (1993a 1994b)

2 Measurements as listed in the CRC Handbook of Chem-istry and Physics (2006) (black crosses)

3 Calculations according to Eq (20) using CRC-solubilitymeasurements to deriveνw by assuming complete dis-sociation(νs) (blue dots)

4 Same as (3) but considering effective dissociation(νe)

(red crosses)

5 Estimates based on calculated solubility assuming thatthe solubility approximates one minus the ratio of ini-tial molar volumes of water(Vw) and solute(Vs)ie ws=1minusVwVs with Vw=Mwρw andVs=Msρs whereρw andρs denote the density [gcm3] of waterand solute respectively (turquoise squares)

Figure 2a shows that Eq (20) based onνe is in excellentagreement with the measurement data used in various EQMseg ISORROPIA (Nenes et al 1998) and SCAPE (Kim etal 1993a b 1995) SCAPE2 (Meng et al 1995) as pre-viously used by Metzger et al (2006) and with inferredmeasurements from the CRC Handbook The single solutemolalities plotted against RH start from saturation water ac-tivity ie the relative humidity of deliquescence (RHD) upto water vapor saturation (RH=1) Note that usually the sin-gle solute molalities are plotted against water activity whichin this case equals RH For all cases where water activitydata from SCAPE2 were available also RHD values were de-rived For all other cases the single solute molalities areplotted over the entire RH range and only for some com-pounds a comparison with the inferred CRC measurementsis possible All other data should be regarded as predictionsfor which we assumed complete dissociation(νs) so that thetwo lines (blue dots and red crosses) are identical

wwwatmos-chem-physnet731632007 Atmos Chem Phys 7 3163ndash3193 2007

3174 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

Table 2 Relative humidity of deliquescence (RHD) as used in various EQMs (see text for details) All values correspond toT =298 K

1

Table 2) ldquoTabulatedrdquo relative humidity of deliquescence (RHD) as used in various EQMs (see text for details) Values of those compounds

of Table 1 are listed which are currently used All values correspond to T=298K

PO43- SO4

2- HSO4- NO3

- Cl- Br- I- CO32- HCO3

- OH- CHO2- C2H3O2

- C2O42- C6H5O7

3-

H+ -- -- -- -- -- -- -- -- -- -- -- -- -- --

NH4+ -- 07997 04000 06183 07710 -- -- -- -- -- -- -- -- --

Na+ -- 09300 05200 07379 07528 -- -- 08977 09640 -- -- -- -- --

K+ -- 09751 -- 09300 08426 -- -- -- -- -- -- -- -- --

Ca2+ -- 09700 -- 04906 02830 -- -- -- -- -- -- -- -- --

Mg2+ -- 08613 -- -- 03284 -- -- -- -- -- -- -- -- --

Fe3+ -- -- -- -- -- -- -- -- -- -- -- -- -- --

-- -- -- -- -- -- -- -- -- -- -- -- -- --

Table 3) Calculated RHD based on

w from measured solubility and effective dissociation

(e) according to Eq 21 Note that values are

given when solubility measurements were available (listed in Table 1) and that these values strongly depend on the solubility values

PO43- SO4

2- HSO4- NO3

- Cl- Br- I- CO32- HCO3

- OH- CHO2- C2H3O2

- C2O42- C6H5O7

3-

H+ 00011 00939 -- 07816 06908 08366 08886 -- -- 00000 04370 07818 09400 00784

NH4+ 08581 07980 03999 06067 07659 07839 07572 02182 08610 00019 05907 06387 09960 --

Na+ 09985 09390 09285 07476 07540 07713 07593 09051 09486 05160 06732 07965 09989 05927

K+ 01699 09827 08836 09279 08429 08363 08075 03334 08704 05850 05154 05964 08187 --

Ca2+ -- 10000 -- 04806 03228 03922 04559 -- -- -- 08990 -- -- --

Mg2+ -- 08950 -- 07161 03508 04945 05676 -- -- -- -- 05117 -- --

Fe3+ -- 00060 -- 03338 01126 00248 09992 -- -- -- 01994 -- -- --

09415 07255 00018 00000 00032 08776 00000 -- -- -- -- -- -- --

Table 3 Calculated RHD based onνw from measured solubility and effective dissociation(νe) according to Eq (21) Note that values aregiven when solubility measurements were available (listed in Table 1) and that these values strongly depend on the solubility

1

Table 2) ldquoTabulatedrdquo relative humidity of deliquescence (RHD) as used in various EQMs (see text for details) Values of those compounds

of Table 1 are listed which are currently used All values correspond to T=298K

PO43- SO4

2- HSO4- NO3

- Cl- Br- I- CO32- HCO3

- OH- CHO2- C2H3O2

- C2O42- C6H5O7

3-

H+ -- -- -- -- -- -- -- -- -- -- -- -- -- --

NH4+ -- 07997 04000 06183 07710 -- -- -- -- -- -- -- -- --

Na+ -- 09300 05200 07379 07528 -- -- 08977 09640 -- -- -- -- --

K+ -- 09751 -- 09300 08426 -- -- -- -- -- -- -- -- --

Ca2+ -- 09700 -- 04906 02830 -- -- -- -- -- -- -- -- --

Mg2+ -- 08613 -- -- 03284 -- -- -- -- -- -- -- -- --

Fe3+ -- -- -- -- -- -- -- -- -- -- -- -- -- --

-- -- -- -- -- -- -- -- -- -- -- -- -- --

Table 3) Calculated RHD based on

w from measured solubility and effective dissociation

(e) according to Eq 21 Note that values are

given when solubility measurements were available (listed in Table 1) and that these values strongly depend on the solubility values

PO43- SO4

2- HSO4- NO3

- Cl- Br- I- CO32- HCO3

- OH- CHO2- C2H3O2

- C2O42- C6H5O7

3-

H+ 00011 00939 -- 07816 06908 08366 08886 -- -- 00000 04370 07818 09400 00784

NH4+ 08581 07980 03999 06067 07659 07839 07572 02182 08610 00019 05907 06387 09960 --

Na+ 09985 09390 09285 07476 07540 07713 07593 09051 09486 05160 06732 07965 09989 05927

K+ 01699 09827 08836 09279 08429 08363 08075 03334 08704 05850 05154 05964 08187 --

Ca2+ -- 10000 -- 04806 03228 03922 04559 -- -- -- 08990 -- -- --

Mg2+ -- 08950 -- 07161 03508 04945 05676 -- -- -- -- 05117 -- --

Fe3+ -- 00060 -- 03338 01126 00248 09992 -- -- -- 01994 -- -- --

09415 07255 00018 00000 00032 08776 00000 -- -- -- -- -- -- --

Only in case of EQM water activity measurements wecould determine the effective dissociation(νe) by using anoptimal fit of Eq (20) to the measurements where necessaryas for instance for NH4Cl The accuracy of the results ofEq (20) is then dependent on the accuracy of these measure-ments However strong electrolytes practically completelydissociate NaNO3 and NaCl have almost identicalνs andνewhich provides some confidence in both the measurementsand Eq (20) For cases whereνs andνe differ we can seethe sensitivity of Eq (20) to these parameters Similarly thesingle solute molality estimates based on simple solubilityapproximations (turquoise squares) additionally indicate thesensitivity ofmss(RH) to uncertainties in the solubility dataIt is important to note that only concentration independentconstants have been used over the entire concentration rangeto predict the single solute molalities of various solutes forall cases1

1The single solute molality measurements in the CRC Hand-book are listed as a function of solubility(Ws=100timesws) ratherthan water activity(aw) We therefore plotted the molality (blackcrosses) against the water activity only when the solubility val-ues matched those derived from Eq (20) Although this can leadto a bias in the comparison in particular for the steepness of themss (RH) functions we can evaluate the accuracy of this compari-son for all cases where we additionally have measurements avail-able as used in EQMs Since the agreement is rather good andsince the CRC solubility measurements and those derived fromEq (20) must match at the saturation water activity (ie at theRHD) which fixes the steepness of themss (RH) function wehave included the CRC measurements also for cases where we donot have independent measurements Especially the steepness ofthe mss (RH) functions of the 7 non-electrolytes (Table 1 for fig-

36 Relative humidity of deliquescence (RHD)

The relative humidity of deliquescence (RHD) describes therelative humidity at which a solid salt deliquesces throughwater uptake At equilibrium a solution is saturated and thecorresponding RH equals the RHD of the salt The RHD cantherefore be directly computed with Eq (20) RearrangingEq (20) and solving for RH ie with RH=RHD

RHD =

(νeνw m

νeνwsssat5551 + 1

)minus1 (21)

whereby we obtain the saturation molality (msssat) from thesolutersquos solubility according to Eq (18c)

Table 2 lists RHD values used by various EQMs (seeeg Metzger 2000 for details) and previously applied byMetzger et al (2002 2006) while Table 3 lists all (predicted)RHD values obtained with Eq (21) for all compounds listed

ures see electronic supplementhttpwwwatmos-chem-physnet731632007acp-7-3163-2007-supplementzip) strongly depend onthe assumptions forνe and νs By assuming a value of one foreither νe or νs the steepness of allmss (RH) functions of the 7non-electrolytes increase as strongly as the one of ammonia (NH3)which seems unrealistic for the alcohols and sugars as it would in-dicate a very low solubility Furthermore the agreement with CRCmeasurements is then quite poor as only the very first water activ-ity measurements near unity match (not shown) Only for theνe

andνs values given in Table 1 the relatively best agreement withthe CRC measurements is achieved in terms of a maximum num-ber of solubility data points that matches This indicates howeverthat ndash probably as a rule of thumb ndash approximately each fractionalgroup or oxygen atom becomes hydrated so that egνe=11 for D-mannitol and sucrose

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics 3175

in Table 1 Note that Table 2 only contains RHD values forthose compounds of Table 1 that are included in the citedEQMs Note further that Eq (21) further allows to calculateefflorescence and deliquescence RHDs of single or mixedsalt solutions (see Sect 41 point 5 and 9) whereby these val-ues provide additional support for the accuracy of Eq (20)

37 Aerosol associated water mass

The water mass associated with atmospheric aerosols can beobtained for single solute or mixed solutions from the defini-tion of molality (mss=5551nsnw) using Eq (20)

mwss = nsmss

= ns times [νwνe 5551 (1RHminus1)]minusνwνe (22)

The total water mass associated with a mixed solution con-tainingn-single solutes is in case of osmotic pressure addi-tivity the sum of the water masses associated with all singlesolute solutions

mw =

nsumj=1

nsjmssj (23)

Figure 2b shows (according to Fig 2a) the aerosol watermass as a function of RH for four selected compounds calcu-lated with Eq (22) for single solute solutions containing 1microgof solute atT =25C For the cases where RH approachesunity the aerosol water mass is limited by the saturation wa-ter vapor mass which is a function of temperature For allother cases (RHlt1) the water mass is limited by the avail-able water vapor and depends on RH

The use of equivalent solute masses indicates that in con-trast to the calculated single solute molalities the associatedwater mass is much less sensitive to uncertainties in solu-bility Different hygroscopicities of salt solutes cause (a) adifferent amount of water uptake at a given RH and (b) obeya different RHD which determines (i) the RH at which a solu-tion is saturated with respect to the dissolved salt and (ii) theRH range over which water is associated Less soluble saltscan take up water only over a smaller RH range ie they fol-low a higher RHD so that they precipitate more rapidly fromthe solution as the water activity deviates from unity

Note this is very important for aerosol optical and air pol-lution aspects For instance the deliquescence behavior ofnatural aerosol compounds which include eg NaCl andMgCl2 changes considerably through the mixing with airpollution whereby the chlorides are often replaced by ni-trates and sulfates which have different RHDs (see Table 3)Air pollution can thus drastically alter the RHD of sea saltaerosol particles reducing their equilibrium radius and thusmodify their scattering properties and the efficiency by whichthe particles can grow into cloud droplets Furthermore thewater mass associated with a certain amount of solute alsodepends on the salt component Lighter salt compounds typ-ically bind a larger mass of water For instance 1microg of

ammonium chloride (NH4Cl) in air pollution fixes approx-imately the same amount of water as NaCl According toFig 2b at RH=80 both would be associated with approxi-mately 30microg of water while NaNO3 and ammonium nitrate(NH4NO3) would fix only about half that amount thoughover a wider range of RH conditions

4 Equilibrium model

41 EQSAM3

The theoretical considerations of the previous sections havebeen incorporated into the third version of the thermody-namic Equilibrium Simplified Aerosol Model EQSAM3building on earlier versions of Metzger et al (2002 2006)Our aim is to apply EQSAM3 in regional and globalchemistry-transport and climate models It computes thegasliquidsolid partitioning of all compounds listed in Ta-ble 1 whereby only the measured (or estimated) solubilityis required as input for each compound Note that extensionwith additional compounds can be easily accomplished Pre-vious EQSAM versions instead used equilibrium constantsand tabulated RHD values and a division in certain chemicaldomains and sub-domains similar to other EQMs EQSAM3analytically solves the gasliquidsolid partitioning of almost100 compounds without further constraints on the aerosolsystem

The model set up is as follows

1 The model is initialized using the thermodynamic dataprovided in Table 1 whereby the stoichiometric con-stants for water(νw) and solute(νe) can be either pre-scribed orνw can be computed online with Eq (19)from the compoundrsquos solubility (Ws) by accounting forits temperature dependency

2 Since the underlying physical principles are those of anosmotic system for which the gas-solution analogy isappropriate (see Sect 212) we assume that the tem-perature dependency is described by the gas law ie thatit is sufficient for most compounds to divideWs byToT To is the temperature at which the solubilitylisted in Table 1 has been measured (for most com-poundsTo=29815 [K]) Note that this describes thetemperature dependency of the gasaerosol system as awhole since the solubility is used to calculate all otherthermodynamic properties (νw mss mw RHD)

3 From νw andνe the single solute molalities (mss) arederived with Eq (20) as a function of RH andT for allcompounds listed in Table 1

4 The water mass (mw) of single solutes and mixed solu-tions is computed according to Eqs (22) and (23) Notethat Eq (23) directly follows from the first principlesunderlying an osmotic system (Sect 2) ie from the

wwwatmos-chem-physnet731632007 Atmos Chem Phys 7 3163ndash3193 2007

3176 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

Figure 2 b)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Phosphoric acid - H3PO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Sulfuric acid - H2SO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Hydrogen - dummy 03

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Nitric acid - HNO3

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Hydrogen chloride - HCl

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Hydrogen bromide - HBr

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Hydrogen iodide - HI

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Ammonium phosphate - (NH4)3PO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Ammonium sulfate - (NH4)2SO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Ammonium hydrogen sulfate - NH4HSO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass [ug m

-3(a

ir)]

Relative humidity (RH) []

Ammonium nitrate - NH4NO3

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass [ug m

-3(a

ir)]

Relative humidity (RH) []

Ammonium chloride - NH4Cl

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Ammonium bromide - NH4Br

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Ammonium iodide - NH4I

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Sodium phosphate - Na3PO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Sodium sulfate - Na2SO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Sodium hydrogen sulfate - NaHSO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass [ug m

-3(a

ir)]

Relative humidity (RH) []

Sodium nitrate - NaNO3

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass [ug m

-3(a

ir)]

Relative humidity (RH) []

Sodium chloride - NaCl

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Sodium bromide - NaBr

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Sodium iodide - NaI

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Potassium phosphate - K3PO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Potassium sulfate - K2SO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Potassium hydrogen sulfate - KHSO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Potassium nitrate - KNO3

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Potassium chloride - KCl

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Potassium bromide - KBr

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Potassium iodide - KI

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

Figure 1 Single solute water mass

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Calcium phosphate - Ca3(PO4)2

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Calcium sulfate - CaSO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Calcium - dummy 03

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Calcium nitrate - Ca(NO3)2

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Calcium chloride - CaCl2

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Calcium bromide - CaBr2

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Calcium iodide - CaI2

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Magnesium phosphate - Mg3(PO4)2

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Magnesium sulfate - MgSO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Magnesium - dummy 03

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Magnesium nitrate - Mg(NO3)2

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Magnesium chloride - MgCl2

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Magnesium bromide - MgBr2

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Magnesium iodide - MgI2

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Iron(III) phosphate - FePO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Iron(III) sulfate - Fe2(SO4)3

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Iron(III) - dummy 03

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Iron(III) nitrate - Fe(NO3)3

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Iron(III) chloride - FeCl3

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Iron(III) bromide - FeBr3

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Iron(III) iodide - FeI3

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

Fig 2b Continued

additivity of partial pressures (implying molar volumeadditivity for salt compounds) being a consequence ofthe gas-solution analogy Note further that Eq (23) isequivalent to the ZSR-relation an assumption about theadditivity of partial water masses widely used in atmo-spheric modeling as empirically established accordingto Zdanovskii (1948) Stokes and Robinson (1966)

5 The relative humidity of deliquescence (RHD)is calculated from Eq (21) for single solute so-lutions and mixed solutions For the latter theRHD is currently approximated by summariz-ing νw νe and mss over all compounds presentin the solution so that Eq (21) yields RHD=l(aq)sum

j=1νe

l(aq)sumj=1

νw

l(aq)sumj=1

m

l(aq)sumj=1

νe

l(aq)sumj=1

νw

ss 5551 + 1

minus1

For instance for a mixed solution containing(NH4)2SO4 + Na2SO4 + NH4Cl we obtain a mixedsolution RHD=0522 which is lower than the RHDsof the individual compounds (which are accordingto Table 3 0798 0939 07659 respectively) Thevalues used in ISORROPIA (Nenes et al 1998) are054 (mixed solution) and according to Table 207997 093 0771 respectively for the individualcompounds However the RHD of mixed solutionscould be calculated more explicitly with Eq (21) if theactual solubilities of solutes in mixed solutions are usedto deriveνw (and subsequentlymss mw) at a givenT ndasha subject that will be investigated further

6 The reaction order can be either prescribed or deter-mined automatically based on the RHD of the solutes(a) In case the reaction order is not prescribed we con-sider that the reaction order is primarily determined bythe solubility Compounds with a low solubility precip-itate from solution already at relatively high RH so that

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics 3177

these ions are not available for further reactions Forinstance the solubility of calcium sulfate (CaSO4) isvery low (lt1) which leads to precipitation of CaSO4at a RH close to 100 (Table 2) CaSO4 and otherlow-soluble salt compounds are therefore regarded aspure solids over the entire RH range (b) In case thereaction order is prescribed we rank the ions towardstheir ability of neutralization according to the Hofmeis-ter series (Hofmeister 1888) to account for the degreeto which ions bind water (salting-out effect) This in-creases the effective concentration of other ions (in theremaining ldquofreerdquo water) so that they precipitate thusreleasing low entropy surface water (for details seeeghttpwwwlsbuacukwater)

We assume the following neutralization order for theions of the single solutes currently considered wherebythe ions to the left become preferentially neutralized

ndash Anions PO3minus

4 gtSO2minus

4 gtHSOminus

4 gtNOminus

3 gt

ClminusgtBrminusgtIminusgtCO2minus

3 gt

HCOminus

3 gtOHminusgtCHOminus

2 gt

C2H3Ominus

2 gtC2O2minus

4 gtC6H5O3minus

7

ndash Cations Fe3+gtMg2+gtCa2+gtNa+gt

K+gtNH+

4 gtH+

Note that this neutralization order is preliminary Mea-surements are required for validation

7 Based on the reaction order (prescribed or automaticallydetermined for givenT RH) the compounds in the solu-tion and the non-neutralized ldquofreerdquo ions are computedThe H+ concentration and the pH of the solution is ex-plicitly calculated starting from electroneutrality by ac-counting for the auto-dissociation of water and option-ally for atmospheric CO2 Based on the RHD of thesingle solutes in the (mixed) solution liquidsolid parti-tioning is calculated whereby all compounds for whichthe RH is below the RHD are assumed to be precipi-tated so that a solid and liquid phase can co-exist

8 We distinguish between the wetting and drying processof the particle following a hysteresis loop While forthe former case (lower tail of the hysteresis loop) am-bient aerosols are wetted as the RH increases above thedeliquescence of salt compounds in the solid phase (bywhich more hygroscopic compounds take up water firstie at a lower RH) the latter case (upper tail of thehysteresis loop) is calculated by considering the com-pounds efflorescence RHs whereby the particle is as-sumed to be a pure solid if all salt compounds are crys-tallized The RH at which salt compounds crystallizescan be much lower than its deliquescence RH This di-rectly results from the fact that in case the RH decreasesbelow the RHD the compound precipitates out of the

solution until its solubility product is re-establised hav-ing the same water activity as at the RHD but with lesswater according to the lower solute mass This processcontinues until the solute has been completely precip-itated whereby the solute can be assumed to be crys-tallized if no water is left for hydration This exactlyhappens at the efflorescence RHcr Depending on thesolute hygroscopicity the final particle crystallizationcan occur at a very low RH below 10

9 Efflorescence humidities (RHcr) are calculated withEq (21) Instead of the saturation molality (msssat)we use the actual (supersaturation) molality tocalculate the water uptake as a function of RHExpressing the molality in terms of the remainingsolute mass fraction similar to Eq (18c) but withmss=1000Ms(1wsminus1) we can calculate uponsubstitution into Eq (21) the humidity at which thesolute is completely precipitated (crystallized) from

RHcr=1minus[( νe

νw[1000Ms(1ws minus 1)]

νeνw 5551)+1]

minus1For instance for three pure salt compounds (1)NH4NO3 (2) NH4HSO4 and (3) (NH4)2SO4we obtain the following efflorescence humidi-ties RHcr1=011(010) RHcr2=007(002) andRHcr3=034(039) respectively values in brackets arereported in the literature by ten Brink et al (1996) forNH4NO3 and Tang and Munkelwitz (1994) for the twoother salts

10 Gasliquid partitioning is calculated for all (semi-)volatile compounds (first row of Table 1) except phos-phoric and sulfuric acid which are treated as non-volatile due to their very low vapor pressures

11 For (semi-) volatile compounds such as NH4NO3 andNH4Cl activity coefficients are used Non-volatilecompounds remain in the particulate phase independentof the solute concentration whereby the liquidsolidpartitioning is merely determined by the solute sol-ubility Since the water mass is proportional to thesolute mass (at a givenT RH) activity coefficients arenot needed for non-volatile compounds For (semi-)volatile compounds which can be driven out of theaerosol in the gas phase activity coefficients are neededand computed from Eqs (14ndash16) and (20) whereby weaccount for the charge density of the solution (Metzgeret al 2002a) The mean ion-pair activity coefficients ofvolatile compounds are thus obtained in EQSAM3 from

γ plusmn

sj=

(RHminus

νwνe [νwνe 5551 (1RH minus 1)]

νwνe

) 2ξsj

whereby we divide the mean ion-pair activity coeffi-cient by the solutersquos density (ρsj ) to obtain the the meanmolar binary activity coefficientγ plusmn

s(molar)j=γ plusmn

sjρsj

[l(H2O)mol(solute)] and additionally multiply it bythe density of water (ρw) to obtain it on the molal scaleγ plusmn

s(molal)j = γ plusmn

sjtimesρwρsj [kg(H2O)mol(solute)]

wwwatmos-chem-physnet731632007 Atmos Chem Phys 7 3163ndash3193 2007

3176 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

Figure 2 b)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Phosphoric acid - H3PO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Sulfuric acid - H2SO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Hydrogen - dummy 03

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Nitric acid - HNO3

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Hydrogen chloride - HCl

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Hydrogen bromide - HBr

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Hydrogen iodide - HI

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Ammonium phosphate - (NH4)3PO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Ammonium sulfate - (NH4)2SO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Ammonium hydrogen sulfate - NH4HSO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass [ug m

-3(a

ir)]

Relative humidity (RH) []

Ammonium nitrate - NH4NO3

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass [ug m

-3(a

ir)]

Relative humidity (RH) []

Ammonium chloride - NH4Cl

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Ammonium bromide - NH4Br

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Ammonium iodide - NH4I

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Sodium phosphate - Na3PO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Sodium sulfate - Na2SO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Sodium hydrogen sulfate - NaHSO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass [ug m

-3(a

ir)]

Relative humidity (RH) []

Sodium nitrate - NaNO3

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass [ug m

-3(a

ir)]

Relative humidity (RH) []

Sodium chloride - NaCl

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Sodium bromide - NaBr

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Sodium iodide - NaI

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Potassium phosphate - K3PO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Potassium sulfate - K2SO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Potassium hydrogen sulfate - KHSO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Potassium nitrate - KNO3

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Potassium chloride - KCl

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Potassium bromide - KBr

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Potassium iodide - KI

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

Figure 1 Single solute water mass

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Calcium phosphate - Ca3(PO4)2

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Calcium sulfate - CaSO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Calcium - dummy 03

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Calcium nitrate - Ca(NO3)2

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Calcium chloride - CaCl2

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Calcium bromide - CaBr2

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Calcium iodide - CaI2

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Magnesium phosphate - Mg3(PO4)2

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Magnesium sulfate - MgSO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Magnesium - dummy 03

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Magnesium nitrate - Mg(NO3)2

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Magnesium chloride - MgCl2

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Magnesium bromide - MgBr2

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Magnesium iodide - MgI2

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Iron(III) phosphate - FePO4

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Iron(III) sulfate - Fe2(SO4)3

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Iron(III) - dummy 03

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Iron(III) nitrate - Fe(NO3)3

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Iron(III) chloride - FeCl3

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Iron(III) bromide - FeBr3

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

00001

001

1

100

10000

1e+06

1e+08

20 40 60 80 100

Aero

sol w

ate

r m

ass m

w(1

ug m

s)

[ug m

-3(a

ir)]

Relative humidity (RH) []

Iron(III) iodide - FeI3

Measurements (EQM data)Measurements (CRC data)

Calculations (Sol - ns)Calculations (Sol - ne)

Estimates (cal Sol)

Fig 2b Continued

additivity of partial pressures (implying molar volumeadditivity for salt compounds) being a consequence ofthe gas-solution analogy Note further that Eq (23) isequivalent to the ZSR-relation an assumption about theadditivity of partial water masses widely used in atmo-spheric modeling as empirically established accordingto Zdanovskii (1948) Stokes and Robinson (1966)

5 The relative humidity of deliquescence (RHD)is calculated from Eq (21) for single solute so-lutions and mixed solutions For the latter theRHD is currently approximated by summariz-ing νw νe and mss over all compounds presentin the solution so that Eq (21) yields RHD=l(aq)sum

j=1νe

l(aq)sumj=1

νw

l(aq)sumj=1

m

l(aq)sumj=1

νe

l(aq)sumj=1

νw

ss 5551 + 1

minus1

For instance for a mixed solution containing(NH4)2SO4 + Na2SO4 + NH4Cl we obtain a mixedsolution RHD=0522 which is lower than the RHDsof the individual compounds (which are accordingto Table 3 0798 0939 07659 respectively) Thevalues used in ISORROPIA (Nenes et al 1998) are054 (mixed solution) and according to Table 207997 093 0771 respectively for the individualcompounds However the RHD of mixed solutionscould be calculated more explicitly with Eq (21) if theactual solubilities of solutes in mixed solutions are usedto deriveνw (and subsequentlymss mw) at a givenT ndasha subject that will be investigated further

6 The reaction order can be either prescribed or deter-mined automatically based on the RHD of the solutes(a) In case the reaction order is not prescribed we con-sider that the reaction order is primarily determined bythe solubility Compounds with a low solubility precip-itate from solution already at relatively high RH so that

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics 3177

these ions are not available for further reactions Forinstance the solubility of calcium sulfate (CaSO4) isvery low (lt1) which leads to precipitation of CaSO4at a RH close to 100 (Table 2) CaSO4 and otherlow-soluble salt compounds are therefore regarded aspure solids over the entire RH range (b) In case thereaction order is prescribed we rank the ions towardstheir ability of neutralization according to the Hofmeis-ter series (Hofmeister 1888) to account for the degreeto which ions bind water (salting-out effect) This in-creases the effective concentration of other ions (in theremaining ldquofreerdquo water) so that they precipitate thusreleasing low entropy surface water (for details seeeghttpwwwlsbuacukwater)

We assume the following neutralization order for theions of the single solutes currently considered wherebythe ions to the left become preferentially neutralized

ndash Anions PO3minus

4 gtSO2minus

4 gtHSOminus

4 gtNOminus

3 gt

ClminusgtBrminusgtIminusgtCO2minus

3 gt

HCOminus

3 gtOHminusgtCHOminus

2 gt

C2H3Ominus

2 gtC2O2minus

4 gtC6H5O3minus

7

ndash Cations Fe3+gtMg2+gtCa2+gtNa+gt

K+gtNH+

4 gtH+

Note that this neutralization order is preliminary Mea-surements are required for validation

7 Based on the reaction order (prescribed or automaticallydetermined for givenT RH) the compounds in the solu-tion and the non-neutralized ldquofreerdquo ions are computedThe H+ concentration and the pH of the solution is ex-plicitly calculated starting from electroneutrality by ac-counting for the auto-dissociation of water and option-ally for atmospheric CO2 Based on the RHD of thesingle solutes in the (mixed) solution liquidsolid parti-tioning is calculated whereby all compounds for whichthe RH is below the RHD are assumed to be precipi-tated so that a solid and liquid phase can co-exist

8 We distinguish between the wetting and drying processof the particle following a hysteresis loop While forthe former case (lower tail of the hysteresis loop) am-bient aerosols are wetted as the RH increases above thedeliquescence of salt compounds in the solid phase (bywhich more hygroscopic compounds take up water firstie at a lower RH) the latter case (upper tail of thehysteresis loop) is calculated by considering the com-pounds efflorescence RHs whereby the particle is as-sumed to be a pure solid if all salt compounds are crys-tallized The RH at which salt compounds crystallizescan be much lower than its deliquescence RH This di-rectly results from the fact that in case the RH decreasesbelow the RHD the compound precipitates out of the

solution until its solubility product is re-establised hav-ing the same water activity as at the RHD but with lesswater according to the lower solute mass This processcontinues until the solute has been completely precip-itated whereby the solute can be assumed to be crys-tallized if no water is left for hydration This exactlyhappens at the efflorescence RHcr Depending on thesolute hygroscopicity the final particle crystallizationcan occur at a very low RH below 10

9 Efflorescence humidities (RHcr) are calculated withEq (21) Instead of the saturation molality (msssat)we use the actual (supersaturation) molality tocalculate the water uptake as a function of RHExpressing the molality in terms of the remainingsolute mass fraction similar to Eq (18c) but withmss=1000Ms(1wsminus1) we can calculate uponsubstitution into Eq (21) the humidity at which thesolute is completely precipitated (crystallized) from

RHcr=1minus[( νe

νw[1000Ms(1ws minus 1)]

νeνw 5551)+1]

minus1For instance for three pure salt compounds (1)NH4NO3 (2) NH4HSO4 and (3) (NH4)2SO4we obtain the following efflorescence humidi-ties RHcr1=011(010) RHcr2=007(002) andRHcr3=034(039) respectively values in brackets arereported in the literature by ten Brink et al (1996) forNH4NO3 and Tang and Munkelwitz (1994) for the twoother salts

10 Gasliquid partitioning is calculated for all (semi-)volatile compounds (first row of Table 1) except phos-phoric and sulfuric acid which are treated as non-volatile due to their very low vapor pressures

11 For (semi-) volatile compounds such as NH4NO3 andNH4Cl activity coefficients are used Non-volatilecompounds remain in the particulate phase independentof the solute concentration whereby the liquidsolidpartitioning is merely determined by the solute sol-ubility Since the water mass is proportional to thesolute mass (at a givenT RH) activity coefficients arenot needed for non-volatile compounds For (semi-)volatile compounds which can be driven out of theaerosol in the gas phase activity coefficients are neededand computed from Eqs (14ndash16) and (20) whereby weaccount for the charge density of the solution (Metzgeret al 2002a) The mean ion-pair activity coefficients ofvolatile compounds are thus obtained in EQSAM3 from

γ plusmn

sj=

(RHminus

νwνe [νwνe 5551 (1RH minus 1)]

νwνe

) 2ξsj

whereby we divide the mean ion-pair activity coeffi-cient by the solutersquos density (ρsj ) to obtain the the meanmolar binary activity coefficientγ plusmn

s(molar)j=γ plusmn

sjρsj

[l(H2O)mol(solute)] and additionally multiply it bythe density of water (ρw) to obtain it on the molal scaleγ plusmn

s(molal)j = γ plusmn

sjtimesρwρsj [kg(H2O)mol(solute)]

wwwatmos-chem-physnet731632007 Atmos Chem Phys 7 3163ndash3193 2007

S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics 3177

these ions are not available for further reactions Forinstance the solubility of calcium sulfate (CaSO4) isvery low (lt1) which leads to precipitation of CaSO4at a RH close to 100 (Table 2) CaSO4 and otherlow-soluble salt compounds are therefore regarded aspure solids over the entire RH range (b) In case thereaction order is prescribed we rank the ions towardstheir ability of neutralization according to the Hofmeis-ter series (Hofmeister 1888) to account for the degreeto which ions bind water (salting-out effect) This in-creases the effective concentration of other ions (in theremaining ldquofreerdquo water) so that they precipitate thusreleasing low entropy surface water (for details seeeghttpwwwlsbuacukwater)

We assume the following neutralization order for theions of the single solutes currently considered wherebythe ions to the left become preferentially neutralized

ndash Anions PO3minus

4 gtSO2minus

4 gtHSOminus

4 gtNOminus

3 gt

ClminusgtBrminusgtIminusgtCO2minus

3 gt

HCOminus

3 gtOHminusgtCHOminus

2 gt

C2H3Ominus

2 gtC2O2minus

4 gtC6H5O3minus

7

ndash Cations Fe3+gtMg2+gtCa2+gtNa+gt

K+gtNH+

4 gtH+

Note that this neutralization order is preliminary Mea-surements are required for validation

7 Based on the reaction order (prescribed or automaticallydetermined for givenT RH) the compounds in the solu-tion and the non-neutralized ldquofreerdquo ions are computedThe H+ concentration and the pH of the solution is ex-plicitly calculated starting from electroneutrality by ac-counting for the auto-dissociation of water and option-ally for atmospheric CO2 Based on the RHD of thesingle solutes in the (mixed) solution liquidsolid parti-tioning is calculated whereby all compounds for whichthe RH is below the RHD are assumed to be precipi-tated so that a solid and liquid phase can co-exist

8 We distinguish between the wetting and drying processof the particle following a hysteresis loop While forthe former case (lower tail of the hysteresis loop) am-bient aerosols are wetted as the RH increases above thedeliquescence of salt compounds in the solid phase (bywhich more hygroscopic compounds take up water firstie at a lower RH) the latter case (upper tail of thehysteresis loop) is calculated by considering the com-pounds efflorescence RHs whereby the particle is as-sumed to be a pure solid if all salt compounds are crys-tallized The RH at which salt compounds crystallizescan be much lower than its deliquescence RH This di-rectly results from the fact that in case the RH decreasesbelow the RHD the compound precipitates out of the

solution until its solubility product is re-establised hav-ing the same water activity as at the RHD but with lesswater according to the lower solute mass This processcontinues until the solute has been completely precip-itated whereby the solute can be assumed to be crys-tallized if no water is left for hydration This exactlyhappens at the efflorescence RHcr Depending on thesolute hygroscopicity the final particle crystallizationcan occur at a very low RH below 10

9 Efflorescence humidities (RHcr) are calculated withEq (21) Instead of the saturation molality (msssat)we use the actual (supersaturation) molality tocalculate the water uptake as a function of RHExpressing the molality in terms of the remainingsolute mass fraction similar to Eq (18c) but withmss=1000Ms(1wsminus1) we can calculate uponsubstitution into Eq (21) the humidity at which thesolute is completely precipitated (crystallized) from

RHcr=1minus[( νe

νw[1000Ms(1ws minus 1)]

νeνw 5551)+1]

minus1For instance for three pure salt compounds (1)NH4NO3 (2) NH4HSO4 and (3) (NH4)2SO4we obtain the following efflorescence humidi-ties RHcr1=011(010) RHcr2=007(002) andRHcr3=034(039) respectively values in brackets arereported in the literature by ten Brink et al (1996) forNH4NO3 and Tang and Munkelwitz (1994) for the twoother salts

10 Gasliquid partitioning is calculated for all (semi-)volatile compounds (first row of Table 1) except phos-phoric and sulfuric acid which are treated as non-volatile due to their very low vapor pressures

11 For (semi-) volatile compounds such as NH4NO3 andNH4Cl activity coefficients are used Non-volatilecompounds remain in the particulate phase independentof the solute concentration whereby the liquidsolidpartitioning is merely determined by the solute sol-ubility Since the water mass is proportional to thesolute mass (at a givenT RH) activity coefficients arenot needed for non-volatile compounds For (semi-)volatile compounds which can be driven out of theaerosol in the gas phase activity coefficients are neededand computed from Eqs (14ndash16) and (20) whereby weaccount for the charge density of the solution (Metzgeret al 2002a) The mean ion-pair activity coefficients ofvolatile compounds are thus obtained in EQSAM3 from

γ plusmn

sj=

(RHminus

νwνe [νwνe 5551 (1RH minus 1)]

νwνe

) 2ξsj

whereby we divide the mean ion-pair activity coeffi-cient by the solutersquos density (ρsj ) to obtain the the meanmolar binary activity coefficientγ plusmn

s(molar)j=γ plusmn

sjρsj

[l(H2O)mol(solute)] and additionally multiply it bythe density of water (ρw) to obtain it on the molal scaleγ plusmn

s(molal)j = γ plusmn

sjtimesρwρsj [kg(H2O)mol(solute)]

wwwatmos-chem-physnet731632007 Atmos Chem Phys 7 3163ndash3193 2007

3178 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

0001

001

01

1

10

100

280701 040801 110801 180801 250801

Aero

sol W

ate

r [u

gm

^3]

Time [270720011300 - 25082001400]

MINOS Campaign Fine Aerosol Fraction (gasliquid partitioning) NaCl-NH3NH4-HNO3NO3-H2SO4SO4-H2O-System

EQSAM3ISORROPIA

SCAPE2EQSAM2

0001

001

01

1

10

100

280701 040801 110801 180801 250801

Aero

sol W

ate

r [u

gm

^3]

Time [270720011300 - 25082001400]

MINOS Campaign Coarse Aerosol Fraction (gasliquid partitioning) NaCl-NH3NH4-HNO3NO3-H2SO4SO4-H2O-System

EQSAM3ISORROPIA

SCAPE2EQSAM2

0

100

200

300

400

500

600

280701 040801 110801 180801 250801

tota

l P

M [nm

ol m

-3 (

air)]

Time [270720011300 - 25082001400]

MINOS Campaign Fine Aerosol Fraction (gasliquid partitioning) NaCl-NH3NH4-HNO3NO3-H2SO4SO4-H2O-System

Meas tPMEQSAM3

ISORROPIASCAPE2EQSAM2

0

100

200

300

400

500

600

280701 040801 110801 180801 250801

tota

l P

M [nm

ol m

-3 (

air)]

Time [270720011300 - 25082001400]

MINOS Campaign Coarse Aerosol Fraction (gasliquid partitioning) NaCl-NH3NH4-HNO3NO3-H2SO4SO4-H2O-System

Meas tPMEQSAM3

ISORROPIASCAPE2EQSAM2

Figure 3 a)Fig 3a Mixed solution properties and model comparison for the MINOS campaign (Metzger et al 2006)(a) Aerosol water (top) totalnumber of moles of particulate matter (PM) (bottom) for fine (left) and coarse mode (right)(b) fine mode total aerosol mass (top left)total solid mass (top right) aerosol ammonium nitrate activity coefficient (bottom left) pH (bottom right)(c) residual gaseous ammonia(top left) residual gaseous nitric acid (top right) aerosol fine mode ammonium (bottom left) aerosol fine mode nitrate (bottom right) Allpanels show time series for the period 28 Julyndash25 August 2001 of the ammoniumsulfatenitratechloridesodiumwater system comparingmeasurements (black solid line) and results of EQSAM3 (red crosses) ISORROPIA (yellow closed triangles) SCAPE2 (green closedsquares) EQSAM2 (blue small crosses) Note that this model comparison and chemical system is identical to the model comparison forchemical system F2C2 of Metzger et al (2006) with EQSAM2 denoted as EQSAM2

Activity coefficients of the corresponding cations(γs+)

and anions(γsminus) can be computed from Eq (16)

assumingγν+e

s+ =γνminuse

sminus in accord with Eq (15)

ξsj expresses the effective ion charge of the hy-drateddissociated solute relative to the charge of thewater ions involved in the hydration (which act as adielectricum reducing the electrical forces of the so-lute cation and anions) Thusξsj=Nplusmnνez

plusmn

sj with

zplusmn

sj=zν+e

s+j+zνminuse

sminusj the total charge of the ion-pair of

the j th-compound withNplusmn=kkplusmn

plusmn wherebykplusmn=2 andaccounts for the fact that 2 moles of water are consumedfor each mole of H3O+ produced (assuming electroneu-

trality for dissociation reactions) For instance for(semi-) volatile single charged compounds such ammo-nium nitrate (j=NH4NO3) where zplusmn

sj=1ν+e +1νminus

e =2we obtain with the thermodynamic data provided in Ta-ble 1 (atT =29815 [K]) ρw=0997 [gcm3] ρs=172[gcm3] νe=197 Ws=6805 [] andνw=1839 (de-rived from Ws by Eq 19) for ξsj=4times1972=394and for the mean molal binary activity coefficient atsaturation (RH=RHD=06067 according to Eq 21)γ plusmn

s(molal)NH4NO3=γ plusmn

sjtimesρwρsj=0239times0997172 =

01389 [kg(H2O)mol(solute)] For a comparisonHamer and Wu (1972) give a value of 0131 which isdiscussed in Mozurkewich (1993)

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics 3179

0

5

10

15

20

25

280701 040801 110801 180801 250801

tota

l P

M (

liquid

s amp

solid

s)

[ugm

^3]

Time [270720011300 - 25082001400]

MINOS Campaign Fine Aerosol Fraction (gasliquid partitioning) NaCl-NH3NH4-HNO3NO3-H2SO4SO4-H2O-System

EQSAM3ISORROPIA

SCAPE2EQSAM2

0

2

4

6

8

10

12

14

16

18

280701 040801 110801 180801 250801

tota

l P

M (

solid

) [u

gm

^3]

Time [270720011300 - 25082001400]

MINOS Campaign Fine Aerosol Fraction (gasliquidsolid partitioning) NaCl-NH3NH4-HNO3NO3-H2SO4SO4-H2O-System

EQSAM3ISORROPIA

SCAPE2EQSAM2

Figure 3 b)

0

1

2

3

4

5

280701 040801 110801 180801 250801

pH

[-]

Time [270720011300 - 25082001400]

MINOS Campaign Fine Aerosol Fraction (gasliquid partitioning) NaCl-NH3NH4-HNO3NO3-H2SO4SO4-H2O-System

EQSAM3ISORROPIA

SCAPE2EQSAM2

0

02

04

06

08

1

280701 040801 110801 180801 250801

NH

4N

O3 a

ctivity c

oeffic

ient [-

]

Time [270720011300 - 25082001400]

MINOS Campaign Fine Aerosol Fraction (gasliquid partitioning) NaCl-NH3NH4-HNO3NO3-H2SO4SO4-H2O-System

EQSAM3ISORROPIA

SCAPE2EQSAM2

Fig 3b Continued

12 The residual gases and acids (first row of Table 1)are computed from the remaining cations and anionswhereby (semi-)volatile acids are assumed to remain inthe gas phase if not neutralized or taken up directlyfrom the aqueous solution (which however yields onlysmall amounts relative to the total particulate matter)

13 Non-electrolyte solutes (last row of Table 1) are exceptammonia (NH3) not directly considered for the deter-mination of the reaction order nor are they assumed tobe involved in neutralization reactions However theycontribute to the aerosol mass and as long as they re-main hydrated also to the aerosol water mass

14 Various aerosol properties can be computed and storedfor diagnosis including aerosol properties that are bothdifficult to measure and to model such as the solution

pH= minus logl(aq)sumj=1

(ns+jmw) and the ionic strength of

binary and mixed solutionsZ=05times(Zs++Zsminus)mw

with Zs+=

l(aq)sumj=1

zν+e

s+j and Zsminus=

l(aq)sumj=1

zνminuse

sminusj the total

charge of cations and anions respectively Aerosolproperties such as mass and number of moles andthe associated water mass can be stored for each com-pound yielding listings similar to Table 3 Addition-ally the total particulate matter (PM) including solidsand ions can be expressed as both the total number of

moles PM=l(aq)sumj=1

ns(aq)j +

l(cr)sumj=1

ns(cr)j the total mass

PMt=l(aq)sumj=1

ns(aq)jMsj+

l(cr)sumj=1

ns(cr)jMsj respective the

total dry mass PMs=l(cr)sumj=1

ns(cr)jMsj whereby mass

and water fractions of all individual compounds are ex-plicitly summarized (for PMt and PMs using the indi-vidual molar massesMsj aerosol water according toEq (23))

wwwatmos-chem-physnet731632007 Atmos Chem Phys 7 3163ndash3193 2007

3180 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

0

10

20

30

40

50

60

70

80

280701 040801 110801 180801 250801

resid

ual H

NO

3 [nm

ol m

-3 (

air)]

Time [270720011300 - 25082001400]

MINOS Campaign Fine Aerosol Fraction (gasliquidsolid partitioning) NaCl-NH3NH4-HNO3NO3-H2SO4SO4-H2O-System

Meas HNO3EQSAM3

ISORROPIASCAPE2EQSAM2

0

50

100

150

200

250

300

350

400

450

500

280701 040801 110801 180801 250801

resid

ual N

H3 [nm

ol m

-3 (

air)]

Time [270720011300 - 25082001400]

MINOS Campaign Fine Aerosol Fraction (gasliquidsolid partitioning) NaCl-NH3NH4-HNO3NO3-H2SO4SO4-H2O-System

Meas NH3EQSAM3

ISORROPIASCAPE2EQSAM2

0

50

100

150

200

250

300

350

400

450

500

280701 040801 110801 180801 250801

NH

4+

[nm

ol m

-3 (

air)]

Time [270720011300 - 25082001400]

MINOS Campaign Fine Aerosol Fraction (gasliquidsolid partitioning) NaCl-NH3NH4-HNO3NO3-H2SO4SO4-H2O-System

Meas NH4+EQSAM3

ISORROPIASCAPE2EQSAM2

0

5

10

15

20

25

30

35

40

45

280701 040801 110801 180801 250801

NO

3-

[nm

ol m

-3 (

air)]

Time [270720011300 - 25082001400]

MINOS Campaign Fine Aerosol Fraction (gasliquidsolid partitioning) NaCl-NH3NH4-HNO3NO3-H2SO4SO4-H2O-System

Meas NO3-EQSAM3

ISORROPIASCAPE2EQSAM2

Figure 3 c)Fig 3c Continued

Note that the structure of EQSAM2 (Metzger et al 2006)has been adopted also in EQSAM3 however EQSAM2 wasnot based on solubilities Instead it used equilibrium con-stants and prescribed RHDs as listed in Metzger (2000)and activity coefficients for volatile compounds according toMetzger et al (2002) The underlying physical principles arenevertheless the same An example application of EQSAM2and EQSAM3 is given in the next section Both model ver-sions are available for the scientific community upon request

42 Comparison with measurements

We apply EQSAM3 to measurements obtained from MINOS(Mediterranean INtensive Oxidant Study) in Crete in the pe-riod 27 July to 25 August 2001 (Lelieveld et al 2002) by ex-tending the model-data comparison of Metzger et al (2006)For a general description of the measurements and the com-parison set-up we refer to that article To compare EQSAM3with other EQMs (EQSAM2 ISORROPIA SCAPE2) we

apply all models to mixed solutions at the same level ofcomplexity by which we focus on the chemical systemF2C2 as defined in Metzger et al (2006) ie the ammo-niumsulfatenitratechloridesodiumwater system

Figure 3 shows 4-weekly time series of various model cal-culated mixed solution properties observations are includedwhere available Figure 3a shows that the total fine andcoarse mode aerosol water mass is consistently predicted bythe EQMs assuming metastable aerosols (gasliquid parti-tioning) with EQSAM3 EQSAM2 and ISORROPIA Partic-ularly the results of EQSAM3 and ISORROPIA are ratherclose SCAPE2 deviates most significantly for the dry pe-riods because the assumption of metastable aerosols breaksdown Instead SCAPE2 calculates the full gasliquidsolidpartitioning These results (in particular the deviations) pro-vide an indication of the relative importance of deliquescencethresholds (RHDs) crucial for the liquid-solid partitioning

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics 3181

All model predictions of the total number of moles in thesolid and aqueous phase (particulate matter PM) are in goodagreement for both the fine and coarse mode EQSAM2shows relatively largest deviations for the fine mode Theresults of EQSAM3 appear to be closest to ISORROPIAwhich is coded to achieve the highest possible degree of nu-merical accuracy (though being much more CPU-time ex-pensive than EQSAM3)

Focusing further on the fine mode Fig 3b shows that allmodels consistently predict the total aerosol mass (total PM)and because all models account for the full gasliquidsolidpartitioning also the associated dry aerosol mass fractionNote that the total PM in terms of mass is more sensitiveto failures in predicting the aerosol composition than the to-tal PM in terms of moles since each individual molar massof the compounds (in the solid and liquid phase) is explic-itly accounted for (summarized) Again the relatively largestdeviations can be attributed to differences in the RHD calcu-lations except SCAPE2 all models account for the RHD ofmixed solutions whereby EQSAM2 uses a combination ofRHD values from both ISORROPIA and SCAPE2 ie mu-tual deliquescence RHDs of ISORROPIA when availableand RHDs of SCAPE2 for all mineral compounds (not con-sidered in ISORROPIA) EQSAM3 computes all RHDs (ofsingle solute or mixed solutions) from Eq (21) as describedabove Figure 3b furthermore shows that even the predic-tions of very sensitive aerosol properties such as the meanbinary activity coefficients (shown is the one of ammoniumnitrate in the mixed solution) and the solution pH are in gen-eral agreement

Figure 3c demonstrates that all EQMs predict the resid-ual gaseous ammonia and nitric acid and the correspondingaerosol ammonium and nitrate Especially the calculations ofthe lowest measured aerosol nitrate concentrations are mostaccurate with EQSAM3 being quite sensitive to the activ-ity coefficient of ammonium nitrate Note that the modelsdo not necessarily need to be in agreement with all obser-vations for ammoniaammonium The reason is that mineralcations and organic acids are omitted in the EQM compar-ison because ISORROPIA does not account for these com-pounds Metzger et al (2006) showed that the presence ofammonium in the aerosol phase is dependent on the presenceof organic acids (eg from biomass burning) in cases wherealkali-cations (eg mineral dust) are present in excess of inor-ganic acids In fact the consistent inclusion of alkali-cationsand organic acids is important for the gasaerosol partition-ing of reactive nitrogen compounds for both fine and coarsemode particles In contrast to the cation ammonium the an-ion nitrate is less affected in the fine mode than in the coarsemode so that the aerosol nitrate predictions of EQSAM3which are closest to the observations for this sensitive casealso give evidence for its applicability

Outlook

Figure 4

10

100

1000

10000

100000

041003 111003 181003 251003 011103 0

05

1

15

2

25

3

35

4

45

Vis

ibili

ty [

m]

Ra

in [

mm

]

Time [011020030000 - 311020032300]

MOHp October 2003 60 nm Aerosol Particles total PM as Ammonium Sulfate

Meas VisMeas Rain

EQSAM3

10

100

1000

10000

100000

041003 111003 181003 251003 011103 0

05

1

15

2

25

3

35

4

45

Vis

ibili

ty [

m]

Ra

in [

mm

]

Time [011020030000 - 311020032300]

MOHp October 2003 120 nm Aerosol Particles total PM as Ammonium Sulfate

Meas VisMeas Rain

EQSAM3

10

100

1000

10000

100000

041003 111003 181003 251003 011103 0

05

1

15

2

25

3

35

4

45

Vis

ibili

ty [

m]

Ra

in [

mm

]

Time [011020030000 - 311020032300]

MOHp October 2003 1 um Aerosol Particles total PM as Ammonium Sulfate

Meas VisMeas Rain

EQSAM3

10

100

1000

10000

100000

041003 111003 181003 251003 011103 0

05

1

15

2

25

3

35

4

45

Vis

ibili

ty [

m]

Ra

in [

mm

]

Time [011020030000 - 311020032300]

MOHp October 2003 120nm Aerosol Particles 50 of total PM as Ammonium Sulfate and LMW Organic Acids

Meas VisMeas Rain

EQSAM3

Fig 4 Outlook Visibility predictions with EQSAM3 (red) assum-ing 120 nm particles with 50 ammonium sulfate and low molec-ular weight (LMW) organic acids Visibility measurements of theMeteorological Observatory Hohenpeiszligenberg (MOHp) Germanyare shown for October 2003 in black (solid line) precipitation (righty-axes) in blue (dotted)

43 Application outlook

To indicate the potential of our theoretical considerations andimplications for atmospheric pollution and climate modelingwe present in this section a preview of some applications inprogress with EQSAM3 including visibility predictions andaerosol-cloud interactions For details we refer to future pub-lications

431 Visibility predictions

Under humid conditions hygroscopic aerosol particles cansubstantially reduce atmospheric visibility Figure 4 presentsvisibility predictions with EQSAM3 compared with obser-vations at the Meteorological Observatory Hohenpeiszligenberg(MOHp) Germany for October 2003 The ldquotranslationrdquo ofPM concentration and particle size into visibility based onaerosol optical parameters will be described elsewhere

Visibility calculations based on 120 nm sized particlescomposed of 50 ammonium sulfate (AS) and 50 of lowmolecular weight (LMW) organic acids (eg formic andacetic acid) are comparable to the observations while sameor smaller sized particles composed out of either only AS ororganics show poorer agreement in particular for high andlow visibility (not shown) Remarkably even in cases whereprecipitation was observed the model predicts low visibilityThis provides a first indication that there exists overlap be-tween conditions of high aerosol water and cloud formation

432 Global applications

Our preliminary global modeling applications focus on therole of the aerosol water mass being highly relevant for

wwwatmos-chem-physnet731632007 Atmos Chem Phys 7 3163ndash3193 2007

3182 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamicsPHOENICS

Aerosol mass (tropospheric burden) [ppb]

9th September 2000 1200 hr GMT

ECHAM5MESSy

Aerosol species (atmospheric burden) [ppb]9 September 2000 1200 hr GMT

E5M1-EQSAM3

Outlook

Figure 5 a)Fig 5 Outlook Global aerosol distributions for GMT noon 9 September 2000 (snap shot)(a) Atmospheric burden (fine mode) Aerosolwater mass (AW) in [microgm2] aerosol nitrate ammonium sulfate primary organics and sea salt (from top left to bottom right) AW(b)tropospheric burden(c) AW upper troposphere lower stratosphere (UTLS)(d) cloud coverage derived from AW(e)cloud cover comparisonfor the US ECHAM5 standard calculations (top left) AW based (regional selection of panel d) satellite observations (GEOS channel 4)The global calculations were obtained with the chemistry-climate model E5M1 (httpwwwmessy-interfaceorg)

climate forcing estimates Note that present climate mod-els that include aerosols do not explicitly calculate aerosolwater Here we aim to show that EQSAM3 can provide acomputationally efficient alternative that does not only accu-rately simulates the aerosol chemical composition but alsothe most abundant aerosol species water

We apply the general circulation model ECHAM5 (Roeck-ner et al 2003) at T63 (sim19 degree) resolution extendedwith the comprehensive Modular Earth Submodel System(MESSy) to account for atmospheric chemistry (Jockel et al2006) The model will be abbreviated in the following asE5M1 It has been additionally extended by the MESSy ver-sion of EQSAM (httpwwwmessy-interfaceorg) whichaccounts for 7 aerosol modes including four soluble andthree insoluble nucleation aitken accumulation and coarsewhereby the latter three modes are used to distinguish be-tween primary insoluble and soluble aerosol species such asblack carbon or certain mineral dust compounds (similar asin Vignati et al 2004) Details will be provided in a follow-up publication

Various aerosol species

Figure 5a presents a ldquosnapshotrdquo of the model results for9 September 2000 (1200 GMT) by showing the spatialdistribution (vertical integral) of various fine mode aerosolspecies including aerosol water mass [microgm2] aerosol ni-trate ammonium sulfate organic carbon and sea salt [ppbv](from top left to bottom right) While certain aerosol speciessuch as nitrates and organics are largely confined to conti-nental areas associated with the localized emission sourcesother species such as ammonium and sulfate show muchwider dispersion Fine mode ammonium is highly corre-lated with sulfate and can be transported over long distancesNitrate compounds are volatile and they are also found oncoarse mode particles (not shown) such as sea salt and min-eral dust which are more efficiently deposited

Figure 5a shows that the spatial distribution of aerosol wa-ter (RHlt095) strongly correlates with that of sea salt sul-fate and ammonium and that at this arbitrary RH limit theaerosol water mass already dominates the total aerosol loadFigure 5b shows the tropospheric aerosol water mass whileFig 5c presents the corresponding aerosol water in the up-per troposphere - lower stratosphere (UTLS) region These

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics 3179

0

5

10

15

20

25

280701 040801 110801 180801 250801

tota

l P

M (

liquid

s amp

solid

s)

[ugm

^3]

Time [270720011300 - 25082001400]

MINOS Campaign Fine Aerosol Fraction (gasliquid partitioning) NaCl-NH3NH4-HNO3NO3-H2SO4SO4-H2O-System

EQSAM3ISORROPIA

SCAPE2EQSAM2

0

2

4

6

8

10

12

14

16

18

280701 040801 110801 180801 250801

tota

l P

M (

solid

) [u

gm

^3]

Time [270720011300 - 25082001400]

MINOS Campaign Fine Aerosol Fraction (gasliquidsolid partitioning) NaCl-NH3NH4-HNO3NO3-H2SO4SO4-H2O-System

EQSAM3ISORROPIA

SCAPE2EQSAM2

Figure 3 b)

0

1

2

3

4

5

280701 040801 110801 180801 250801

pH

[-]

Time [270720011300 - 25082001400]

MINOS Campaign Fine Aerosol Fraction (gasliquid partitioning) NaCl-NH3NH4-HNO3NO3-H2SO4SO4-H2O-System

EQSAM3ISORROPIA

SCAPE2EQSAM2

0

02

04

06

08

1

280701 040801 110801 180801 250801

NH

4N

O3 a

ctivity c

oeffic

ient [-

]

Time [270720011300 - 25082001400]

MINOS Campaign Fine Aerosol Fraction (gasliquid partitioning) NaCl-NH3NH4-HNO3NO3-H2SO4SO4-H2O-System

EQSAM3ISORROPIA

SCAPE2EQSAM2

Fig 3b Continued

12 The residual gases and acids (first row of Table 1)are computed from the remaining cations and anionswhereby (semi-)volatile acids are assumed to remain inthe gas phase if not neutralized or taken up directlyfrom the aqueous solution (which however yields onlysmall amounts relative to the total particulate matter)

13 Non-electrolyte solutes (last row of Table 1) are exceptammonia (NH3) not directly considered for the deter-mination of the reaction order nor are they assumed tobe involved in neutralization reactions However theycontribute to the aerosol mass and as long as they re-main hydrated also to the aerosol water mass

14 Various aerosol properties can be computed and storedfor diagnosis including aerosol properties that are bothdifficult to measure and to model such as the solution

pH= minus logl(aq)sumj=1

(ns+jmw) and the ionic strength of

binary and mixed solutionsZ=05times(Zs++Zsminus)mw

with Zs+=

l(aq)sumj=1

zν+e

s+j and Zsminus=

l(aq)sumj=1

zνminuse

sminusj the total

charge of cations and anions respectively Aerosolproperties such as mass and number of moles andthe associated water mass can be stored for each com-pound yielding listings similar to Table 3 Addition-ally the total particulate matter (PM) including solidsand ions can be expressed as both the total number of

moles PM=l(aq)sumj=1

ns(aq)j +

l(cr)sumj=1

ns(cr)j the total mass

PMt=l(aq)sumj=1

ns(aq)jMsj+

l(cr)sumj=1

ns(cr)jMsj respective the

total dry mass PMs=l(cr)sumj=1

ns(cr)jMsj whereby mass

and water fractions of all individual compounds are ex-plicitly summarized (for PMt and PMs using the indi-vidual molar massesMsj aerosol water according toEq (23))

wwwatmos-chem-physnet731632007 Atmos Chem Phys 7 3163ndash3193 2007

3180 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

0

10

20

30

40

50

60

70

80

280701 040801 110801 180801 250801

resid

ual H

NO

3 [nm

ol m

-3 (

air)]

Time [270720011300 - 25082001400]

MINOS Campaign Fine Aerosol Fraction (gasliquidsolid partitioning) NaCl-NH3NH4-HNO3NO3-H2SO4SO4-H2O-System

Meas HNO3EQSAM3

ISORROPIASCAPE2EQSAM2

0

50

100

150

200

250

300

350

400

450

500

280701 040801 110801 180801 250801

resid

ual N

H3 [nm

ol m

-3 (

air)]

Time [270720011300 - 25082001400]

MINOS Campaign Fine Aerosol Fraction (gasliquidsolid partitioning) NaCl-NH3NH4-HNO3NO3-H2SO4SO4-H2O-System

Meas NH3EQSAM3

ISORROPIASCAPE2EQSAM2

0

50

100

150

200

250

300

350

400

450

500

280701 040801 110801 180801 250801

NH

4+

[nm

ol m

-3 (

air)]

Time [270720011300 - 25082001400]

MINOS Campaign Fine Aerosol Fraction (gasliquidsolid partitioning) NaCl-NH3NH4-HNO3NO3-H2SO4SO4-H2O-System

Meas NH4+EQSAM3

ISORROPIASCAPE2EQSAM2

0

5

10

15

20

25

30

35

40

45

280701 040801 110801 180801 250801

NO

3-

[nm

ol m

-3 (

air)]

Time [270720011300 - 25082001400]

MINOS Campaign Fine Aerosol Fraction (gasliquidsolid partitioning) NaCl-NH3NH4-HNO3NO3-H2SO4SO4-H2O-System

Meas NO3-EQSAM3

ISORROPIASCAPE2EQSAM2

Figure 3 c)Fig 3c Continued

Note that the structure of EQSAM2 (Metzger et al 2006)has been adopted also in EQSAM3 however EQSAM2 wasnot based on solubilities Instead it used equilibrium con-stants and prescribed RHDs as listed in Metzger (2000)and activity coefficients for volatile compounds according toMetzger et al (2002) The underlying physical principles arenevertheless the same An example application of EQSAM2and EQSAM3 is given in the next section Both model ver-sions are available for the scientific community upon request

42 Comparison with measurements

We apply EQSAM3 to measurements obtained from MINOS(Mediterranean INtensive Oxidant Study) in Crete in the pe-riod 27 July to 25 August 2001 (Lelieveld et al 2002) by ex-tending the model-data comparison of Metzger et al (2006)For a general description of the measurements and the com-parison set-up we refer to that article To compare EQSAM3with other EQMs (EQSAM2 ISORROPIA SCAPE2) we

apply all models to mixed solutions at the same level ofcomplexity by which we focus on the chemical systemF2C2 as defined in Metzger et al (2006) ie the ammo-niumsulfatenitratechloridesodiumwater system

Figure 3 shows 4-weekly time series of various model cal-culated mixed solution properties observations are includedwhere available Figure 3a shows that the total fine andcoarse mode aerosol water mass is consistently predicted bythe EQMs assuming metastable aerosols (gasliquid parti-tioning) with EQSAM3 EQSAM2 and ISORROPIA Partic-ularly the results of EQSAM3 and ISORROPIA are ratherclose SCAPE2 deviates most significantly for the dry pe-riods because the assumption of metastable aerosols breaksdown Instead SCAPE2 calculates the full gasliquidsolidpartitioning These results (in particular the deviations) pro-vide an indication of the relative importance of deliquescencethresholds (RHDs) crucial for the liquid-solid partitioning

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics 3181

All model predictions of the total number of moles in thesolid and aqueous phase (particulate matter PM) are in goodagreement for both the fine and coarse mode EQSAM2shows relatively largest deviations for the fine mode Theresults of EQSAM3 appear to be closest to ISORROPIAwhich is coded to achieve the highest possible degree of nu-merical accuracy (though being much more CPU-time ex-pensive than EQSAM3)

Focusing further on the fine mode Fig 3b shows that allmodels consistently predict the total aerosol mass (total PM)and because all models account for the full gasliquidsolidpartitioning also the associated dry aerosol mass fractionNote that the total PM in terms of mass is more sensitiveto failures in predicting the aerosol composition than the to-tal PM in terms of moles since each individual molar massof the compounds (in the solid and liquid phase) is explic-itly accounted for (summarized) Again the relatively largestdeviations can be attributed to differences in the RHD calcu-lations except SCAPE2 all models account for the RHD ofmixed solutions whereby EQSAM2 uses a combination ofRHD values from both ISORROPIA and SCAPE2 ie mu-tual deliquescence RHDs of ISORROPIA when availableand RHDs of SCAPE2 for all mineral compounds (not con-sidered in ISORROPIA) EQSAM3 computes all RHDs (ofsingle solute or mixed solutions) from Eq (21) as describedabove Figure 3b furthermore shows that even the predic-tions of very sensitive aerosol properties such as the meanbinary activity coefficients (shown is the one of ammoniumnitrate in the mixed solution) and the solution pH are in gen-eral agreement

Figure 3c demonstrates that all EQMs predict the resid-ual gaseous ammonia and nitric acid and the correspondingaerosol ammonium and nitrate Especially the calculations ofthe lowest measured aerosol nitrate concentrations are mostaccurate with EQSAM3 being quite sensitive to the activ-ity coefficient of ammonium nitrate Note that the modelsdo not necessarily need to be in agreement with all obser-vations for ammoniaammonium The reason is that mineralcations and organic acids are omitted in the EQM compar-ison because ISORROPIA does not account for these com-pounds Metzger et al (2006) showed that the presence ofammonium in the aerosol phase is dependent on the presenceof organic acids (eg from biomass burning) in cases wherealkali-cations (eg mineral dust) are present in excess of inor-ganic acids In fact the consistent inclusion of alkali-cationsand organic acids is important for the gasaerosol partition-ing of reactive nitrogen compounds for both fine and coarsemode particles In contrast to the cation ammonium the an-ion nitrate is less affected in the fine mode than in the coarsemode so that the aerosol nitrate predictions of EQSAM3which are closest to the observations for this sensitive casealso give evidence for its applicability

Outlook

Figure 4

10

100

1000

10000

100000

041003 111003 181003 251003 011103 0

05

1

15

2

25

3

35

4

45

Vis

ibili

ty [

m]

Ra

in [

mm

]

Time [011020030000 - 311020032300]

MOHp October 2003 60 nm Aerosol Particles total PM as Ammonium Sulfate

Meas VisMeas Rain

EQSAM3

10

100

1000

10000

100000

041003 111003 181003 251003 011103 0

05

1

15

2

25

3

35

4

45

Vis

ibili

ty [

m]

Ra

in [

mm

]

Time [011020030000 - 311020032300]

MOHp October 2003 120 nm Aerosol Particles total PM as Ammonium Sulfate

Meas VisMeas Rain

EQSAM3

10

100

1000

10000

100000

041003 111003 181003 251003 011103 0

05

1

15

2

25

3

35

4

45

Vis

ibili

ty [

m]

Ra

in [

mm

]

Time [011020030000 - 311020032300]

MOHp October 2003 1 um Aerosol Particles total PM as Ammonium Sulfate

Meas VisMeas Rain

EQSAM3

10

100

1000

10000

100000

041003 111003 181003 251003 011103 0

05

1

15

2

25

3

35

4

45

Vis

ibili

ty [

m]

Ra

in [

mm

]

Time [011020030000 - 311020032300]

MOHp October 2003 120nm Aerosol Particles 50 of total PM as Ammonium Sulfate and LMW Organic Acids

Meas VisMeas Rain

EQSAM3

Fig 4 Outlook Visibility predictions with EQSAM3 (red) assum-ing 120 nm particles with 50 ammonium sulfate and low molec-ular weight (LMW) organic acids Visibility measurements of theMeteorological Observatory Hohenpeiszligenberg (MOHp) Germanyare shown for October 2003 in black (solid line) precipitation (righty-axes) in blue (dotted)

43 Application outlook

To indicate the potential of our theoretical considerations andimplications for atmospheric pollution and climate modelingwe present in this section a preview of some applications inprogress with EQSAM3 including visibility predictions andaerosol-cloud interactions For details we refer to future pub-lications

431 Visibility predictions

Under humid conditions hygroscopic aerosol particles cansubstantially reduce atmospheric visibility Figure 4 presentsvisibility predictions with EQSAM3 compared with obser-vations at the Meteorological Observatory Hohenpeiszligenberg(MOHp) Germany for October 2003 The ldquotranslationrdquo ofPM concentration and particle size into visibility based onaerosol optical parameters will be described elsewhere

Visibility calculations based on 120 nm sized particlescomposed of 50 ammonium sulfate (AS) and 50 of lowmolecular weight (LMW) organic acids (eg formic andacetic acid) are comparable to the observations while sameor smaller sized particles composed out of either only AS ororganics show poorer agreement in particular for high andlow visibility (not shown) Remarkably even in cases whereprecipitation was observed the model predicts low visibilityThis provides a first indication that there exists overlap be-tween conditions of high aerosol water and cloud formation

432 Global applications

Our preliminary global modeling applications focus on therole of the aerosol water mass being highly relevant for

wwwatmos-chem-physnet731632007 Atmos Chem Phys 7 3163ndash3193 2007

3182 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamicsPHOENICS

Aerosol mass (tropospheric burden) [ppb]

9th September 2000 1200 hr GMT

ECHAM5MESSy

Aerosol species (atmospheric burden) [ppb]9 September 2000 1200 hr GMT

E5M1-EQSAM3

Outlook

Figure 5 a)Fig 5 Outlook Global aerosol distributions for GMT noon 9 September 2000 (snap shot)(a) Atmospheric burden (fine mode) Aerosolwater mass (AW) in [microgm2] aerosol nitrate ammonium sulfate primary organics and sea salt (from top left to bottom right) AW(b)tropospheric burden(c) AW upper troposphere lower stratosphere (UTLS)(d) cloud coverage derived from AW(e)cloud cover comparisonfor the US ECHAM5 standard calculations (top left) AW based (regional selection of panel d) satellite observations (GEOS channel 4)The global calculations were obtained with the chemistry-climate model E5M1 (httpwwwmessy-interfaceorg)

climate forcing estimates Note that present climate mod-els that include aerosols do not explicitly calculate aerosolwater Here we aim to show that EQSAM3 can provide acomputationally efficient alternative that does not only accu-rately simulates the aerosol chemical composition but alsothe most abundant aerosol species water

We apply the general circulation model ECHAM5 (Roeck-ner et al 2003) at T63 (sim19 degree) resolution extendedwith the comprehensive Modular Earth Submodel System(MESSy) to account for atmospheric chemistry (Jockel et al2006) The model will be abbreviated in the following asE5M1 It has been additionally extended by the MESSy ver-sion of EQSAM (httpwwwmessy-interfaceorg) whichaccounts for 7 aerosol modes including four soluble andthree insoluble nucleation aitken accumulation and coarsewhereby the latter three modes are used to distinguish be-tween primary insoluble and soluble aerosol species such asblack carbon or certain mineral dust compounds (similar asin Vignati et al 2004) Details will be provided in a follow-up publication

Various aerosol species

Figure 5a presents a ldquosnapshotrdquo of the model results for9 September 2000 (1200 GMT) by showing the spatialdistribution (vertical integral) of various fine mode aerosolspecies including aerosol water mass [microgm2] aerosol ni-trate ammonium sulfate organic carbon and sea salt [ppbv](from top left to bottom right) While certain aerosol speciessuch as nitrates and organics are largely confined to conti-nental areas associated with the localized emission sourcesother species such as ammonium and sulfate show muchwider dispersion Fine mode ammonium is highly corre-lated with sulfate and can be transported over long distancesNitrate compounds are volatile and they are also found oncoarse mode particles (not shown) such as sea salt and min-eral dust which are more efficiently deposited

Figure 5a shows that the spatial distribution of aerosol wa-ter (RHlt095) strongly correlates with that of sea salt sul-fate and ammonium and that at this arbitrary RH limit theaerosol water mass already dominates the total aerosol loadFigure 5b shows the tropospheric aerosol water mass whileFig 5c presents the corresponding aerosol water in the up-per troposphere - lower stratosphere (UTLS) region These

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

3180 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

0

10

20

30

40

50

60

70

80

280701 040801 110801 180801 250801

resid

ual H

NO

3 [nm

ol m

-3 (

air)]

Time [270720011300 - 25082001400]

MINOS Campaign Fine Aerosol Fraction (gasliquidsolid partitioning) NaCl-NH3NH4-HNO3NO3-H2SO4SO4-H2O-System

Meas HNO3EQSAM3

ISORROPIASCAPE2EQSAM2

0

50

100

150

200

250

300

350

400

450

500

280701 040801 110801 180801 250801

resid

ual N

H3 [nm

ol m

-3 (

air)]

Time [270720011300 - 25082001400]

MINOS Campaign Fine Aerosol Fraction (gasliquidsolid partitioning) NaCl-NH3NH4-HNO3NO3-H2SO4SO4-H2O-System

Meas NH3EQSAM3

ISORROPIASCAPE2EQSAM2

0

50

100

150

200

250

300

350

400

450

500

280701 040801 110801 180801 250801

NH

4+

[nm

ol m

-3 (

air)]

Time [270720011300 - 25082001400]

MINOS Campaign Fine Aerosol Fraction (gasliquidsolid partitioning) NaCl-NH3NH4-HNO3NO3-H2SO4SO4-H2O-System

Meas NH4+EQSAM3

ISORROPIASCAPE2EQSAM2

0

5

10

15

20

25

30

35

40

45

280701 040801 110801 180801 250801

NO

3-

[nm

ol m

-3 (

air)]

Time [270720011300 - 25082001400]

MINOS Campaign Fine Aerosol Fraction (gasliquidsolid partitioning) NaCl-NH3NH4-HNO3NO3-H2SO4SO4-H2O-System

Meas NO3-EQSAM3

ISORROPIASCAPE2EQSAM2

Figure 3 c)Fig 3c Continued

Note that the structure of EQSAM2 (Metzger et al 2006)has been adopted also in EQSAM3 however EQSAM2 wasnot based on solubilities Instead it used equilibrium con-stants and prescribed RHDs as listed in Metzger (2000)and activity coefficients for volatile compounds according toMetzger et al (2002) The underlying physical principles arenevertheless the same An example application of EQSAM2and EQSAM3 is given in the next section Both model ver-sions are available for the scientific community upon request

42 Comparison with measurements

We apply EQSAM3 to measurements obtained from MINOS(Mediterranean INtensive Oxidant Study) in Crete in the pe-riod 27 July to 25 August 2001 (Lelieveld et al 2002) by ex-tending the model-data comparison of Metzger et al (2006)For a general description of the measurements and the com-parison set-up we refer to that article To compare EQSAM3with other EQMs (EQSAM2 ISORROPIA SCAPE2) we

apply all models to mixed solutions at the same level ofcomplexity by which we focus on the chemical systemF2C2 as defined in Metzger et al (2006) ie the ammo-niumsulfatenitratechloridesodiumwater system

Figure 3 shows 4-weekly time series of various model cal-culated mixed solution properties observations are includedwhere available Figure 3a shows that the total fine andcoarse mode aerosol water mass is consistently predicted bythe EQMs assuming metastable aerosols (gasliquid parti-tioning) with EQSAM3 EQSAM2 and ISORROPIA Partic-ularly the results of EQSAM3 and ISORROPIA are ratherclose SCAPE2 deviates most significantly for the dry pe-riods because the assumption of metastable aerosols breaksdown Instead SCAPE2 calculates the full gasliquidsolidpartitioning These results (in particular the deviations) pro-vide an indication of the relative importance of deliquescencethresholds (RHDs) crucial for the liquid-solid partitioning

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics 3181

All model predictions of the total number of moles in thesolid and aqueous phase (particulate matter PM) are in goodagreement for both the fine and coarse mode EQSAM2shows relatively largest deviations for the fine mode Theresults of EQSAM3 appear to be closest to ISORROPIAwhich is coded to achieve the highest possible degree of nu-merical accuracy (though being much more CPU-time ex-pensive than EQSAM3)

Focusing further on the fine mode Fig 3b shows that allmodels consistently predict the total aerosol mass (total PM)and because all models account for the full gasliquidsolidpartitioning also the associated dry aerosol mass fractionNote that the total PM in terms of mass is more sensitiveto failures in predicting the aerosol composition than the to-tal PM in terms of moles since each individual molar massof the compounds (in the solid and liquid phase) is explic-itly accounted for (summarized) Again the relatively largestdeviations can be attributed to differences in the RHD calcu-lations except SCAPE2 all models account for the RHD ofmixed solutions whereby EQSAM2 uses a combination ofRHD values from both ISORROPIA and SCAPE2 ie mu-tual deliquescence RHDs of ISORROPIA when availableand RHDs of SCAPE2 for all mineral compounds (not con-sidered in ISORROPIA) EQSAM3 computes all RHDs (ofsingle solute or mixed solutions) from Eq (21) as describedabove Figure 3b furthermore shows that even the predic-tions of very sensitive aerosol properties such as the meanbinary activity coefficients (shown is the one of ammoniumnitrate in the mixed solution) and the solution pH are in gen-eral agreement

Figure 3c demonstrates that all EQMs predict the resid-ual gaseous ammonia and nitric acid and the correspondingaerosol ammonium and nitrate Especially the calculations ofthe lowest measured aerosol nitrate concentrations are mostaccurate with EQSAM3 being quite sensitive to the activ-ity coefficient of ammonium nitrate Note that the modelsdo not necessarily need to be in agreement with all obser-vations for ammoniaammonium The reason is that mineralcations and organic acids are omitted in the EQM compar-ison because ISORROPIA does not account for these com-pounds Metzger et al (2006) showed that the presence ofammonium in the aerosol phase is dependent on the presenceof organic acids (eg from biomass burning) in cases wherealkali-cations (eg mineral dust) are present in excess of inor-ganic acids In fact the consistent inclusion of alkali-cationsand organic acids is important for the gasaerosol partition-ing of reactive nitrogen compounds for both fine and coarsemode particles In contrast to the cation ammonium the an-ion nitrate is less affected in the fine mode than in the coarsemode so that the aerosol nitrate predictions of EQSAM3which are closest to the observations for this sensitive casealso give evidence for its applicability

Outlook

Figure 4

10

100

1000

10000

100000

041003 111003 181003 251003 011103 0

05

1

15

2

25

3

35

4

45

Vis

ibili

ty [

m]

Ra

in [

mm

]

Time [011020030000 - 311020032300]

MOHp October 2003 60 nm Aerosol Particles total PM as Ammonium Sulfate

Meas VisMeas Rain

EQSAM3

10

100

1000

10000

100000

041003 111003 181003 251003 011103 0

05

1

15

2

25

3

35

4

45

Vis

ibili

ty [

m]

Ra

in [

mm

]

Time [011020030000 - 311020032300]

MOHp October 2003 120 nm Aerosol Particles total PM as Ammonium Sulfate

Meas VisMeas Rain

EQSAM3

10

100

1000

10000

100000

041003 111003 181003 251003 011103 0

05

1

15

2

25

3

35

4

45

Vis

ibili

ty [

m]

Ra

in [

mm

]

Time [011020030000 - 311020032300]

MOHp October 2003 1 um Aerosol Particles total PM as Ammonium Sulfate

Meas VisMeas Rain

EQSAM3

10

100

1000

10000

100000

041003 111003 181003 251003 011103 0

05

1

15

2

25

3

35

4

45

Vis

ibili

ty [

m]

Ra

in [

mm

]

Time [011020030000 - 311020032300]

MOHp October 2003 120nm Aerosol Particles 50 of total PM as Ammonium Sulfate and LMW Organic Acids

Meas VisMeas Rain

EQSAM3

Fig 4 Outlook Visibility predictions with EQSAM3 (red) assum-ing 120 nm particles with 50 ammonium sulfate and low molec-ular weight (LMW) organic acids Visibility measurements of theMeteorological Observatory Hohenpeiszligenberg (MOHp) Germanyare shown for October 2003 in black (solid line) precipitation (righty-axes) in blue (dotted)

43 Application outlook

To indicate the potential of our theoretical considerations andimplications for atmospheric pollution and climate modelingwe present in this section a preview of some applications inprogress with EQSAM3 including visibility predictions andaerosol-cloud interactions For details we refer to future pub-lications

431 Visibility predictions

Under humid conditions hygroscopic aerosol particles cansubstantially reduce atmospheric visibility Figure 4 presentsvisibility predictions with EQSAM3 compared with obser-vations at the Meteorological Observatory Hohenpeiszligenberg(MOHp) Germany for October 2003 The ldquotranslationrdquo ofPM concentration and particle size into visibility based onaerosol optical parameters will be described elsewhere

Visibility calculations based on 120 nm sized particlescomposed of 50 ammonium sulfate (AS) and 50 of lowmolecular weight (LMW) organic acids (eg formic andacetic acid) are comparable to the observations while sameor smaller sized particles composed out of either only AS ororganics show poorer agreement in particular for high andlow visibility (not shown) Remarkably even in cases whereprecipitation was observed the model predicts low visibilityThis provides a first indication that there exists overlap be-tween conditions of high aerosol water and cloud formation

432 Global applications

Our preliminary global modeling applications focus on therole of the aerosol water mass being highly relevant for

wwwatmos-chem-physnet731632007 Atmos Chem Phys 7 3163ndash3193 2007

3182 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamicsPHOENICS

Aerosol mass (tropospheric burden) [ppb]

9th September 2000 1200 hr GMT

ECHAM5MESSy

Aerosol species (atmospheric burden) [ppb]9 September 2000 1200 hr GMT

E5M1-EQSAM3

Outlook

Figure 5 a)Fig 5 Outlook Global aerosol distributions for GMT noon 9 September 2000 (snap shot)(a) Atmospheric burden (fine mode) Aerosolwater mass (AW) in [microgm2] aerosol nitrate ammonium sulfate primary organics and sea salt (from top left to bottom right) AW(b)tropospheric burden(c) AW upper troposphere lower stratosphere (UTLS)(d) cloud coverage derived from AW(e)cloud cover comparisonfor the US ECHAM5 standard calculations (top left) AW based (regional selection of panel d) satellite observations (GEOS channel 4)The global calculations were obtained with the chemistry-climate model E5M1 (httpwwwmessy-interfaceorg)

climate forcing estimates Note that present climate mod-els that include aerosols do not explicitly calculate aerosolwater Here we aim to show that EQSAM3 can provide acomputationally efficient alternative that does not only accu-rately simulates the aerosol chemical composition but alsothe most abundant aerosol species water

We apply the general circulation model ECHAM5 (Roeck-ner et al 2003) at T63 (sim19 degree) resolution extendedwith the comprehensive Modular Earth Submodel System(MESSy) to account for atmospheric chemistry (Jockel et al2006) The model will be abbreviated in the following asE5M1 It has been additionally extended by the MESSy ver-sion of EQSAM (httpwwwmessy-interfaceorg) whichaccounts for 7 aerosol modes including four soluble andthree insoluble nucleation aitken accumulation and coarsewhereby the latter three modes are used to distinguish be-tween primary insoluble and soluble aerosol species such asblack carbon or certain mineral dust compounds (similar asin Vignati et al 2004) Details will be provided in a follow-up publication

Various aerosol species

Figure 5a presents a ldquosnapshotrdquo of the model results for9 September 2000 (1200 GMT) by showing the spatialdistribution (vertical integral) of various fine mode aerosolspecies including aerosol water mass [microgm2] aerosol ni-trate ammonium sulfate organic carbon and sea salt [ppbv](from top left to bottom right) While certain aerosol speciessuch as nitrates and organics are largely confined to conti-nental areas associated with the localized emission sourcesother species such as ammonium and sulfate show muchwider dispersion Fine mode ammonium is highly corre-lated with sulfate and can be transported over long distancesNitrate compounds are volatile and they are also found oncoarse mode particles (not shown) such as sea salt and min-eral dust which are more efficiently deposited

Figure 5a shows that the spatial distribution of aerosol wa-ter (RHlt095) strongly correlates with that of sea salt sul-fate and ammonium and that at this arbitrary RH limit theaerosol water mass already dominates the total aerosol loadFigure 5b shows the tropospheric aerosol water mass whileFig 5c presents the corresponding aerosol water in the up-per troposphere - lower stratosphere (UTLS) region These

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics 3181

All model predictions of the total number of moles in thesolid and aqueous phase (particulate matter PM) are in goodagreement for both the fine and coarse mode EQSAM2shows relatively largest deviations for the fine mode Theresults of EQSAM3 appear to be closest to ISORROPIAwhich is coded to achieve the highest possible degree of nu-merical accuracy (though being much more CPU-time ex-pensive than EQSAM3)

Focusing further on the fine mode Fig 3b shows that allmodels consistently predict the total aerosol mass (total PM)and because all models account for the full gasliquidsolidpartitioning also the associated dry aerosol mass fractionNote that the total PM in terms of mass is more sensitiveto failures in predicting the aerosol composition than the to-tal PM in terms of moles since each individual molar massof the compounds (in the solid and liquid phase) is explic-itly accounted for (summarized) Again the relatively largestdeviations can be attributed to differences in the RHD calcu-lations except SCAPE2 all models account for the RHD ofmixed solutions whereby EQSAM2 uses a combination ofRHD values from both ISORROPIA and SCAPE2 ie mu-tual deliquescence RHDs of ISORROPIA when availableand RHDs of SCAPE2 for all mineral compounds (not con-sidered in ISORROPIA) EQSAM3 computes all RHDs (ofsingle solute or mixed solutions) from Eq (21) as describedabove Figure 3b furthermore shows that even the predic-tions of very sensitive aerosol properties such as the meanbinary activity coefficients (shown is the one of ammoniumnitrate in the mixed solution) and the solution pH are in gen-eral agreement

Figure 3c demonstrates that all EQMs predict the resid-ual gaseous ammonia and nitric acid and the correspondingaerosol ammonium and nitrate Especially the calculations ofthe lowest measured aerosol nitrate concentrations are mostaccurate with EQSAM3 being quite sensitive to the activ-ity coefficient of ammonium nitrate Note that the modelsdo not necessarily need to be in agreement with all obser-vations for ammoniaammonium The reason is that mineralcations and organic acids are omitted in the EQM compar-ison because ISORROPIA does not account for these com-pounds Metzger et al (2006) showed that the presence ofammonium in the aerosol phase is dependent on the presenceof organic acids (eg from biomass burning) in cases wherealkali-cations (eg mineral dust) are present in excess of inor-ganic acids In fact the consistent inclusion of alkali-cationsand organic acids is important for the gasaerosol partition-ing of reactive nitrogen compounds for both fine and coarsemode particles In contrast to the cation ammonium the an-ion nitrate is less affected in the fine mode than in the coarsemode so that the aerosol nitrate predictions of EQSAM3which are closest to the observations for this sensitive casealso give evidence for its applicability

Outlook

Figure 4

10

100

1000

10000

100000

041003 111003 181003 251003 011103 0

05

1

15

2

25

3

35

4

45

Vis

ibili

ty [

m]

Ra

in [

mm

]

Time [011020030000 - 311020032300]

MOHp October 2003 60 nm Aerosol Particles total PM as Ammonium Sulfate

Meas VisMeas Rain

EQSAM3

10

100

1000

10000

100000

041003 111003 181003 251003 011103 0

05

1

15

2

25

3

35

4

45

Vis

ibili

ty [

m]

Ra

in [

mm

]

Time [011020030000 - 311020032300]

MOHp October 2003 120 nm Aerosol Particles total PM as Ammonium Sulfate

Meas VisMeas Rain

EQSAM3

10

100

1000

10000

100000

041003 111003 181003 251003 011103 0

05

1

15

2

25

3

35

4

45

Vis

ibili

ty [

m]

Ra

in [

mm

]

Time [011020030000 - 311020032300]

MOHp October 2003 1 um Aerosol Particles total PM as Ammonium Sulfate

Meas VisMeas Rain

EQSAM3

10

100

1000

10000

100000

041003 111003 181003 251003 011103 0

05

1

15

2

25

3

35

4

45

Vis

ibili

ty [

m]

Ra

in [

mm

]

Time [011020030000 - 311020032300]

MOHp October 2003 120nm Aerosol Particles 50 of total PM as Ammonium Sulfate and LMW Organic Acids

Meas VisMeas Rain

EQSAM3

Fig 4 Outlook Visibility predictions with EQSAM3 (red) assum-ing 120 nm particles with 50 ammonium sulfate and low molec-ular weight (LMW) organic acids Visibility measurements of theMeteorological Observatory Hohenpeiszligenberg (MOHp) Germanyare shown for October 2003 in black (solid line) precipitation (righty-axes) in blue (dotted)

43 Application outlook

To indicate the potential of our theoretical considerations andimplications for atmospheric pollution and climate modelingwe present in this section a preview of some applications inprogress with EQSAM3 including visibility predictions andaerosol-cloud interactions For details we refer to future pub-lications

431 Visibility predictions

Under humid conditions hygroscopic aerosol particles cansubstantially reduce atmospheric visibility Figure 4 presentsvisibility predictions with EQSAM3 compared with obser-vations at the Meteorological Observatory Hohenpeiszligenberg(MOHp) Germany for October 2003 The ldquotranslationrdquo ofPM concentration and particle size into visibility based onaerosol optical parameters will be described elsewhere

Visibility calculations based on 120 nm sized particlescomposed of 50 ammonium sulfate (AS) and 50 of lowmolecular weight (LMW) organic acids (eg formic andacetic acid) are comparable to the observations while sameor smaller sized particles composed out of either only AS ororganics show poorer agreement in particular for high andlow visibility (not shown) Remarkably even in cases whereprecipitation was observed the model predicts low visibilityThis provides a first indication that there exists overlap be-tween conditions of high aerosol water and cloud formation

432 Global applications

Our preliminary global modeling applications focus on therole of the aerosol water mass being highly relevant for

wwwatmos-chem-physnet731632007 Atmos Chem Phys 7 3163ndash3193 2007

3182 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamicsPHOENICS

Aerosol mass (tropospheric burden) [ppb]

9th September 2000 1200 hr GMT

ECHAM5MESSy

Aerosol species (atmospheric burden) [ppb]9 September 2000 1200 hr GMT

E5M1-EQSAM3

Outlook

Figure 5 a)Fig 5 Outlook Global aerosol distributions for GMT noon 9 September 2000 (snap shot)(a) Atmospheric burden (fine mode) Aerosolwater mass (AW) in [microgm2] aerosol nitrate ammonium sulfate primary organics and sea salt (from top left to bottom right) AW(b)tropospheric burden(c) AW upper troposphere lower stratosphere (UTLS)(d) cloud coverage derived from AW(e)cloud cover comparisonfor the US ECHAM5 standard calculations (top left) AW based (regional selection of panel d) satellite observations (GEOS channel 4)The global calculations were obtained with the chemistry-climate model E5M1 (httpwwwmessy-interfaceorg)

climate forcing estimates Note that present climate mod-els that include aerosols do not explicitly calculate aerosolwater Here we aim to show that EQSAM3 can provide acomputationally efficient alternative that does not only accu-rately simulates the aerosol chemical composition but alsothe most abundant aerosol species water

We apply the general circulation model ECHAM5 (Roeck-ner et al 2003) at T63 (sim19 degree) resolution extendedwith the comprehensive Modular Earth Submodel System(MESSy) to account for atmospheric chemistry (Jockel et al2006) The model will be abbreviated in the following asE5M1 It has been additionally extended by the MESSy ver-sion of EQSAM (httpwwwmessy-interfaceorg) whichaccounts for 7 aerosol modes including four soluble andthree insoluble nucleation aitken accumulation and coarsewhereby the latter three modes are used to distinguish be-tween primary insoluble and soluble aerosol species such asblack carbon or certain mineral dust compounds (similar asin Vignati et al 2004) Details will be provided in a follow-up publication

Various aerosol species

Figure 5a presents a ldquosnapshotrdquo of the model results for9 September 2000 (1200 GMT) by showing the spatialdistribution (vertical integral) of various fine mode aerosolspecies including aerosol water mass [microgm2] aerosol ni-trate ammonium sulfate organic carbon and sea salt [ppbv](from top left to bottom right) While certain aerosol speciessuch as nitrates and organics are largely confined to conti-nental areas associated with the localized emission sourcesother species such as ammonium and sulfate show muchwider dispersion Fine mode ammonium is highly corre-lated with sulfate and can be transported over long distancesNitrate compounds are volatile and they are also found oncoarse mode particles (not shown) such as sea salt and min-eral dust which are more efficiently deposited

Figure 5a shows that the spatial distribution of aerosol wa-ter (RHlt095) strongly correlates with that of sea salt sul-fate and ammonium and that at this arbitrary RH limit theaerosol water mass already dominates the total aerosol loadFigure 5b shows the tropospheric aerosol water mass whileFig 5c presents the corresponding aerosol water in the up-per troposphere - lower stratosphere (UTLS) region These

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

3182 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamicsPHOENICS

Aerosol mass (tropospheric burden) [ppb]

9th September 2000 1200 hr GMT

ECHAM5MESSy

Aerosol species (atmospheric burden) [ppb]9 September 2000 1200 hr GMT

E5M1-EQSAM3

Outlook

Figure 5 a)Fig 5 Outlook Global aerosol distributions for GMT noon 9 September 2000 (snap shot)(a) Atmospheric burden (fine mode) Aerosolwater mass (AW) in [microgm2] aerosol nitrate ammonium sulfate primary organics and sea salt (from top left to bottom right) AW(b)tropospheric burden(c) AW upper troposphere lower stratosphere (UTLS)(d) cloud coverage derived from AW(e)cloud cover comparisonfor the US ECHAM5 standard calculations (top left) AW based (regional selection of panel d) satellite observations (GEOS channel 4)The global calculations were obtained with the chemistry-climate model E5M1 (httpwwwmessy-interfaceorg)

climate forcing estimates Note that present climate mod-els that include aerosols do not explicitly calculate aerosolwater Here we aim to show that EQSAM3 can provide acomputationally efficient alternative that does not only accu-rately simulates the aerosol chemical composition but alsothe most abundant aerosol species water

We apply the general circulation model ECHAM5 (Roeck-ner et al 2003) at T63 (sim19 degree) resolution extendedwith the comprehensive Modular Earth Submodel System(MESSy) to account for atmospheric chemistry (Jockel et al2006) The model will be abbreviated in the following asE5M1 It has been additionally extended by the MESSy ver-sion of EQSAM (httpwwwmessy-interfaceorg) whichaccounts for 7 aerosol modes including four soluble andthree insoluble nucleation aitken accumulation and coarsewhereby the latter three modes are used to distinguish be-tween primary insoluble and soluble aerosol species such asblack carbon or certain mineral dust compounds (similar asin Vignati et al 2004) Details will be provided in a follow-up publication

Various aerosol species

Figure 5a presents a ldquosnapshotrdquo of the model results for9 September 2000 (1200 GMT) by showing the spatialdistribution (vertical integral) of various fine mode aerosolspecies including aerosol water mass [microgm2] aerosol ni-trate ammonium sulfate organic carbon and sea salt [ppbv](from top left to bottom right) While certain aerosol speciessuch as nitrates and organics are largely confined to conti-nental areas associated with the localized emission sourcesother species such as ammonium and sulfate show muchwider dispersion Fine mode ammonium is highly corre-lated with sulfate and can be transported over long distancesNitrate compounds are volatile and they are also found oncoarse mode particles (not shown) such as sea salt and min-eral dust which are more efficiently deposited

Figure 5a shows that the spatial distribution of aerosol wa-ter (RHlt095) strongly correlates with that of sea salt sul-fate and ammonium and that at this arbitrary RH limit theaerosol water mass already dominates the total aerosol loadFigure 5b shows the tropospheric aerosol water mass whileFig 5c presents the corresponding aerosol water in the up-per troposphere - lower stratosphere (UTLS) region These

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics 3183

figures are complementary and illustrate the dependence onsynoptic weather conditions Atmospheric dynamics deter-mine the temperature and the water vapor mass available forcondensation which subsequently determines the amount ofaerosol water

Cloud cover

In a case for which we do not limit the RH to 95 as in theprevious example aerosol water strongly increases when RHapproaches unity This directly follows from the thermody-namic principles described in Sect 2 and can be best seenfrom Eq (5) Note that the aerosol water mass calculatedfrom Eq (23) involves Eq (20) based on Eq (5) Howeverapplying Eq (5) to atmospheric conditions requires that thecondensation of water vapor is limited by the availability ofwater vapor which is determined by RH and the saturationwater vapor Thus we use the following thermodynamic con-straints to limit the condensation of water vapor and hencethe aerosol water mass for atmospheric applications

1 Limiting the amount of condensing water by the avail-ability of water vapor this encompasses all conditionswith RHlt1

2 Limiting the amount of condensing water by the amountof water vapor constrained by the saturation water vapormass being only a function of temperature thus encom-passing cases with RHge1

3 Adjusting the water vapor concentration for the con-densed amount of water vapor (aerosol water mass)which becomes important for all cases where RH ap-proaches unity in particular for the UTLS region butalso for all regions where aerosol water overlaps withcloud waterice presence

Subsequently we directly compute the cloud cover from thetotal aerosol load (including water) by assuming total cover-age of the relevant model grid cells (ie a cloud cover frac-tion of unity) in case the aerosol load exceeds an amount thatis determined by the saturation water vapor mass at a giventemperature The cloud cover therefore depends on

1 The amount of total particulate aerosol matter (PMt)according to Eqs (22) (23) the aerosol water mass isproportional to PMt (illustrated in Fig 1b)

2 The type of PMt according to Eqs (20) (21) theaerosol water mass depends on the type of solute (il-lustrated in Figs 2a b)

3 The temperature dependent saturation water vapor massthat determines the maximum amount of water avail-able for condensation leading to hygroscopic growth ofaerosols hazy conditions are favored by high aerosolload (dominated by hygroscopic salt solutes) and low

(b)

Aerosol associated water mass (tropospheric burden) [gm2]9 September 2000 1200 hr GMT

PHOENICS

Aerosol associated water mass (tropospheric burden) [gm2]

9th September 2000 1200 hr GMT

ECHAM5MESSyE5M1-EQSAM3

Outlook

Figure 5 b)

(c)

Aerosol associated water mass in the UTLS region (burden) [gm2]9 September 2000 1200 hr GMT

PHOENICS

Aerosol associated water mass in the UTLS region (burden) [microgm2]9th September 2000 1200 hr GMT

ECHAM5MESSyE5M1-EQSAM3

Outlook

Figure 5 c)

(d)

Cloud coverage based on aerosol associated water massfor troposphere and UTLS region (burden)

9 September 2000 1200 hr GMT PHOENICS

Cloud coverage based on aerosol associated water massfor troposphere and UTLS region (burden) [microgm2]

9th September 2000 1200 hr GMT

ECHAM5MESSyE5M1-EQSAM3

Outlook

Figure 5 d)

Fig 5 Continued

temperatures When the ambient temperature drops be-low the dew point temperature the water vapor concen-tration exceeds the amount of water vapor that can betaken up by the air so that it is saturated with respect towater vapor additional water vapor directly condenseswhich leads to fog haze and clouds Note that this

wwwatmos-chem-physnet731632007 Atmos Chem Phys 7 3163ndash3193 2007

3184 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

(e)

Cloud Cover - 9 September 2000PHOENICS

Cloud Cover - 9 September 2000

ECHAM5 ECHAM5MESSy

GEOS CH 4

E5M1 E5M1-EQSAM3

OutlookCloud coverage based on aerosol associated water mass

for troposphere and UTLS region (burden) 9 September 2000 1200 hr GMT

PHOENICS

Cloud coverage based on aerosol associated water massfor troposphere and UTLS region (burden) [microgm2]

9th September 2000 1200 hr GMT

ECHAM5MESSyE5M1-EQSAM3

Outlook

Figure 5 d)

Cloud coverage based on aerosol associated water massfor troposphere and UTLS region (burden)

9 September 2000 1200 hr GMT PHOENICS

Cloud coverage based on aerosol associated water massfor troposphere and UTLS region (burden) [microgm2]

9th September 2000 1200 hr GMT

ECHAM5MESSyE5M1-EQSAM3

Outlook

Figure 5 d)Clou

d co

vera

ge b

ased

on

aero

sol a

ssoc

iated

wat

er m

ass

for

trop

osph

ere

and

UTL

S re

gion

(bur

den)

9 S

epte

mbe

r 20

00 1

200

hr

GMT

PHOENICS

Clo

ud

co

vera

ge

bas

ed o

n a

ero

sol a

sso

ciat

ed w

ater

mas

sfo

r tr

op

osp

her

e an

d U

TL

S r

egio

n (

bu

rden

) [microg

m2 ]

9th

Sep

tem

ber

200

0 1

200

hr

GM

T

ECHAM5MESSy

E5M1-EQSAM3

Outlo

ok

Figu

re 5

d)

Fig 5 Continued

temperature dependency implicitly accounts for all as-pects of atmospheric dynamics (to the extent resolvedby the model) including moisture and temperature fluc-tuations in the UTLS

Figure 5d shows that the cloud cover coincides with theaerosol water mass following the synoptic pattern shown inFigs 5andashc since the cloud cover has been diagnosed fromthe total aerosol load as described above (without consider-ing aerosol-cloud and radiation feedbacks) Figure 5e showsa regional comparison of these results with GEOS satelliteobservations for the USA The lower panel shows the cloudcover as observed from space (GEOS channel 4) while theupper left panel shows the cloud cover prediction of the baseE5M1 model in the upper right panel the cloud cover isbased on the EQSAM3 water uptake calculations This qual-itative comparison with satellite observations indicates that

especially optically thin clouds are better resolved if aerosolwater is accounted for Note that the cloud cover computedwith the combination of E5M1 and EQSAM3 is higher overthe region of Florida and southern California northern Mex-ico than with the original E5M1 cloud cover scheme andcompare favorably with GEOS This issue deserves more at-tention including additional model tests and optimizationand will be considered in future work

Cloud ice

Our results thus indicate that especially optically thin cloudsmay be described by aerosol water model predictions Ifaerosols are treated comprehensively as described abovethere is actually no principal physical difference betweenaerosol water cloud water and cloud ice ndash the subdivisionis to some degree arbitrary as determined by the droplet size

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics 3185

Outlook

Cloud Ice (CI) - 3 June 2004 (noon)

Figure 6

E5M1 - Total vertically integrated CI [gm2] E5M1-EQSAM3 - Total vertically integrated CI [gm2] E5M1-EQSAM3 (T=243K) - Total vertically integrated CI [gm2]

E5M1 - UTLS vertically integrated CI [mgm2] E5M1-EQSAM3 - UTLS vertically integrated CI [mgm2] E5M1-EQSAM3 (T=243K) - UTLS vertically integrated CI [mgm2]

Fig 6 Comparison of cloud ice content predictions ndash vertically integrated (top) UTLS (bottom) E5M1 standard calculations (left) E5M1-EQSAM3 based on AW and PM with explicit freezing point depression (right) Model similar to Fig 5

To investigate to what extent the cloud ice content can becalculated directly Fig 6 presents an additional snap shotfor 3 June 2004 (1200 GMT) comparing the E5M1 basemodel (left two panels) with that including EQSAM3 for theentire troposphere (top panels) and the UTLS region (bot-tom panels) The calculations including EQSAM3 assumethat the excess aerosol water with respect to the limit basedon the saturation water mass represents the condensed cloudwater mass In addition the freezing point depression asso-ciated with salt solutes was taken into account and computedfrom the total number of moles of solute in solution and thecryoscopic constant of water which is 186 [K molminus1]

Figure 6 (right panels) shows that the ice content anddistribution predicted by EQSAM3 compares qualitativelythough remarkably well with the original E5M1 parameter-ized ice clouds in particularly for the UTLS region Con-trasting the explicit treatment of freezing point depressionwith a simpler constraint ie a constant freezing point tem-

perature (T =243 [K]) for the same aerosol load shows muchpoorer agreement which is most noticeable for the UTLS(not shown) This comparison ndash though not rigorous ndash indi-cates that

1 The explicit prediction of aerosol water mass enablesthe direct calculation of large scale (ie model gridscale) cloud properties

2 Cloud properties are sensitive to aerosol chemistry thelatter determines the particle hygroscopic growth andwater uptake which in turn controls the ambient sizedistributions of aerosols and cloud droplets at their ini-tial formation

To consistently apply these concepts in regional and globalcloud modeling studies it will be necessary to fully coupleEQSAM3 with other cloud processes such as the collisionand coalescence of droplets and precipitation formation

wwwatmos-chem-physnet731632007 Atmos Chem Phys 7 3163ndash3193 2007

3186 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

5 Summary and discussion

51 From laboratory to atmosphere

Under controlled laboratory conditions the water mass isfixed so that solution properties such as the solute molarityor molality can be measured upon which non-ideal solutionproperties such as activity coefficients can be defined Appli-cation to the atmosphere requires the transformation of thesesolution properties ie they need to be a function of RH

In the atmosphere aerosol and cloud water both dependon the water vapor mass In turn the water vapor mass isto a large extent controlled by the ambient temperature (T )which determines both the amount of water vapor availablethrough evaporation and the maximum amount of water va-por that the air can contain atT At RHge1 excess water va-por directly condenses to the aqueous or ice phase while atRHlt1 only a fraction of the saturation water mass is in equi-librium with the aqueous phase At subsaturation the max-imum amount of aerosol water is limited only by the avail-able water vapor mass and at saturation or supersaturationby theT -dependent maximum water vapor mass Under allconditions for which thermodynamic equilibrium can be as-sumed the condensed water mass is proportional to the massof solute(s) since the RH determines the water activity of thesolution that must hence remain constant

Although dynamic processes such as cloud formation im-ply non-equilibrium conditions quasi-equilibrium might beassumed for grid-scale processes being a reasonable approx-imation for most regional and large scale modeling applica-tions The reason is that the condensation of water vapor onaerosol or cloud droplets is a relatively fast process (order ofseconds) compared to the modeling time steps that are usu-ally much longer (order of minutes or more)

52 Conceptual difficulty

The ldquoclassicalrdquo equilibrium thermodynamics (eg Den-bigh 1981 Seinfeld and Pandis 1998 Wexler and Po-tukuchi 1998) does not explicitly account for the amount ofwater involved by hydration and the dissociation of eg saltsolutes although hydration drives the hygroscopic growth ofnatural and man-made aerosols Water is only consideredwhen explicitly consumed or produced In addition the wa-ter mass used to define eg aerosol activity coefficients iskept constant which is reasonable for laboratory but not foratmospheric conditions

We overcome these limitation by reformulating equilib-rium thermodynamics and hygroscopic growth of atmo-spheric aerosols into fog haze and clouds by consistentlyaccounting for the water associated with the hydration of so-lutes and by constraining the condensed water by either theavailable water vapor mass or the saturation limit The un-derlying physical principles that govern hydration have beengeneralized based on first principles that explain osmosis

53 Osmosis

Following Arrheniusrsquo theory of partial dissociation and theoriginal description of osmosis by vanrsquot Hoff and Ostwaldwe extended vanrsquot Hoffrsquos gas-solution analogy to non-idealsolutions by introducing (a) a stoichiometric coefficient forwater

(νw=ν+

w+νminusw

)to account for the actual number of

moles of water causing hydration and (b) an associated ef-fective coefficient for the solute

(νe=ν+

e +νminuse

)to account for

the effective number of moles arising from partial or com-plete solute dissociation

The advantage of using these stoichiometric coefficients isthat they allow to analytically solve the gasliquidsolid equi-librium partitioning and associated water uptake This con-siderably simplifies the numerical solution and limits CPUtime Note that activity coefficients applied in standard meth-ods vary with the concentration range which depends onboth the solute concentration and on aerosol water whichin turn is used to determine the solute concentration Thesedependencies require an CPU demanding iterative numericalsolution Moreover their use is restricted to compounds forwhich comprehensive thermodynamic data are available

In contrastνw and νe can be easily determined for anycompound from its solubility and used to analytically solvethe gas-aerosol partitioning of mixtures because these co-efficients can be assumed constant over the entire activityrange For a further discussion about the state of dissocia-tion of electrolytes in solutions we refer the interested readerto Heyrovska (1989)

54 Solubility

Common with Heyrovska (1989) we use solute specific co-efficients to calculate the vapor pressure reduction over awide concentration range In contrast to Heyrovska (1989)who did not explicitly account for the water mass consumedby hydration we additionally use a stoichiometric constantfor waterνw which can be derived from the solute solubil-ity according to Eq (19) The advantage is that only onendash easily measured and commonly used ndash solubility value isneeded The disadvantage is that for salt solutes that do notdissociate completely the effective dissociation constantνe

must be known Howeverνe can be determined withνw byEq (20) if compared to water activity measurements

55 Water activity

Although the discussion of water activity is intricate beingrelevant to many areas including for instance pharmaceuti-cal and food industry (see eg the web portal by M Chaplinhttpwwwlsbuacukwateractivityhtml) atmospheric ap-plications allow for simplifications as shown in this workOne reason is that at equilibrium the osmotic pressure if ex-pressed in terms of water eg associated with the hydrationof a salt solute equals the partial vapor pressure of water

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

3184 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

(e)

Cloud Cover - 9 September 2000PHOENICS

Cloud Cover - 9 September 2000

ECHAM5 ECHAM5MESSy

GEOS CH 4

E5M1 E5M1-EQSAM3

OutlookCloud coverage based on aerosol associated water mass

for troposphere and UTLS region (burden) 9 September 2000 1200 hr GMT

PHOENICS

Cloud coverage based on aerosol associated water massfor troposphere and UTLS region (burden) [microgm2]

9th September 2000 1200 hr GMT

ECHAM5MESSyE5M1-EQSAM3

Outlook

Figure 5 d)

Cloud coverage based on aerosol associated water massfor troposphere and UTLS region (burden)

9 September 2000 1200 hr GMT PHOENICS

Cloud coverage based on aerosol associated water massfor troposphere and UTLS region (burden) [microgm2]

9th September 2000 1200 hr GMT

ECHAM5MESSyE5M1-EQSAM3

Outlook

Figure 5 d)Clou

d co

vera

ge b

ased

on

aero

sol a

ssoc

iated

wat

er m

ass

for

trop

osph

ere

and

UTL

S re

gion

(bur

den)

9 S

epte

mbe

r 20

00 1

200

hr

GMT

PHOENICS

Clo

ud

co

vera

ge

bas

ed o

n a

ero

sol a

sso

ciat

ed w

ater

mas

sfo

r tr

op

osp

her

e an

d U

TL

S r

egio

n (

bu

rden

) [microg

m2 ]

9th

Sep

tem

ber

200

0 1

200

hr

GM

T

ECHAM5MESSy

E5M1-EQSAM3

Outlo

ok

Figu

re 5

d)

Fig 5 Continued

temperature dependency implicitly accounts for all as-pects of atmospheric dynamics (to the extent resolvedby the model) including moisture and temperature fluc-tuations in the UTLS

Figure 5d shows that the cloud cover coincides with theaerosol water mass following the synoptic pattern shown inFigs 5andashc since the cloud cover has been diagnosed fromthe total aerosol load as described above (without consider-ing aerosol-cloud and radiation feedbacks) Figure 5e showsa regional comparison of these results with GEOS satelliteobservations for the USA The lower panel shows the cloudcover as observed from space (GEOS channel 4) while theupper left panel shows the cloud cover prediction of the baseE5M1 model in the upper right panel the cloud cover isbased on the EQSAM3 water uptake calculations This qual-itative comparison with satellite observations indicates that

especially optically thin clouds are better resolved if aerosolwater is accounted for Note that the cloud cover computedwith the combination of E5M1 and EQSAM3 is higher overthe region of Florida and southern California northern Mex-ico than with the original E5M1 cloud cover scheme andcompare favorably with GEOS This issue deserves more at-tention including additional model tests and optimizationand will be considered in future work

Cloud ice

Our results thus indicate that especially optically thin cloudsmay be described by aerosol water model predictions Ifaerosols are treated comprehensively as described abovethere is actually no principal physical difference betweenaerosol water cloud water and cloud ice ndash the subdivisionis to some degree arbitrary as determined by the droplet size

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics 3185

Outlook

Cloud Ice (CI) - 3 June 2004 (noon)

Figure 6

E5M1 - Total vertically integrated CI [gm2] E5M1-EQSAM3 - Total vertically integrated CI [gm2] E5M1-EQSAM3 (T=243K) - Total vertically integrated CI [gm2]

E5M1 - UTLS vertically integrated CI [mgm2] E5M1-EQSAM3 - UTLS vertically integrated CI [mgm2] E5M1-EQSAM3 (T=243K) - UTLS vertically integrated CI [mgm2]

Fig 6 Comparison of cloud ice content predictions ndash vertically integrated (top) UTLS (bottom) E5M1 standard calculations (left) E5M1-EQSAM3 based on AW and PM with explicit freezing point depression (right) Model similar to Fig 5

To investigate to what extent the cloud ice content can becalculated directly Fig 6 presents an additional snap shotfor 3 June 2004 (1200 GMT) comparing the E5M1 basemodel (left two panels) with that including EQSAM3 for theentire troposphere (top panels) and the UTLS region (bot-tom panels) The calculations including EQSAM3 assumethat the excess aerosol water with respect to the limit basedon the saturation water mass represents the condensed cloudwater mass In addition the freezing point depression asso-ciated with salt solutes was taken into account and computedfrom the total number of moles of solute in solution and thecryoscopic constant of water which is 186 [K molminus1]

Figure 6 (right panels) shows that the ice content anddistribution predicted by EQSAM3 compares qualitativelythough remarkably well with the original E5M1 parameter-ized ice clouds in particularly for the UTLS region Con-trasting the explicit treatment of freezing point depressionwith a simpler constraint ie a constant freezing point tem-

perature (T =243 [K]) for the same aerosol load shows muchpoorer agreement which is most noticeable for the UTLS(not shown) This comparison ndash though not rigorous ndash indi-cates that

1 The explicit prediction of aerosol water mass enablesthe direct calculation of large scale (ie model gridscale) cloud properties

2 Cloud properties are sensitive to aerosol chemistry thelatter determines the particle hygroscopic growth andwater uptake which in turn controls the ambient sizedistributions of aerosols and cloud droplets at their ini-tial formation

To consistently apply these concepts in regional and globalcloud modeling studies it will be necessary to fully coupleEQSAM3 with other cloud processes such as the collisionand coalescence of droplets and precipitation formation

wwwatmos-chem-physnet731632007 Atmos Chem Phys 7 3163ndash3193 2007

3186 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

5 Summary and discussion

51 From laboratory to atmosphere

Under controlled laboratory conditions the water mass isfixed so that solution properties such as the solute molarityor molality can be measured upon which non-ideal solutionproperties such as activity coefficients can be defined Appli-cation to the atmosphere requires the transformation of thesesolution properties ie they need to be a function of RH

In the atmosphere aerosol and cloud water both dependon the water vapor mass In turn the water vapor mass isto a large extent controlled by the ambient temperature (T )which determines both the amount of water vapor availablethrough evaporation and the maximum amount of water va-por that the air can contain atT At RHge1 excess water va-por directly condenses to the aqueous or ice phase while atRHlt1 only a fraction of the saturation water mass is in equi-librium with the aqueous phase At subsaturation the max-imum amount of aerosol water is limited only by the avail-able water vapor mass and at saturation or supersaturationby theT -dependent maximum water vapor mass Under allconditions for which thermodynamic equilibrium can be as-sumed the condensed water mass is proportional to the massof solute(s) since the RH determines the water activity of thesolution that must hence remain constant

Although dynamic processes such as cloud formation im-ply non-equilibrium conditions quasi-equilibrium might beassumed for grid-scale processes being a reasonable approx-imation for most regional and large scale modeling applica-tions The reason is that the condensation of water vapor onaerosol or cloud droplets is a relatively fast process (order ofseconds) compared to the modeling time steps that are usu-ally much longer (order of minutes or more)

52 Conceptual difficulty

The ldquoclassicalrdquo equilibrium thermodynamics (eg Den-bigh 1981 Seinfeld and Pandis 1998 Wexler and Po-tukuchi 1998) does not explicitly account for the amount ofwater involved by hydration and the dissociation of eg saltsolutes although hydration drives the hygroscopic growth ofnatural and man-made aerosols Water is only consideredwhen explicitly consumed or produced In addition the wa-ter mass used to define eg aerosol activity coefficients iskept constant which is reasonable for laboratory but not foratmospheric conditions

We overcome these limitation by reformulating equilib-rium thermodynamics and hygroscopic growth of atmo-spheric aerosols into fog haze and clouds by consistentlyaccounting for the water associated with the hydration of so-lutes and by constraining the condensed water by either theavailable water vapor mass or the saturation limit The un-derlying physical principles that govern hydration have beengeneralized based on first principles that explain osmosis

53 Osmosis

Following Arrheniusrsquo theory of partial dissociation and theoriginal description of osmosis by vanrsquot Hoff and Ostwaldwe extended vanrsquot Hoffrsquos gas-solution analogy to non-idealsolutions by introducing (a) a stoichiometric coefficient forwater

(νw=ν+

w+νminusw

)to account for the actual number of

moles of water causing hydration and (b) an associated ef-fective coefficient for the solute

(νe=ν+

e +νminuse

)to account for

the effective number of moles arising from partial or com-plete solute dissociation

The advantage of using these stoichiometric coefficients isthat they allow to analytically solve the gasliquidsolid equi-librium partitioning and associated water uptake This con-siderably simplifies the numerical solution and limits CPUtime Note that activity coefficients applied in standard meth-ods vary with the concentration range which depends onboth the solute concentration and on aerosol water whichin turn is used to determine the solute concentration Thesedependencies require an CPU demanding iterative numericalsolution Moreover their use is restricted to compounds forwhich comprehensive thermodynamic data are available

In contrastνw and νe can be easily determined for anycompound from its solubility and used to analytically solvethe gas-aerosol partitioning of mixtures because these co-efficients can be assumed constant over the entire activityrange For a further discussion about the state of dissocia-tion of electrolytes in solutions we refer the interested readerto Heyrovska (1989)

54 Solubility

Common with Heyrovska (1989) we use solute specific co-efficients to calculate the vapor pressure reduction over awide concentration range In contrast to Heyrovska (1989)who did not explicitly account for the water mass consumedby hydration we additionally use a stoichiometric constantfor waterνw which can be derived from the solute solubil-ity according to Eq (19) The advantage is that only onendash easily measured and commonly used ndash solubility value isneeded The disadvantage is that for salt solutes that do notdissociate completely the effective dissociation constantνe

must be known Howeverνe can be determined withνw byEq (20) if compared to water activity measurements

55 Water activity

Although the discussion of water activity is intricate beingrelevant to many areas including for instance pharmaceuti-cal and food industry (see eg the web portal by M Chaplinhttpwwwlsbuacukwateractivityhtml) atmospheric ap-plications allow for simplifications as shown in this workOne reason is that at equilibrium the osmotic pressure if ex-pressed in terms of water eg associated with the hydrationof a salt solute equals the partial vapor pressure of water

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics 3185

Outlook

Cloud Ice (CI) - 3 June 2004 (noon)

Figure 6

E5M1 - Total vertically integrated CI [gm2] E5M1-EQSAM3 - Total vertically integrated CI [gm2] E5M1-EQSAM3 (T=243K) - Total vertically integrated CI [gm2]

E5M1 - UTLS vertically integrated CI [mgm2] E5M1-EQSAM3 - UTLS vertically integrated CI [mgm2] E5M1-EQSAM3 (T=243K) - UTLS vertically integrated CI [mgm2]

Fig 6 Comparison of cloud ice content predictions ndash vertically integrated (top) UTLS (bottom) E5M1 standard calculations (left) E5M1-EQSAM3 based on AW and PM with explicit freezing point depression (right) Model similar to Fig 5

To investigate to what extent the cloud ice content can becalculated directly Fig 6 presents an additional snap shotfor 3 June 2004 (1200 GMT) comparing the E5M1 basemodel (left two panels) with that including EQSAM3 for theentire troposphere (top panels) and the UTLS region (bot-tom panels) The calculations including EQSAM3 assumethat the excess aerosol water with respect to the limit basedon the saturation water mass represents the condensed cloudwater mass In addition the freezing point depression asso-ciated with salt solutes was taken into account and computedfrom the total number of moles of solute in solution and thecryoscopic constant of water which is 186 [K molminus1]

Figure 6 (right panels) shows that the ice content anddistribution predicted by EQSAM3 compares qualitativelythough remarkably well with the original E5M1 parameter-ized ice clouds in particularly for the UTLS region Con-trasting the explicit treatment of freezing point depressionwith a simpler constraint ie a constant freezing point tem-

perature (T =243 [K]) for the same aerosol load shows muchpoorer agreement which is most noticeable for the UTLS(not shown) This comparison ndash though not rigorous ndash indi-cates that

1 The explicit prediction of aerosol water mass enablesthe direct calculation of large scale (ie model gridscale) cloud properties

2 Cloud properties are sensitive to aerosol chemistry thelatter determines the particle hygroscopic growth andwater uptake which in turn controls the ambient sizedistributions of aerosols and cloud droplets at their ini-tial formation

To consistently apply these concepts in regional and globalcloud modeling studies it will be necessary to fully coupleEQSAM3 with other cloud processes such as the collisionand coalescence of droplets and precipitation formation

wwwatmos-chem-physnet731632007 Atmos Chem Phys 7 3163ndash3193 2007

3186 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

5 Summary and discussion

51 From laboratory to atmosphere

Under controlled laboratory conditions the water mass isfixed so that solution properties such as the solute molarityor molality can be measured upon which non-ideal solutionproperties such as activity coefficients can be defined Appli-cation to the atmosphere requires the transformation of thesesolution properties ie they need to be a function of RH

In the atmosphere aerosol and cloud water both dependon the water vapor mass In turn the water vapor mass isto a large extent controlled by the ambient temperature (T )which determines both the amount of water vapor availablethrough evaporation and the maximum amount of water va-por that the air can contain atT At RHge1 excess water va-por directly condenses to the aqueous or ice phase while atRHlt1 only a fraction of the saturation water mass is in equi-librium with the aqueous phase At subsaturation the max-imum amount of aerosol water is limited only by the avail-able water vapor mass and at saturation or supersaturationby theT -dependent maximum water vapor mass Under allconditions for which thermodynamic equilibrium can be as-sumed the condensed water mass is proportional to the massof solute(s) since the RH determines the water activity of thesolution that must hence remain constant

Although dynamic processes such as cloud formation im-ply non-equilibrium conditions quasi-equilibrium might beassumed for grid-scale processes being a reasonable approx-imation for most regional and large scale modeling applica-tions The reason is that the condensation of water vapor onaerosol or cloud droplets is a relatively fast process (order ofseconds) compared to the modeling time steps that are usu-ally much longer (order of minutes or more)

52 Conceptual difficulty

The ldquoclassicalrdquo equilibrium thermodynamics (eg Den-bigh 1981 Seinfeld and Pandis 1998 Wexler and Po-tukuchi 1998) does not explicitly account for the amount ofwater involved by hydration and the dissociation of eg saltsolutes although hydration drives the hygroscopic growth ofnatural and man-made aerosols Water is only consideredwhen explicitly consumed or produced In addition the wa-ter mass used to define eg aerosol activity coefficients iskept constant which is reasonable for laboratory but not foratmospheric conditions

We overcome these limitation by reformulating equilib-rium thermodynamics and hygroscopic growth of atmo-spheric aerosols into fog haze and clouds by consistentlyaccounting for the water associated with the hydration of so-lutes and by constraining the condensed water by either theavailable water vapor mass or the saturation limit The un-derlying physical principles that govern hydration have beengeneralized based on first principles that explain osmosis

53 Osmosis

Following Arrheniusrsquo theory of partial dissociation and theoriginal description of osmosis by vanrsquot Hoff and Ostwaldwe extended vanrsquot Hoffrsquos gas-solution analogy to non-idealsolutions by introducing (a) a stoichiometric coefficient forwater

(νw=ν+

w+νminusw

)to account for the actual number of

moles of water causing hydration and (b) an associated ef-fective coefficient for the solute

(νe=ν+

e +νminuse

)to account for

the effective number of moles arising from partial or com-plete solute dissociation

The advantage of using these stoichiometric coefficients isthat they allow to analytically solve the gasliquidsolid equi-librium partitioning and associated water uptake This con-siderably simplifies the numerical solution and limits CPUtime Note that activity coefficients applied in standard meth-ods vary with the concentration range which depends onboth the solute concentration and on aerosol water whichin turn is used to determine the solute concentration Thesedependencies require an CPU demanding iterative numericalsolution Moreover their use is restricted to compounds forwhich comprehensive thermodynamic data are available

In contrastνw and νe can be easily determined for anycompound from its solubility and used to analytically solvethe gas-aerosol partitioning of mixtures because these co-efficients can be assumed constant over the entire activityrange For a further discussion about the state of dissocia-tion of electrolytes in solutions we refer the interested readerto Heyrovska (1989)

54 Solubility

Common with Heyrovska (1989) we use solute specific co-efficients to calculate the vapor pressure reduction over awide concentration range In contrast to Heyrovska (1989)who did not explicitly account for the water mass consumedby hydration we additionally use a stoichiometric constantfor waterνw which can be derived from the solute solubil-ity according to Eq (19) The advantage is that only onendash easily measured and commonly used ndash solubility value isneeded The disadvantage is that for salt solutes that do notdissociate completely the effective dissociation constantνe

must be known Howeverνe can be determined withνw byEq (20) if compared to water activity measurements

55 Water activity

Although the discussion of water activity is intricate beingrelevant to many areas including for instance pharmaceuti-cal and food industry (see eg the web portal by M Chaplinhttpwwwlsbuacukwateractivityhtml) atmospheric ap-plications allow for simplifications as shown in this workOne reason is that at equilibrium the osmotic pressure if ex-pressed in terms of water eg associated with the hydrationof a salt solute equals the partial vapor pressure of water

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

3186 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

5 Summary and discussion

51 From laboratory to atmosphere

Under controlled laboratory conditions the water mass isfixed so that solution properties such as the solute molarityor molality can be measured upon which non-ideal solutionproperties such as activity coefficients can be defined Appli-cation to the atmosphere requires the transformation of thesesolution properties ie they need to be a function of RH

In the atmosphere aerosol and cloud water both dependon the water vapor mass In turn the water vapor mass isto a large extent controlled by the ambient temperature (T )which determines both the amount of water vapor availablethrough evaporation and the maximum amount of water va-por that the air can contain atT At RHge1 excess water va-por directly condenses to the aqueous or ice phase while atRHlt1 only a fraction of the saturation water mass is in equi-librium with the aqueous phase At subsaturation the max-imum amount of aerosol water is limited only by the avail-able water vapor mass and at saturation or supersaturationby theT -dependent maximum water vapor mass Under allconditions for which thermodynamic equilibrium can be as-sumed the condensed water mass is proportional to the massof solute(s) since the RH determines the water activity of thesolution that must hence remain constant

Although dynamic processes such as cloud formation im-ply non-equilibrium conditions quasi-equilibrium might beassumed for grid-scale processes being a reasonable approx-imation for most regional and large scale modeling applica-tions The reason is that the condensation of water vapor onaerosol or cloud droplets is a relatively fast process (order ofseconds) compared to the modeling time steps that are usu-ally much longer (order of minutes or more)

52 Conceptual difficulty

The ldquoclassicalrdquo equilibrium thermodynamics (eg Den-bigh 1981 Seinfeld and Pandis 1998 Wexler and Po-tukuchi 1998) does not explicitly account for the amount ofwater involved by hydration and the dissociation of eg saltsolutes although hydration drives the hygroscopic growth ofnatural and man-made aerosols Water is only consideredwhen explicitly consumed or produced In addition the wa-ter mass used to define eg aerosol activity coefficients iskept constant which is reasonable for laboratory but not foratmospheric conditions

We overcome these limitation by reformulating equilib-rium thermodynamics and hygroscopic growth of atmo-spheric aerosols into fog haze and clouds by consistentlyaccounting for the water associated with the hydration of so-lutes and by constraining the condensed water by either theavailable water vapor mass or the saturation limit The un-derlying physical principles that govern hydration have beengeneralized based on first principles that explain osmosis

53 Osmosis

Following Arrheniusrsquo theory of partial dissociation and theoriginal description of osmosis by vanrsquot Hoff and Ostwaldwe extended vanrsquot Hoffrsquos gas-solution analogy to non-idealsolutions by introducing (a) a stoichiometric coefficient forwater

(νw=ν+

w+νminusw

)to account for the actual number of

moles of water causing hydration and (b) an associated ef-fective coefficient for the solute

(νe=ν+

e +νminuse

)to account for

the effective number of moles arising from partial or com-plete solute dissociation

The advantage of using these stoichiometric coefficients isthat they allow to analytically solve the gasliquidsolid equi-librium partitioning and associated water uptake This con-siderably simplifies the numerical solution and limits CPUtime Note that activity coefficients applied in standard meth-ods vary with the concentration range which depends onboth the solute concentration and on aerosol water whichin turn is used to determine the solute concentration Thesedependencies require an CPU demanding iterative numericalsolution Moreover their use is restricted to compounds forwhich comprehensive thermodynamic data are available

In contrastνw and νe can be easily determined for anycompound from its solubility and used to analytically solvethe gas-aerosol partitioning of mixtures because these co-efficients can be assumed constant over the entire activityrange For a further discussion about the state of dissocia-tion of electrolytes in solutions we refer the interested readerto Heyrovska (1989)

54 Solubility

Common with Heyrovska (1989) we use solute specific co-efficients to calculate the vapor pressure reduction over awide concentration range In contrast to Heyrovska (1989)who did not explicitly account for the water mass consumedby hydration we additionally use a stoichiometric constantfor waterνw which can be derived from the solute solubil-ity according to Eq (19) The advantage is that only onendash easily measured and commonly used ndash solubility value isneeded The disadvantage is that for salt solutes that do notdissociate completely the effective dissociation constantνe

must be known Howeverνe can be determined withνw byEq (20) if compared to water activity measurements

55 Water activity

Although the discussion of water activity is intricate beingrelevant to many areas including for instance pharmaceuti-cal and food industry (see eg the web portal by M Chaplinhttpwwwlsbuacukwateractivityhtml) atmospheric ap-plications allow for simplifications as shown in this workOne reason is that at equilibrium the osmotic pressure if ex-pressed in terms of water eg associated with the hydrationof a salt solute equals the partial vapor pressure of water

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics 3187

so that the water activity remains constant and equal to RH(Sect 2) One important consequence is that the amount ofwater required for hydration is directly proportional to theamount of solute

56 Kelvin-term

Another consequence not addressed thus far is that the va-por pressure reduction associated with dissolution hydrationand dissociation of a solute equals the osmotic pressure dif-ference independently of the surface curvature (see Fig 1)At equilibrium the osmotic pressure of the solute or solvent(in solution) equals the corresponding partial vapor pressuresof solute or solvent above the solution In contrast to the the-oretical solvent partial pressure in solution the (measurable)osmotic pressure is an effective pressure and hence implicitlyaccounts for surface tension and non-ideality effects Notethat Raoultrsquos law describes this for the solvent and Henryrsquoslaw for the solute and both can be generalized to accountfor solution non-idealities by including either activity coeffi-cients or the stoichiometric coefficientsνw andνe Note thateg surface tension or droplet curvature affects the solubilityand at equilibrium is implicitly accounted for byνw

The advantage of our method is that the water uptake ofatmospheric aerosols can be directly calculated for any par-ticle size from the solute solubility (if known) The so calledldquoKelvin-termrdquo that accounts for the curvature of nanometersized particles is not needed in case of equilibrium Suchsmall particles are rarely in equilibrium with the ambient airrather they grow relatively fast due to hygroscopic growthwhereby their equilibrium size is maintained by RH Thusexcept for the calculation of nucleation grow rates (non-equilibrium conditions) surface curvature corrections are notrequired

57 Kohler-equation

Another consequence is that in equilibrium the widely usedKohler-equation becomes redundant since the Kelvin-termis not required for the calculation of the aerosol particlesize if aerosol water is consistently treated In fact theuse of the Kohler-equation is somewhat misleading sincemost approximations of the 1rminus1r3 ndash radius dependencyof the ldquosurface-(Kelvin)rdquo and ldquovolume (Raoult)rdquo term in theKohler equation are inaccurate Usually neither concentra-tion changes in the surface tension are accounted for nor isthe fact that the volume of water is not constant but depen-dent on relative humidity

By explicitly including aerosol water the 1rminus1r3 ra-dius dependency is automatically accounted for Note thatr is usually approximated as the ambient radius (eg Prup-pacher and Klett 1997) while the more explicit formulation(see eg Dufour and Defay 1963) actually yields a differ-ence between the ambient (wet) radius and the dry radius ofthe solute expressing the radius of the aerosol water which

however cancels out if the volume occupied by water is notassumed to be constant

58 Cloud condensation nuclei

Nevertheless the Kohler-equation has been successfullyused in many applications to calculate the activation ofaerosol particles as cloud condensation nuclei (CCN) and todescribe cloud droplet size spectra Is this really neededsince the definition of CCN requires that a specific thougharbitrary supersaturation is prescribed As illustrated inSect 43 aerosol particles can grow by water uptake intocloud droplets simply by allowing ambient conditions toinclude RHge1 Obviously the growing particles competefor the available water vapor and the likelihood for largerdroplets to collect the available water vapor is highest sothat they grow at the expense of smaller ones Hence atequilibrium a bimodal size distribution establishes as ob-served with larger droplets and smaller ones that have col-lected more and less water vapor respectively The largerones can settle gravitationally dependent on their mass andlocal vertical air velocities and by collision and coalescencethey can ultimately grow into rain drops The equilibriumdroplet size distribution to a large extent depends on thecloud dynamics whereas the initial size is determined by theaerosol water mass Therefore our method eliminates theneed to define CCN It furthermore allows to directly relatethe chemical properties of aerosol particles to cloud dropletsbeing a requirement to explicitly link emission sources of at-mospheric trace constituents in models to the physical prop-erties of aerosol particles fog haze and clouds

59 Final comments

Our new concept developed in Sect 2 uses the fundamen-tal equations and ideas proposed more than a century ago toexplain the principles of osmosis Our contribution is thatwe have applied and transformed these ideas by consistentlyaccounting for water involved in the hydration of salt or non-electrolyte solutes by the introduction of the stoichiometricconstantsνw andνe This yields both a) a more general for-mulation of the principles that govern osmosis that now ex-tends to non-ideal solutions and b) a consistent calculationof the hygroscopic growth of atmospheric aerosols

591 Mixed solutions

It is worth mentioning that the use ofνw νe is not onlylimited to the calculation of single solute solution molalitiesor dilute solutions As demonstrated by the application ofEQSAM3 to observations of sea salt desert dust and pol-lution particles over the Mediterranean Seaνw νe can alsobe assumed constant for highly concentrated mixed solutions(see Fig 3)

wwwatmos-chem-physnet731632007 Atmos Chem Phys 7 3163ndash3193 2007

3188 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

592 Gibbs free energy

The consistent use ofνw νe also eliminates the need to itera-tively minimize the Gibbs free energy in model applicationssince at equilibrium the total Gibbs free energy change mustbe zero when water is included in the summation As a con-sequence the gasliquidsolid partitioning can be solved an-alytically which saves a considerable amount of CPU-time(see also Metzger et al 2002) so that applications in com-plex regional or global models become feasible

593 Equilibrium constants

The consistent use ofνw νe furthermore substitutes the useof equilibrium constants since the Gibbs free energy is zeroandνw implicitly includes the relevant information to com-pute the equilibrium phase partitioning as it is derived fromthe solute solubility The successful application of EQSAM3demonstrates that this approach suffices (Fig 3) Note thatEQSAM3 merely uses solubilities as measured for each so-lute in Table 1

594 Relative humidity of deliquescence

The use of solubility measurements together withνw νe

yields the relative humidity of deliquescence (RHD) whichcorresponds to the RH at which the solution is saturated withrespect to a solute Any increase of the solute then resultsin its precipitation whereby the solid is in equilibrium withits ions in the aqueous phase If Eq (21) is used complexcomputations are redundant Even efflorescence and deli-quescence RHDs of single or mixed salt solutions can be cal-culated from Eq (21) (see Sect 41 point 5 and 9) Notethat all these values provide independent evidence of the ac-curacy of Eq (20) and the applicability ofνw νe to concen-trated mixed solutions Previous approaches were dependenton activity coefficients

595 Activity coefficients

With the successful application of EQSAM3 we furtherdemonstrate that activity coefficients are only required for(semi-) volatile compounds and that they can be directlycalculated from RH whenνw νe are known ie by substi-tuting Eq (20) into Eq (14) and considering Eqs (15) and(16) (Sect 41) Note that (semi-) volatile compounds canbe additionally driven out of the aqueous phase into the gasphase in contrast to non-volatile compounds which can onlyprecipitate into the solid phase when the RH drops and thewater activity decreases The gasliquid partitioning is thusdetermined by the solute solubility ifνw νe are used to cal-culate the aerosol water mass since at equilibrium a solutionis always saturated and because the RH determines the wateractivity whereby RHlt1 is already a correction for solutionnon-ideality

For instance the mean molal binary activity coefficientof ammonium nitrate used in EQSAM3 compares well withthat of ISORROPIA also for concentrated mixed solutions(Fig 3) Even very sensitive aerosol properties such as thesolution pH compare well and low aerosol nitrate concentra-tions can be effectively predicted Although very low aerosolnitrate concentrations could not be predicted with EQSAM2(mainly since the use of the ammonium nitrate equilibriumconstant is insufficient for certain mixed aerosol systems) themean binary activity coefficient of ammonium nitrate is com-parable in both versions

Note that EQSAM2 used activity coefficients that havebeen derived according to the method described in Metzgeret al (2002) Zaveri et al (2005) compared this method withvarious other and well established activity coefficient calcu-lation methods and recognized that this was the first time toexpress binary mean activity coefficients of individual elec-trolytes directly as a function of water activity Unfortu-nately Zaveri et al (2005) have overlooked that the methodof Metzger et al (2002) is not limited to dilute binary sys-tems but also extends to multicomponent activity coefficientsof concentrated mixed solutions as demonstrated in Fig 3The reason is that Zaveri et al (2005) applied these meth-ods to laboratory conditions where activity coefficients aremeasured as function of solute concentration but not of theequilibrium aerosol water mass being itself a function of RHin the atmosphere ndash thus a conceptual difficulty we hope tohave eliminated with our present work

596 Equilibrium assumption

The equilibrium assumption is an essential and central ele-ment in physical chemistry For instance Henryrsquos Law con-stant describes the partitioning of a gas between the atmo-sphere and the aqueous phase It can be based on direct mea-surements or calculated as the ratio of the pure compoundvapor pressure to the solubility The latter approximation isreliable only for compounds of very low solubility In factvalues of Henryrsquos law constant found in the literature fre-quently differ substantially Despite that Henryrsquos law con-stants are determined on the basis of equilibrium they areused to solve chemical systems which are not in equilibrium

Similarly our formulas can be applied to non-equilibriumconditions by solving the chemical system dynamically Theaccuracy achievable with our equations eg Eqs (20) (21)or (23) merely depends on the accuracy and applicability ofthe solubility measurements from which the solute specificconstantsνw νe have been derived Note that this also in-cludes the temperature dependency The approximation usedin EQSAM3 might not necessarily suffice for all compoundsand explicit temperature dependencies can be easily invokedto derive the solute specific constantsνw andνe The assump-tion that droplet growth on aerosol particles can be approxi-mated by an equilibrium approach can be conceptually testedby observing the steady state conditions of eg stratocumulus

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics 3189

and cumulus clouds Even though droplets continually growand evaporate by local moisture variations the cloud appear-ance (ie the mean cloud properties) varies little unless theoverall dynamical or thermodynamical boundary conditionschange

597 Importance of aerosol water

The applicability ofνw νe to mixed solutions as determinedfrom single solubility measurements is particularly usefulfor atmospheric applications since complete knowledge ofthe actual solute composition is often not available Espe-cially the efflorescence and deliquescence relative humidi-ties of single solute and mixed solutions which can be cal-culated fromνw νe by Eq (21) are crucial as they determineat which RH ambient aerosol particles grow due to water up-take into the size range of efficient solar radiation scatteringand subsequently of cloud formation

Aerosol water is hence central in modeling atmosphericchemistry visibility weather and climate It is determinedby the hygroscopicity of natural and man-made aerosolswhereby the hygroscopicity can be altered by air pollutionAerosol water can thus provide a direct link between air pol-lution weather and climate

598 Limitation

Our new method is based on thermodynamic equilibrium andhence does not directly apply to non-equilibrium conditionsthat can occur in the atmosphere However for most mod-eling applications thermodynamic equilibrium suffices sinceequilibrium involving water is usually established in secondswhile model time-steps usually exceed a few minutes

For instance our method allows to predict the RHD ofpure salt compounds which are required to calculate the par-titioning between the solid and liquid phase and hence theRH andT at which a salt takes up water from the atmo-sphere The equilibrium assumption implies that a solutionis saturated if the RH equals the RHD if the RH drops thesolution becomes supersaturated which causes solute precip-itation until the solution is saturated again (so that the satu-ration molality remains constant) Although most salts del-iquesce nearly immediately certain salt mixtures as foundin the polluted atmosphere might alter or even suppress thedeliquescence behavior of natural aerosols so that the com-pounds deliquesces over a different RH range In this casethe equilibrium assumption can break down However theRHD calculations with Eq (21) might still be valid if theactual molality of such an aerosol mixture is used along theRH trajectory or in case it is approximated by using effectivesolubilities Therefore even the drying process of aerosolscan be modeled according to Sect 41 (point 5 8 9) wherethe aerosols follow the upper tail of the so-called hysteresisloop the lower tail is described by the wetting process anddeliquescence

Nevertheless the water uptake is mainly determined bysalt compounds for which the equilibrium assumption andthe derived additivity of the osmotic partial pressures is validNote that the pressure additivity is a consequence of the gas-solution analogy and implies molar volume additivity butthat the additivity might not always be a good assumptionFor instance non-salt solutes and compounds whose molec-ular structure either collapses due to hydration or capillareffects arising from the particles morphology or particle co-agulation might temporarily lead to non-equilibrium condi-tions and hence to a failure in the assumption of additivity (PWinkler personal communication) This would hence affectthe summation of the partial aerosol water masses and theRHD of the mixed aerosol phase since in our methodνe andνw are summarized in this case

However such compounds contribute usually only littleto the overall water uptake so that the aerosol water massof ambient aerosols might still be approximated sufficientlyaccurately Similarly our method may be used to model theinitial cloud conditions by assuming quasi-equilibrium sincethe water uptake is limited by the actually available water va-por mass as determined by atmospheric dynamics and trans-port

6 Conclusions

Based on first thermodynamics principles we conceptuallyapplied osmotic pressure to aerosols fogs hazes and cloudsWhen transformed from laboratory to atmospheric condi-tions as summarized by Eqs (6)ndash(10) it follows that thewater needed for hydration is proportional to the amount ofsolute and governed by the type of solute and RH To accountfor the moles of water needed for hydration and dissociationwe introduced the solute specific effective dissociation coef-ficient νe and the coefficient for waterνw Both coefficientsare independent of the solute concentration and can directlybe computed from the solute solubility of weak and strongelectrolytes (organic and inorganic salts) or non-electrolytes(eg sugars alcohols or dissolved gases)

We demonstrated the applicability of this concept in a newthermodynamic equilibrium model (EQSAM3) for mixedsolutions The results of EQSAM3 compare well to fieldmeasurements and other thermodynamic equilibrium models(EQMs) such as ISORROPIA and SCAPE2 The latter twomodels use comprehensive (and CPU demanding) algorithmsto solve the ldquoclassicalrdquo aerosol thermodynamics (being rathercomplex for mixed solutions) whereas EQSAM3 solves thegasliquidsolid partitioning and hygroscopic growth ana-lytically and non-iteratively EQSAM3 computes variousaerosol properties that are difficult to measure such as size-segregated particle composition efflorescence and deliques-cence relative humidities of singe or mixed solutions solu-tion pH by accounting for major inorganic and organic com-

wwwatmos-chem-physnet731632007 Atmos Chem Phys 7 3163ndash3193 2007

3190 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

pounds (Table 1 lists nearly 100 and the list can be easilyextended)

The advantage of our new model is that it requires onlythe compound solubility to derive all further thermody-namic data whereby the solubility is commonly used eas-ily measured and available for many inorganic and organiccompounds (eg from CRC Handbook of Chemistry andPhysics 2006) Previous EQMs rely on various complexlaboratory measurements (eg water activities activity co-efficients or RHDs) which are not only limited to cer-tain compounds but also involve numerically complex andhence expensive computations Instead the computationalefficiency of EQSAM3 allows application in regional andglobal atmospheric-chemistry and climate models

We further illustrated that the water uptake of aerosols isdirectly proportional to the soluble aerosol load whereby thehygroscopic growth depends on the type of solute We pre-sented calculations of water uptake as a function of RH forselected inorganic salt solutions the electronic supplementextends this to all compounds shown in Table 1 An appli-cation of EQSAM3 to mixed inorganic and organic salt so-lutions is included and indicates the importance of aerosolchemistry for visibility predictions

Our outlook for future applications of EQSAM3 also in-volved global chemistry-transport and meteorological mod-eling including aerosol-cloud interactions The EQSAM3computation of aerosol water ndash without assumptions aboutthe activation of aerosol particles ndash holds promise for theexplicit computation of large scale fogs hazes and cloudsincluding cloud cover the initial cloud water and cloud iceWhen aerosol water is explicitly accounted for aerosol andcloud thermodynamics can be easily coupled and hence sub-stantially simplified

Our new approach will enable model calculations that di-rectly relate various natural and anthropogenic emissions oftrace gases and aerosols to fog haze and cloud formationso that air pollution effects on weather and climate can bestudied explicitly

Appendix A List of acronyms symbols and indices

The following tables list all acronyms symbols andindices used throughout the paper Names of chem-ical substances are listed in the electronic supple-ment (httpwwwatmos-chem-physnet731632007acp-7-3163-2007-supplementzip)

Table A1 List of acronyms and abbreviations

Abbreviation Name

AS Ammonium SulfateCCN Cloud Condensation NucleiECHAM5 Climate Model (Rockner et al 2003)EPA Environmental Protection Agency (USA)EQM Thermodynamic EQuilibrium ModelEQSAM3 EQuilibrium Simplified Aerosol Model version 3

EQM (this work)E5M1 ECHAM5MESSy (Jockel et al 2006)F2C2 FineCoarse Chemical System (Metzger et al 2006)GEOS Geostationary SatelliteIPCC Intergovernmental Panel on Climate ChangeISORROPIA EQM (Nenes et al 1998)LMW Low Molecular WeightMINOS Mediterranean INtensive Oxidant Study

(Lelieveld et al 2002)MESSy Modular Earth Submodel SystemMOHp Meteorological Observatory HohenpeiszligenbergPM Particulate Matterppbv Parts per Billion by volumeSCAPE2 EQM (Meng et al 1995)T63 Climate Model Resolution (sim19 degree)UTLS Upper Troposphere - Lower Stratosphere

Table A2 List of units

Symbol Name Unit

Pa SI-Unit Pascal Nm2

N SI-Unit Newton kg ms2

J SI-Unit Joule Pa m3=Nm2 m3=Nm

R universal gas constant 8314 JmolK

Table A3 List of indices

Abbreviation Name

(cr) crystalline(aq) aqueous(g) gaseous(+) superscript for ldquocationrdquo(minus) superscript for ldquoanionrdquo(plusmn) superscript for ldquoion pairrdquow subscript for ldquowaterrdquos subscript for ldquosoluterdquoe subscript for ldquoeffectiverdquo

i subscript for ldquoith- of k-componentsrdquo

j subscript for ldquoj th-compound or chemical reactionrdquo

k subscript for ldquothe total number of componentsrdquo

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

3188 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

592 Gibbs free energy

The consistent use ofνw νe also eliminates the need to itera-tively minimize the Gibbs free energy in model applicationssince at equilibrium the total Gibbs free energy change mustbe zero when water is included in the summation As a con-sequence the gasliquidsolid partitioning can be solved an-alytically which saves a considerable amount of CPU-time(see also Metzger et al 2002) so that applications in com-plex regional or global models become feasible

593 Equilibrium constants

The consistent use ofνw νe furthermore substitutes the useof equilibrium constants since the Gibbs free energy is zeroandνw implicitly includes the relevant information to com-pute the equilibrium phase partitioning as it is derived fromthe solute solubility The successful application of EQSAM3demonstrates that this approach suffices (Fig 3) Note thatEQSAM3 merely uses solubilities as measured for each so-lute in Table 1

594 Relative humidity of deliquescence

The use of solubility measurements together withνw νe

yields the relative humidity of deliquescence (RHD) whichcorresponds to the RH at which the solution is saturated withrespect to a solute Any increase of the solute then resultsin its precipitation whereby the solid is in equilibrium withits ions in the aqueous phase If Eq (21) is used complexcomputations are redundant Even efflorescence and deli-quescence RHDs of single or mixed salt solutions can be cal-culated from Eq (21) (see Sect 41 point 5 and 9) Notethat all these values provide independent evidence of the ac-curacy of Eq (20) and the applicability ofνw νe to concen-trated mixed solutions Previous approaches were dependenton activity coefficients

595 Activity coefficients

With the successful application of EQSAM3 we furtherdemonstrate that activity coefficients are only required for(semi-) volatile compounds and that they can be directlycalculated from RH whenνw νe are known ie by substi-tuting Eq (20) into Eq (14) and considering Eqs (15) and(16) (Sect 41) Note that (semi-) volatile compounds canbe additionally driven out of the aqueous phase into the gasphase in contrast to non-volatile compounds which can onlyprecipitate into the solid phase when the RH drops and thewater activity decreases The gasliquid partitioning is thusdetermined by the solute solubility ifνw νe are used to cal-culate the aerosol water mass since at equilibrium a solutionis always saturated and because the RH determines the wateractivity whereby RHlt1 is already a correction for solutionnon-ideality

For instance the mean molal binary activity coefficientof ammonium nitrate used in EQSAM3 compares well withthat of ISORROPIA also for concentrated mixed solutions(Fig 3) Even very sensitive aerosol properties such as thesolution pH compare well and low aerosol nitrate concentra-tions can be effectively predicted Although very low aerosolnitrate concentrations could not be predicted with EQSAM2(mainly since the use of the ammonium nitrate equilibriumconstant is insufficient for certain mixed aerosol systems) themean binary activity coefficient of ammonium nitrate is com-parable in both versions

Note that EQSAM2 used activity coefficients that havebeen derived according to the method described in Metzgeret al (2002) Zaveri et al (2005) compared this method withvarious other and well established activity coefficient calcu-lation methods and recognized that this was the first time toexpress binary mean activity coefficients of individual elec-trolytes directly as a function of water activity Unfortu-nately Zaveri et al (2005) have overlooked that the methodof Metzger et al (2002) is not limited to dilute binary sys-tems but also extends to multicomponent activity coefficientsof concentrated mixed solutions as demonstrated in Fig 3The reason is that Zaveri et al (2005) applied these meth-ods to laboratory conditions where activity coefficients aremeasured as function of solute concentration but not of theequilibrium aerosol water mass being itself a function of RHin the atmosphere ndash thus a conceptual difficulty we hope tohave eliminated with our present work

596 Equilibrium assumption

The equilibrium assumption is an essential and central ele-ment in physical chemistry For instance Henryrsquos Law con-stant describes the partitioning of a gas between the atmo-sphere and the aqueous phase It can be based on direct mea-surements or calculated as the ratio of the pure compoundvapor pressure to the solubility The latter approximation isreliable only for compounds of very low solubility In factvalues of Henryrsquos law constant found in the literature fre-quently differ substantially Despite that Henryrsquos law con-stants are determined on the basis of equilibrium they areused to solve chemical systems which are not in equilibrium

Similarly our formulas can be applied to non-equilibriumconditions by solving the chemical system dynamically Theaccuracy achievable with our equations eg Eqs (20) (21)or (23) merely depends on the accuracy and applicability ofthe solubility measurements from which the solute specificconstantsνw νe have been derived Note that this also in-cludes the temperature dependency The approximation usedin EQSAM3 might not necessarily suffice for all compoundsand explicit temperature dependencies can be easily invokedto derive the solute specific constantsνw andνe The assump-tion that droplet growth on aerosol particles can be approxi-mated by an equilibrium approach can be conceptually testedby observing the steady state conditions of eg stratocumulus

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics 3189

and cumulus clouds Even though droplets continually growand evaporate by local moisture variations the cloud appear-ance (ie the mean cloud properties) varies little unless theoverall dynamical or thermodynamical boundary conditionschange

597 Importance of aerosol water

The applicability ofνw νe to mixed solutions as determinedfrom single solubility measurements is particularly usefulfor atmospheric applications since complete knowledge ofthe actual solute composition is often not available Espe-cially the efflorescence and deliquescence relative humidi-ties of single solute and mixed solutions which can be cal-culated fromνw νe by Eq (21) are crucial as they determineat which RH ambient aerosol particles grow due to water up-take into the size range of efficient solar radiation scatteringand subsequently of cloud formation

Aerosol water is hence central in modeling atmosphericchemistry visibility weather and climate It is determinedby the hygroscopicity of natural and man-made aerosolswhereby the hygroscopicity can be altered by air pollutionAerosol water can thus provide a direct link between air pol-lution weather and climate

598 Limitation

Our new method is based on thermodynamic equilibrium andhence does not directly apply to non-equilibrium conditionsthat can occur in the atmosphere However for most mod-eling applications thermodynamic equilibrium suffices sinceequilibrium involving water is usually established in secondswhile model time-steps usually exceed a few minutes

For instance our method allows to predict the RHD ofpure salt compounds which are required to calculate the par-titioning between the solid and liquid phase and hence theRH andT at which a salt takes up water from the atmo-sphere The equilibrium assumption implies that a solutionis saturated if the RH equals the RHD if the RH drops thesolution becomes supersaturated which causes solute precip-itation until the solution is saturated again (so that the satu-ration molality remains constant) Although most salts del-iquesce nearly immediately certain salt mixtures as foundin the polluted atmosphere might alter or even suppress thedeliquescence behavior of natural aerosols so that the com-pounds deliquesces over a different RH range In this casethe equilibrium assumption can break down However theRHD calculations with Eq (21) might still be valid if theactual molality of such an aerosol mixture is used along theRH trajectory or in case it is approximated by using effectivesolubilities Therefore even the drying process of aerosolscan be modeled according to Sect 41 (point 5 8 9) wherethe aerosols follow the upper tail of the so-called hysteresisloop the lower tail is described by the wetting process anddeliquescence

Nevertheless the water uptake is mainly determined bysalt compounds for which the equilibrium assumption andthe derived additivity of the osmotic partial pressures is validNote that the pressure additivity is a consequence of the gas-solution analogy and implies molar volume additivity butthat the additivity might not always be a good assumptionFor instance non-salt solutes and compounds whose molec-ular structure either collapses due to hydration or capillareffects arising from the particles morphology or particle co-agulation might temporarily lead to non-equilibrium condi-tions and hence to a failure in the assumption of additivity (PWinkler personal communication) This would hence affectthe summation of the partial aerosol water masses and theRHD of the mixed aerosol phase since in our methodνe andνw are summarized in this case

However such compounds contribute usually only littleto the overall water uptake so that the aerosol water massof ambient aerosols might still be approximated sufficientlyaccurately Similarly our method may be used to model theinitial cloud conditions by assuming quasi-equilibrium sincethe water uptake is limited by the actually available water va-por mass as determined by atmospheric dynamics and trans-port

6 Conclusions

Based on first thermodynamics principles we conceptuallyapplied osmotic pressure to aerosols fogs hazes and cloudsWhen transformed from laboratory to atmospheric condi-tions as summarized by Eqs (6)ndash(10) it follows that thewater needed for hydration is proportional to the amount ofsolute and governed by the type of solute and RH To accountfor the moles of water needed for hydration and dissociationwe introduced the solute specific effective dissociation coef-ficient νe and the coefficient for waterνw Both coefficientsare independent of the solute concentration and can directlybe computed from the solute solubility of weak and strongelectrolytes (organic and inorganic salts) or non-electrolytes(eg sugars alcohols or dissolved gases)

We demonstrated the applicability of this concept in a newthermodynamic equilibrium model (EQSAM3) for mixedsolutions The results of EQSAM3 compare well to fieldmeasurements and other thermodynamic equilibrium models(EQMs) such as ISORROPIA and SCAPE2 The latter twomodels use comprehensive (and CPU demanding) algorithmsto solve the ldquoclassicalrdquo aerosol thermodynamics (being rathercomplex for mixed solutions) whereas EQSAM3 solves thegasliquidsolid partitioning and hygroscopic growth ana-lytically and non-iteratively EQSAM3 computes variousaerosol properties that are difficult to measure such as size-segregated particle composition efflorescence and deliques-cence relative humidities of singe or mixed solutions solu-tion pH by accounting for major inorganic and organic com-

wwwatmos-chem-physnet731632007 Atmos Chem Phys 7 3163ndash3193 2007

3190 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

pounds (Table 1 lists nearly 100 and the list can be easilyextended)

The advantage of our new model is that it requires onlythe compound solubility to derive all further thermody-namic data whereby the solubility is commonly used eas-ily measured and available for many inorganic and organiccompounds (eg from CRC Handbook of Chemistry andPhysics 2006) Previous EQMs rely on various complexlaboratory measurements (eg water activities activity co-efficients or RHDs) which are not only limited to cer-tain compounds but also involve numerically complex andhence expensive computations Instead the computationalefficiency of EQSAM3 allows application in regional andglobal atmospheric-chemistry and climate models

We further illustrated that the water uptake of aerosols isdirectly proportional to the soluble aerosol load whereby thehygroscopic growth depends on the type of solute We pre-sented calculations of water uptake as a function of RH forselected inorganic salt solutions the electronic supplementextends this to all compounds shown in Table 1 An appli-cation of EQSAM3 to mixed inorganic and organic salt so-lutions is included and indicates the importance of aerosolchemistry for visibility predictions

Our outlook for future applications of EQSAM3 also in-volved global chemistry-transport and meteorological mod-eling including aerosol-cloud interactions The EQSAM3computation of aerosol water ndash without assumptions aboutthe activation of aerosol particles ndash holds promise for theexplicit computation of large scale fogs hazes and cloudsincluding cloud cover the initial cloud water and cloud iceWhen aerosol water is explicitly accounted for aerosol andcloud thermodynamics can be easily coupled and hence sub-stantially simplified

Our new approach will enable model calculations that di-rectly relate various natural and anthropogenic emissions oftrace gases and aerosols to fog haze and cloud formationso that air pollution effects on weather and climate can bestudied explicitly

Appendix A List of acronyms symbols and indices

The following tables list all acronyms symbols andindices used throughout the paper Names of chem-ical substances are listed in the electronic supple-ment (httpwwwatmos-chem-physnet731632007acp-7-3163-2007-supplementzip)

Table A1 List of acronyms and abbreviations

Abbreviation Name

AS Ammonium SulfateCCN Cloud Condensation NucleiECHAM5 Climate Model (Rockner et al 2003)EPA Environmental Protection Agency (USA)EQM Thermodynamic EQuilibrium ModelEQSAM3 EQuilibrium Simplified Aerosol Model version 3

EQM (this work)E5M1 ECHAM5MESSy (Jockel et al 2006)F2C2 FineCoarse Chemical System (Metzger et al 2006)GEOS Geostationary SatelliteIPCC Intergovernmental Panel on Climate ChangeISORROPIA EQM (Nenes et al 1998)LMW Low Molecular WeightMINOS Mediterranean INtensive Oxidant Study

(Lelieveld et al 2002)MESSy Modular Earth Submodel SystemMOHp Meteorological Observatory HohenpeiszligenbergPM Particulate Matterppbv Parts per Billion by volumeSCAPE2 EQM (Meng et al 1995)T63 Climate Model Resolution (sim19 degree)UTLS Upper Troposphere - Lower Stratosphere

Table A2 List of units

Symbol Name Unit

Pa SI-Unit Pascal Nm2

N SI-Unit Newton kg ms2

J SI-Unit Joule Pa m3=Nm2 m3=Nm

R universal gas constant 8314 JmolK

Table A3 List of indices

Abbreviation Name

(cr) crystalline(aq) aqueous(g) gaseous(+) superscript for ldquocationrdquo(minus) superscript for ldquoanionrdquo(plusmn) superscript for ldquoion pairrdquow subscript for ldquowaterrdquos subscript for ldquosoluterdquoe subscript for ldquoeffectiverdquo

i subscript for ldquoith- of k-componentsrdquo

j subscript for ldquoj th-compound or chemical reactionrdquo

k subscript for ldquothe total number of componentsrdquo

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics 3189

and cumulus clouds Even though droplets continually growand evaporate by local moisture variations the cloud appear-ance (ie the mean cloud properties) varies little unless theoverall dynamical or thermodynamical boundary conditionschange

597 Importance of aerosol water

The applicability ofνw νe to mixed solutions as determinedfrom single solubility measurements is particularly usefulfor atmospheric applications since complete knowledge ofthe actual solute composition is often not available Espe-cially the efflorescence and deliquescence relative humidi-ties of single solute and mixed solutions which can be cal-culated fromνw νe by Eq (21) are crucial as they determineat which RH ambient aerosol particles grow due to water up-take into the size range of efficient solar radiation scatteringand subsequently of cloud formation

Aerosol water is hence central in modeling atmosphericchemistry visibility weather and climate It is determinedby the hygroscopicity of natural and man-made aerosolswhereby the hygroscopicity can be altered by air pollutionAerosol water can thus provide a direct link between air pol-lution weather and climate

598 Limitation

Our new method is based on thermodynamic equilibrium andhence does not directly apply to non-equilibrium conditionsthat can occur in the atmosphere However for most mod-eling applications thermodynamic equilibrium suffices sinceequilibrium involving water is usually established in secondswhile model time-steps usually exceed a few minutes

For instance our method allows to predict the RHD ofpure salt compounds which are required to calculate the par-titioning between the solid and liquid phase and hence theRH andT at which a salt takes up water from the atmo-sphere The equilibrium assumption implies that a solutionis saturated if the RH equals the RHD if the RH drops thesolution becomes supersaturated which causes solute precip-itation until the solution is saturated again (so that the satu-ration molality remains constant) Although most salts del-iquesce nearly immediately certain salt mixtures as foundin the polluted atmosphere might alter or even suppress thedeliquescence behavior of natural aerosols so that the com-pounds deliquesces over a different RH range In this casethe equilibrium assumption can break down However theRHD calculations with Eq (21) might still be valid if theactual molality of such an aerosol mixture is used along theRH trajectory or in case it is approximated by using effectivesolubilities Therefore even the drying process of aerosolscan be modeled according to Sect 41 (point 5 8 9) wherethe aerosols follow the upper tail of the so-called hysteresisloop the lower tail is described by the wetting process anddeliquescence

Nevertheless the water uptake is mainly determined bysalt compounds for which the equilibrium assumption andthe derived additivity of the osmotic partial pressures is validNote that the pressure additivity is a consequence of the gas-solution analogy and implies molar volume additivity butthat the additivity might not always be a good assumptionFor instance non-salt solutes and compounds whose molec-ular structure either collapses due to hydration or capillareffects arising from the particles morphology or particle co-agulation might temporarily lead to non-equilibrium condi-tions and hence to a failure in the assumption of additivity (PWinkler personal communication) This would hence affectthe summation of the partial aerosol water masses and theRHD of the mixed aerosol phase since in our methodνe andνw are summarized in this case

However such compounds contribute usually only littleto the overall water uptake so that the aerosol water massof ambient aerosols might still be approximated sufficientlyaccurately Similarly our method may be used to model theinitial cloud conditions by assuming quasi-equilibrium sincethe water uptake is limited by the actually available water va-por mass as determined by atmospheric dynamics and trans-port

6 Conclusions

Based on first thermodynamics principles we conceptuallyapplied osmotic pressure to aerosols fogs hazes and cloudsWhen transformed from laboratory to atmospheric condi-tions as summarized by Eqs (6)ndash(10) it follows that thewater needed for hydration is proportional to the amount ofsolute and governed by the type of solute and RH To accountfor the moles of water needed for hydration and dissociationwe introduced the solute specific effective dissociation coef-ficient νe and the coefficient for waterνw Both coefficientsare independent of the solute concentration and can directlybe computed from the solute solubility of weak and strongelectrolytes (organic and inorganic salts) or non-electrolytes(eg sugars alcohols or dissolved gases)

We demonstrated the applicability of this concept in a newthermodynamic equilibrium model (EQSAM3) for mixedsolutions The results of EQSAM3 compare well to fieldmeasurements and other thermodynamic equilibrium models(EQMs) such as ISORROPIA and SCAPE2 The latter twomodels use comprehensive (and CPU demanding) algorithmsto solve the ldquoclassicalrdquo aerosol thermodynamics (being rathercomplex for mixed solutions) whereas EQSAM3 solves thegasliquidsolid partitioning and hygroscopic growth ana-lytically and non-iteratively EQSAM3 computes variousaerosol properties that are difficult to measure such as size-segregated particle composition efflorescence and deliques-cence relative humidities of singe or mixed solutions solu-tion pH by accounting for major inorganic and organic com-

wwwatmos-chem-physnet731632007 Atmos Chem Phys 7 3163ndash3193 2007

3190 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

pounds (Table 1 lists nearly 100 and the list can be easilyextended)

The advantage of our new model is that it requires onlythe compound solubility to derive all further thermody-namic data whereby the solubility is commonly used eas-ily measured and available for many inorganic and organiccompounds (eg from CRC Handbook of Chemistry andPhysics 2006) Previous EQMs rely on various complexlaboratory measurements (eg water activities activity co-efficients or RHDs) which are not only limited to cer-tain compounds but also involve numerically complex andhence expensive computations Instead the computationalefficiency of EQSAM3 allows application in regional andglobal atmospheric-chemistry and climate models

We further illustrated that the water uptake of aerosols isdirectly proportional to the soluble aerosol load whereby thehygroscopic growth depends on the type of solute We pre-sented calculations of water uptake as a function of RH forselected inorganic salt solutions the electronic supplementextends this to all compounds shown in Table 1 An appli-cation of EQSAM3 to mixed inorganic and organic salt so-lutions is included and indicates the importance of aerosolchemistry for visibility predictions

Our outlook for future applications of EQSAM3 also in-volved global chemistry-transport and meteorological mod-eling including aerosol-cloud interactions The EQSAM3computation of aerosol water ndash without assumptions aboutthe activation of aerosol particles ndash holds promise for theexplicit computation of large scale fogs hazes and cloudsincluding cloud cover the initial cloud water and cloud iceWhen aerosol water is explicitly accounted for aerosol andcloud thermodynamics can be easily coupled and hence sub-stantially simplified

Our new approach will enable model calculations that di-rectly relate various natural and anthropogenic emissions oftrace gases and aerosols to fog haze and cloud formationso that air pollution effects on weather and climate can bestudied explicitly

Appendix A List of acronyms symbols and indices

The following tables list all acronyms symbols andindices used throughout the paper Names of chem-ical substances are listed in the electronic supple-ment (httpwwwatmos-chem-physnet731632007acp-7-3163-2007-supplementzip)

Table A1 List of acronyms and abbreviations

Abbreviation Name

AS Ammonium SulfateCCN Cloud Condensation NucleiECHAM5 Climate Model (Rockner et al 2003)EPA Environmental Protection Agency (USA)EQM Thermodynamic EQuilibrium ModelEQSAM3 EQuilibrium Simplified Aerosol Model version 3

EQM (this work)E5M1 ECHAM5MESSy (Jockel et al 2006)F2C2 FineCoarse Chemical System (Metzger et al 2006)GEOS Geostationary SatelliteIPCC Intergovernmental Panel on Climate ChangeISORROPIA EQM (Nenes et al 1998)LMW Low Molecular WeightMINOS Mediterranean INtensive Oxidant Study

(Lelieveld et al 2002)MESSy Modular Earth Submodel SystemMOHp Meteorological Observatory HohenpeiszligenbergPM Particulate Matterppbv Parts per Billion by volumeSCAPE2 EQM (Meng et al 1995)T63 Climate Model Resolution (sim19 degree)UTLS Upper Troposphere - Lower Stratosphere

Table A2 List of units

Symbol Name Unit

Pa SI-Unit Pascal Nm2

N SI-Unit Newton kg ms2

J SI-Unit Joule Pa m3=Nm2 m3=Nm

R universal gas constant 8314 JmolK

Table A3 List of indices

Abbreviation Name

(cr) crystalline(aq) aqueous(g) gaseous(+) superscript for ldquocationrdquo(minus) superscript for ldquoanionrdquo(plusmn) superscript for ldquoion pairrdquow subscript for ldquowaterrdquos subscript for ldquosoluterdquoe subscript for ldquoeffectiverdquo

i subscript for ldquoith- of k-componentsrdquo

j subscript for ldquoj th-compound or chemical reactionrdquo

k subscript for ldquothe total number of componentsrdquo

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

3190 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

pounds (Table 1 lists nearly 100 and the list can be easilyextended)

The advantage of our new model is that it requires onlythe compound solubility to derive all further thermody-namic data whereby the solubility is commonly used eas-ily measured and available for many inorganic and organiccompounds (eg from CRC Handbook of Chemistry andPhysics 2006) Previous EQMs rely on various complexlaboratory measurements (eg water activities activity co-efficients or RHDs) which are not only limited to cer-tain compounds but also involve numerically complex andhence expensive computations Instead the computationalefficiency of EQSAM3 allows application in regional andglobal atmospheric-chemistry and climate models

We further illustrated that the water uptake of aerosols isdirectly proportional to the soluble aerosol load whereby thehygroscopic growth depends on the type of solute We pre-sented calculations of water uptake as a function of RH forselected inorganic salt solutions the electronic supplementextends this to all compounds shown in Table 1 An appli-cation of EQSAM3 to mixed inorganic and organic salt so-lutions is included and indicates the importance of aerosolchemistry for visibility predictions

Our outlook for future applications of EQSAM3 also in-volved global chemistry-transport and meteorological mod-eling including aerosol-cloud interactions The EQSAM3computation of aerosol water ndash without assumptions aboutthe activation of aerosol particles ndash holds promise for theexplicit computation of large scale fogs hazes and cloudsincluding cloud cover the initial cloud water and cloud iceWhen aerosol water is explicitly accounted for aerosol andcloud thermodynamics can be easily coupled and hence sub-stantially simplified

Our new approach will enable model calculations that di-rectly relate various natural and anthropogenic emissions oftrace gases and aerosols to fog haze and cloud formationso that air pollution effects on weather and climate can bestudied explicitly

Appendix A List of acronyms symbols and indices

The following tables list all acronyms symbols andindices used throughout the paper Names of chem-ical substances are listed in the electronic supple-ment (httpwwwatmos-chem-physnet731632007acp-7-3163-2007-supplementzip)

Table A1 List of acronyms and abbreviations

Abbreviation Name

AS Ammonium SulfateCCN Cloud Condensation NucleiECHAM5 Climate Model (Rockner et al 2003)EPA Environmental Protection Agency (USA)EQM Thermodynamic EQuilibrium ModelEQSAM3 EQuilibrium Simplified Aerosol Model version 3

EQM (this work)E5M1 ECHAM5MESSy (Jockel et al 2006)F2C2 FineCoarse Chemical System (Metzger et al 2006)GEOS Geostationary SatelliteIPCC Intergovernmental Panel on Climate ChangeISORROPIA EQM (Nenes et al 1998)LMW Low Molecular WeightMINOS Mediterranean INtensive Oxidant Study

(Lelieveld et al 2002)MESSy Modular Earth Submodel SystemMOHp Meteorological Observatory HohenpeiszligenbergPM Particulate Matterppbv Parts per Billion by volumeSCAPE2 EQM (Meng et al 1995)T63 Climate Model Resolution (sim19 degree)UTLS Upper Troposphere - Lower Stratosphere

Table A2 List of units

Symbol Name Unit

Pa SI-Unit Pascal Nm2

N SI-Unit Newton kg ms2

J SI-Unit Joule Pa m3=Nm2 m3=Nm

R universal gas constant 8314 JmolK

Table A3 List of indices

Abbreviation Name

(cr) crystalline(aq) aqueous(g) gaseous(+) superscript for ldquocationrdquo(minus) superscript for ldquoanionrdquo(plusmn) superscript for ldquoion pairrdquow subscript for ldquowaterrdquos subscript for ldquosoluterdquoe subscript for ldquoeffectiverdquo

i subscript for ldquoith- of k-componentsrdquo

j subscript for ldquoj th-compound or chemical reactionrdquo

k subscript for ldquothe total number of componentsrdquo

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics 3191

Table A4 List of symbols

Symbol Name Unit

Stoichiometric coefficientνw solvent (water) ndashν+w water cation (H3O+) ndash

νminusw water anion (OHminus) ndash

νs solute (for complete dissociation) ndashνe solute (for effective dissociation) ndashν+s solute cation (complete dissociation) ndash

ν+e solute cation (effective dissociation) ndash

νminuss solute anion (complete dissociation) ndash

νminuse solute anion (effective dissociation) ndash

νij of ith-component ofj th-chem reaction ndash

total number of molesn(g) in the gas phase moln in the aqueous phase mol

n(g)s of solute (gas phase) mol

n(cr)s of solute (crystalline phase) mol

ns of solute (aqueous phase) mol

n(g)w of water (gas phase) mol

nw of water (aqueous phase) molnw of solute-mass equivalent in terms of water mol

n(g)wo of water (gas phase) at ref conditions mol

nwo of water (aqueous phase) at ref conditions molnso of solute (aqueous phase) at ref conditions molzs+j of electrons transferred by 1 mol of cation ndashzsminusj of electrons transferred by 1 mol of anion ndashzplusmn

sjof electrons transferred by 1 mol of ion-pair ndash

kplusmn of water consumed by dissociation ndashinto 1 mol of cationanion

Nplusmn of water consumed by molar dissociation ndash

ms+ cation solute molality molkgmsminus anion solute molality molkgms solute molality molkgmss single solute molality molkgmsssat single solute molality at saturation molkgγs+ cation (molal) activity coeff of solute (s) kgmolγsminus anion (molal) activity coeff of solute (s) kgmolγ plusmns mean (molal) ion-pair activity coefficient kgmol

γ plusmn

s(molal)j mean (molal) binary activity coeff kgmol

γ plusmn

s(molar)j mean (molar) binary activity coeff lmol

ξsj relative charge density of the solution ndashas+ cation solute activity (aqueous phase) 0ndash1asminus anion solute activity (aqueous phase) 0ndash1as solute activity (aqueous phase) 0ndash1aso solute activity (aqueous phase) 0ndash1

at reference conditionsaw+ water cation (H3O+) activity (aq phase) 0ndash1awminus water anion (OHminus) activity (aq phase) 0ndash1aw water activity (aqueous phase) 0ndash1awo water activity (aqueous phase) 0ndash1

at reference conditionsgos (partial) Gibbrsquos free energy of solute (s) kJ

at reference conditionsgow (partial) Gibbrsquos free energy of solvent (water) kJ

at reference conditionsgoij

(partial) Gibbrsquos free energy ofith-component kJ

of j th-chemical reaction at ref conditions

Table A4 List of symbols (continued)

Symbol Name Unit

1 total change in1E total energy kJ1G Gibbrsquos free energy kJ1V (aq) volume (aqueous phase) m3

1ns (total) moles of solute mol1nw (total) moles of water mol

1P(g)w (partial) water vapor pressure (gas phase) Pa

15w (partial) osmotic pressure of solvent (water) Pa15s (partial) osmotic pressure of solute (s) Pa15 (total) osmotic pressure (aqueous phase) Pa1RH (fractional) relative humidity 0ndash11aw (fractional) water activity (aqueous phase) 0ndash1

(partial) vapor pressure of

P(aq)w water (aqueous phase) Pa

P(g)w water (gas phase) Pa

P(g)wsat water at saturation (gas phase) Pa

P(g)wo water (gas phase) at reference conditions Pa

(partial) osmotic pressure (aqueous phase) of5s solute (s) Pa5w solvent (water) Pa5so solute (s) at reference conditions Pa5wo solvent (water) at reference conditions Pa

P (total) atmospheric pressure (gas phase) Pa5 (total) osmotic pressure (aqueous phase) Pa5o (total) osmotic pressure of pure solvent (water) Pa

at reference conditions

χw mole fraction of water 0ndash1χs mole fraction of solute 0ndash1χw generalized mole fraction of water 0ndash1χs generalized mole fraction of solute 0ndash1ρs solute density gcm3

ρw water density gcm3

ws solute mass fraction 0ndash1Ws solute mass fraction ms mass of solute gmw mass of water gMs Molar mass of solute gmolMw Molar mass of water gmolK molal equilibrium constant of reaction (R) ndashKsp solubility product constant (molkg)νs

V (g) total volume (gas phase) m3

V (aq) total volume (aqueous phase) m3

r radius mPM total number of moles (liquid and solid phase) molm3(air)PMt total particulate mass (liquid and solid phase) microgm3(air)PMs total particulate mass (solid phase) microgm3(air)pH acid-base indicator (potentia Hydrogenia)RH fractional relative humidity 0ndash1RHD relative humidity of deliquescence 0ndash1T temperature KTo temperature K

at reference conditionsZ ionic strength of the solution molkg(H2O)Zs+ total molar cation charge ndashZsminus total molar anion charge ndash

wwwatmos-chem-physnet731632007 Atmos Chem Phys 7 3163ndash3193 2007

3192 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

AcknowledgementsThis work was financed through the projectANTISTORM (Anthropogenic Aerosols Triggering and Invigorat-ing Severe Stormshttpantistormisaccnritindexhtml) throughthe European Community program NEST-12444 (New and emerg-ing science and technologyhttpwwwcordislunest)

We acknowledge the German weather service for providing theMOHp measurements and the colleagues at the Max Planck Insti-tute for Meteorology for providing the climate model ECHAM5We are grateful to the MESSy development group of the MaxPlanck Institute for Chemistry for providing E5M1 We furtherthank N Mihalopoulos and the MINOS measurement group for thegasaerosol measurements

Edited by M Dameris

References

Arrhenius S Z Phys Chem 1 631 1887Chan C K Flagan R C and Seinfeld J H Water activities of

NH4NO3(NH4)2SO4 solutions Atmos Environ 26A 1661ndash1673 1992

Cohen M D Flagan R C and Seinfeld J H Studies of concen-trated electrolyte solutions using the electrodynamic balance 1Water activities for single electrolyte solutions J Phys Chem91 4563ndash4574 1987a

Cohen M D Flagan R C and Seinfeld J H Studies of concen-trated electrolyte solutions using the electrodynamic balance 2Water activities for mixed electrolyte solutions J Phys Chem91 4575ndash4582 1987b

Chemical Rubber Company (CRC) Handbook of Chemistry andPhysics 86th Edition Taylor and Francis Group LLC 2004ndash2005 CD-ROM version 2006

Clegg S L Milioto S and Palmer D A Osmotic and ActivityCoefficients of Aqueous (NH4)2SO4 as a Function of Temper-ature and Aqueous (NH4)2SO4-H2SO4 Mixtures at 29815 Kand 32315 K J Chem Eng Data 41 3 455ndash467 1996

Clegg S L and Seinfeld J H Thermodynamic Models ofAqueous Solutions Containing Inorganic Electrolytes and Dicar-boxylic Acids at 29815 K 2 Systems Including DissociationEquilibria J Phys Chem A 110 17 5718ndash5734 2006

Clegg S L Brimblecombe P and Wexler A S ThermodynamicModel of the System H+-NH4+-Na+-SO42ndashNO3ndashClndashH2O at29815 K J Phys Chem A 102 12 2155ndash2171 1998

Denbigh K The Principles of Chemical Equilibrium 4th edCambridge Univ Press New York 1981

Dufour L and Defay R Thermodynamics of Clouds AcademicPress New York 1963 1427ndash1434 1998

Environmental Protection Agency (EPA) Air quality criteria forparticulate matter EPA600P-95001cF 1996

Gibbs J W On the equilibrium of heterogeneous substances(1876) in Collected Works Vol 1 Longmans Green and Co1928

Hamer W J and Wu Y C Osmotic coefficients and mean activitycoefficients of uni-valent electrolytes in water at 25C J PhysChem Ref Data 1 1047ndash1099 1972

Hanel G The size of atmospheric aerosol particles as a functionof the relative humidity Beitr Phys Atmos 43 119ndash132 1970

Hanel G The properties of atmospheric aerosol particles as func-tions of the relative humidity at thermodynamic equilibrium withthe surrounding moist air Adv Geophys Res 19 73ndash188 1976

Heyrovska R A Reappraisal of Arrheniusrsquo Theory of Partial Dis-sociation of Electrolytes American Chem Soc 1989

Hofmeister F Zur Lehre von der Wirkung der Salze Arch ExpPathol Pharmakol (Leipzig) 24 (1888) 247ndash260 translated inZur Lehre von der Wirkung der Salze (about the science of theeffect of salts Franz Hofmeisterrsquos historical papers edited byKunz W Henle J and Ninham B W Curr Opin Coll Inter-face Sci 9 19ndash37 2004

Holgate S T Samet J M Koren H S and Maynard R L Airpollution and health Academic Press San Diego 1999

Intergovernmental Panel on Climate Change (IPCC) ClimateChange 2007 The Physical Science Basis Contribution ofWorking Group I to the Fourth Assessment Report of the Inter-governmental Panel on Climate Change edited by Solomon SQin D Manning M Chen Z Marquis M Averyt K BTignor M and Miller H L Cambridge University Press Cam-bridge UK and New York NY USA 2007

IUPAC Solubility Data Series International Union of Pure and Ap-plied Chemistry Vol 1 to 53 were published by Pergamon PressOxford from 1979 to 1994 subsequent volumes were publishedby Oxford University Press Oxford

Jockel P Tost H Pozzer A Bruhl C Buchholz J GanzeveldL Hoor P Kerkweg A Lawrence M G Sander R SteilB Stiller G Tanarhte M Taraborrelli D van Aardenne Jand Lelieveld J The atmospheric chemistry general circulationmodel ECHAM5MESSy1 consistent simulation of ozone fromthe surface to the mesosphere Atmos Chem Phys 6 5067ndash5104 2006httpwwwatmos-chem-physnet650672006

Kim Y P Seinfeld J H and Saxena P Atmospheric gasaerosolequilibrium I Thermodynamic model Aerosol Sci Technol 19157ndash181 1993a

Kim Y P Seinfeld J H and Saxena P Atmospheric gasaerosolequilibrium II Analysis of common approximations and activitycoefficient calculation methods Aerosol Sci Technol 19 182ndash198 1993b

Kim Y P and Seinfeld J H Atmospheric gasaerosol equilibriumIII Thermodynamics of crustal elements Ca2+ K+ and Mg2+Aerosol Sci Technol 22 93ndash110 1995

Kohler H The nucleus in and the growth of hygroscopic dropletsTrans Faraday Soc 32 1152ndash1162 1936

Lelieveld J Berresheim H Borrmann S Crutzen P J Den-tener F J Fischer H Feichter J Flatau P J HelandJ Holzinger R Korrmann R Lawrence M G Levin ZMarkowicz K M Mihalopoulos N Minikin A RamanathanV de Reus M Roelofs G J Scheeren H A Sciare JSchlager H Schultz M Siegmund P Steil B Stephanou EG Stier P Traub M Warneke C Williams J and ZiereisH Global Air Pollution Crossroads over the Mediterranean Sci-ence 298 794ndash799 2002

Low R D H A generalized equation for the solution effect indroplet growth Atmos Sci 26 608ndash611 1969

Meng Z Seinfeld J H Saxena P and Kim Y P Atmo-spheric gasaerosol equilibrium IV Thermodynamics of carbon-ates Aerosol Sci Technol 131ndash154 1995

Metzger S M GasAerosol Partitioning A simplified Method

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

3192 S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics

AcknowledgementsThis work was financed through the projectANTISTORM (Anthropogenic Aerosols Triggering and Invigorat-ing Severe Stormshttpantistormisaccnritindexhtml) throughthe European Community program NEST-12444 (New and emerg-ing science and technologyhttpwwwcordislunest)

We acknowledge the German weather service for providing theMOHp measurements and the colleagues at the Max Planck Insti-tute for Meteorology for providing the climate model ECHAM5We are grateful to the MESSy development group of the MaxPlanck Institute for Chemistry for providing E5M1 We furtherthank N Mihalopoulos and the MINOS measurement group for thegasaerosol measurements

Edited by M Dameris

References

Arrhenius S Z Phys Chem 1 631 1887Chan C K Flagan R C and Seinfeld J H Water activities of

NH4NO3(NH4)2SO4 solutions Atmos Environ 26A 1661ndash1673 1992

Cohen M D Flagan R C and Seinfeld J H Studies of concen-trated electrolyte solutions using the electrodynamic balance 1Water activities for single electrolyte solutions J Phys Chem91 4563ndash4574 1987a

Cohen M D Flagan R C and Seinfeld J H Studies of concen-trated electrolyte solutions using the electrodynamic balance 2Water activities for mixed electrolyte solutions J Phys Chem91 4575ndash4582 1987b

Chemical Rubber Company (CRC) Handbook of Chemistry andPhysics 86th Edition Taylor and Francis Group LLC 2004ndash2005 CD-ROM version 2006

Clegg S L Milioto S and Palmer D A Osmotic and ActivityCoefficients of Aqueous (NH4)2SO4 as a Function of Temper-ature and Aqueous (NH4)2SO4-H2SO4 Mixtures at 29815 Kand 32315 K J Chem Eng Data 41 3 455ndash467 1996

Clegg S L and Seinfeld J H Thermodynamic Models ofAqueous Solutions Containing Inorganic Electrolytes and Dicar-boxylic Acids at 29815 K 2 Systems Including DissociationEquilibria J Phys Chem A 110 17 5718ndash5734 2006

Clegg S L Brimblecombe P and Wexler A S ThermodynamicModel of the System H+-NH4+-Na+-SO42ndashNO3ndashClndashH2O at29815 K J Phys Chem A 102 12 2155ndash2171 1998

Denbigh K The Principles of Chemical Equilibrium 4th edCambridge Univ Press New York 1981

Dufour L and Defay R Thermodynamics of Clouds AcademicPress New York 1963 1427ndash1434 1998

Environmental Protection Agency (EPA) Air quality criteria forparticulate matter EPA600P-95001cF 1996

Gibbs J W On the equilibrium of heterogeneous substances(1876) in Collected Works Vol 1 Longmans Green and Co1928

Hamer W J and Wu Y C Osmotic coefficients and mean activitycoefficients of uni-valent electrolytes in water at 25C J PhysChem Ref Data 1 1047ndash1099 1972

Hanel G The size of atmospheric aerosol particles as a functionof the relative humidity Beitr Phys Atmos 43 119ndash132 1970

Hanel G The properties of atmospheric aerosol particles as func-tions of the relative humidity at thermodynamic equilibrium withthe surrounding moist air Adv Geophys Res 19 73ndash188 1976

Heyrovska R A Reappraisal of Arrheniusrsquo Theory of Partial Dis-sociation of Electrolytes American Chem Soc 1989

Hofmeister F Zur Lehre von der Wirkung der Salze Arch ExpPathol Pharmakol (Leipzig) 24 (1888) 247ndash260 translated inZur Lehre von der Wirkung der Salze (about the science of theeffect of salts Franz Hofmeisterrsquos historical papers edited byKunz W Henle J and Ninham B W Curr Opin Coll Inter-face Sci 9 19ndash37 2004

Holgate S T Samet J M Koren H S and Maynard R L Airpollution and health Academic Press San Diego 1999

Intergovernmental Panel on Climate Change (IPCC) ClimateChange 2007 The Physical Science Basis Contribution ofWorking Group I to the Fourth Assessment Report of the Inter-governmental Panel on Climate Change edited by Solomon SQin D Manning M Chen Z Marquis M Averyt K BTignor M and Miller H L Cambridge University Press Cam-bridge UK and New York NY USA 2007

IUPAC Solubility Data Series International Union of Pure and Ap-plied Chemistry Vol 1 to 53 were published by Pergamon PressOxford from 1979 to 1994 subsequent volumes were publishedby Oxford University Press Oxford

Jockel P Tost H Pozzer A Bruhl C Buchholz J GanzeveldL Hoor P Kerkweg A Lawrence M G Sander R SteilB Stiller G Tanarhte M Taraborrelli D van Aardenne Jand Lelieveld J The atmospheric chemistry general circulationmodel ECHAM5MESSy1 consistent simulation of ozone fromthe surface to the mesosphere Atmos Chem Phys 6 5067ndash5104 2006httpwwwatmos-chem-physnet650672006

Kim Y P Seinfeld J H and Saxena P Atmospheric gasaerosolequilibrium I Thermodynamic model Aerosol Sci Technol 19157ndash181 1993a

Kim Y P Seinfeld J H and Saxena P Atmospheric gasaerosolequilibrium II Analysis of common approximations and activitycoefficient calculation methods Aerosol Sci Technol 19 182ndash198 1993b

Kim Y P and Seinfeld J H Atmospheric gasaerosol equilibriumIII Thermodynamics of crustal elements Ca2+ K+ and Mg2+Aerosol Sci Technol 22 93ndash110 1995

Kohler H The nucleus in and the growth of hygroscopic dropletsTrans Faraday Soc 32 1152ndash1162 1936

Lelieveld J Berresheim H Borrmann S Crutzen P J Den-tener F J Fischer H Feichter J Flatau P J HelandJ Holzinger R Korrmann R Lawrence M G Levin ZMarkowicz K M Mihalopoulos N Minikin A RamanathanV de Reus M Roelofs G J Scheeren H A Sciare JSchlager H Schultz M Siegmund P Steil B Stephanou EG Stier P Traub M Warneke C Williams J and ZiereisH Global Air Pollution Crossroads over the Mediterranean Sci-ence 298 794ndash799 2002

Low R D H A generalized equation for the solution effect indroplet growth Atmos Sci 26 608ndash611 1969

Meng Z Seinfeld J H Saxena P and Kim Y P Atmo-spheric gasaerosol equilibrium IV Thermodynamics of carbon-ates Aerosol Sci Technol 131ndash154 1995

Metzger S M GasAerosol Partitioning A simplified Method

Atmos Chem Phys 7 3163ndash3193 2007 wwwatmos-chem-physnet731632007

S Metzger and J Lelieveld Reformulating atmospheric aerosol thermodynamics 3193

for Global Modeling PhD Thesis University Utrecht TheNetherlands ISBN90-393-2510-3httpigitur-archivelibraryuunldissertations1930853inhoudhtm 2000

Metzger S M Dentener F J Lelieveld J and PandisS N Gasaerosol partitioning I A computationally effi-cient model J Geophys Res 107(D16) ACH 16-1-24doi1010292001JD001102 2002

Metzger S Mihalopoulos N and Lelieveld J Importance ofmineral cations and organics in gas-aerosol partitioning of reac-tive nitrogen compounds case study based on MINOS resultsAtmos Chem Phys 6 2549ndash2567 2006httpwwwatmos-chem-physnet625492006

Mozurkewich M The dissociation constant of ammonium nitrateand its dependence on temperature relative humidity and particlesize Atmos Environ 27A 261ndash270 1993

Nenes A Pilinis C and Pandis S N Isorropia A newthermodynamic model for multiphase multicomponent inorganicaerosols Aquat Geochem 4 123ndash152 1998

Nernst W Z Phys Chem 4 129 1889Pilinis C Pandis S N and Seinfeld J H Sensitivity of direct

climate forcing by atmospheric aerosols to aerosol size and com-position J Geophys Res 100 18 739ndash18 754 1995

Pfeffer W Pflanzenphysiologie Leipzig Verlag von W Engel-mann 1881

Pruppacher H R and Klett J D Microphysics of Clouds andPrecipitation Dordrecht Holland D Reidel Publ Co 950 p1997

Raoult F M Z Phys Chem 2 353 1888Robinson R A and Stokes R H Electrolyte solutions 2nd Ed

Butterworths London 1965Rockner E Bauml G Bonaventura L Brokopf R Esch

M Giorgetta M Hagemann S Kirchner I KornbluehL Manzini E Rhodin A Schlese U Schulzweida Uand Tompkins A The atmospheric general circulation modelECHAM5 PART I Model description Max Planck Institutefor Meteorology MPI-Report 349httpwwwmpimetmpgdefileadminpublikationenReportsmaxscirep349pdf 2003

Seinfeld J H and Pandis S N Atmospheric chemistry andphysics J Wiley and Sons Inc New York 1998

Stokes R H and Robinson R A Interactions in aqueous non-electrolyte solutions I Solute solvent equilibria J Phys Chem70 2126ndash2130 1966

Tang I N and Munkelwitz H R Water activities densities andrefractive indices of aqueous sulfates and sodium nitrate dropletsof atmospheric importance J Geophys Res 99 18 801ndash18 8081994

ten Brink H M Veefkind J P Waijers-Ijpelaan A and van derHage J C Aerosol light-scattering in The Netherlands AtmosEnviron 30 4251ndash4261 1996

vanrsquot Hoff J H Die Rolle des osmotischen Druckes in der Analo-gie zwischen Losungen und Gasen Z Phys Chem 1 4811887

Vignati E Wilson J and Stier P M7 An efficientsize-resolved aerosol microphysics module for large-scaleaerosol transport models J Geophys Res 109 D22202doi1010292003JD004485 2004

Warneck P Chemistry of the Natural Atmosphere Internat Geo-phys Series 41 Academic Press Inc 1988

Winkler P The growth of atmospheric aerosol particles as a func-tion of the relative humidity ndash II An Improved concept of mixednucleii Aerosol Sci 4 373ndash387 1973

Winkler P The growth of aerosol particles with relative humidityPhysica Scripta 37 223ndash230 1988

Wexler A S and Potukuchi S Kinetics and Thermodynamicsof Tropospheric Aerosols Atmospheric Particles John Wiley ampSons Ltd 1998

Zadanovskii A B New methods of calculating solubilities of elec-trolytes in multicomponent systems Zhu Fiz Khim 22 1475ndash1485 1948

Zaveri R A Easter R C and Wexler A S A new method formulticomponent activity coefficients of electrolytes in aqueousatmospheric aerosols J Geophys Res-Atmos 110 D02201doi1010292004JD004681 2005

wwwatmos-chem-physnet731632007 Atmos Chem Phys 7 3163ndash3193 2007