Chapter 5. Simple mixtures

At equilibrium the chemical potential of a species is the same in every

phase.

How to express the chemical potential of a substance in terms of its mol

e fraction in a mixture in Raoult’s and Henry’s laws.

The effect of a solute on certain thermodynamic properties of a solution

(boiling point; melting point, osmosis ).

To express the chemical potential of a substance in a real mixture in ter

ms of a property is known as the activity.

Chemistry deals with mixtures, including mixtures of substances that ca

n react together.

at first stage, a binary mixtures (a mixture of two components) that do

not react together is considered.

The thermodynamic description of mixtures

5.1 Partial molar quantities:

(a) partial molar volume:

At 25oC, 1 mol H2O was added to a huge volume of water, the volume

increases by 18 cm3 18 cm3 mol-1 is the molar volume of pure water.

When 1 mol H2O was added to a huge volume of ethanol, the volume

increases by only 14 cm3 14 cm3 mol-1 is the molar volume in pure ethanol.

In general, the partial molar volume of a substance A in a mixture is the

change in volume per mole of A added to a large volume of the mixture.

The partial molar volumes of the components of a mixture vary with

composition because the environment of each type of molecule changes as

the compositions from pure A to pure B. modification of the mutual forces.

VJ = (V/nJ)p,T,n’

the subscript n’ signifies the amount of all

other substances present are constant.

Slope of a plot of V vs. nJ. VJ

VJ value is dependent on the composition.

When the composition of the mixture is changed by addition of dnA and dnB

dV = (V/nA)p,T,nBdnA + (V/nB)p,T,nA dnB= VA dnA + VB dnB

When the composition is held constant, as the amount of A and B are in

creased, the final volume of a mixture can be calculated by integration.

V = VA dnA + VB dnB = VA dnA + VB dnB

= VAnA + VBnB

Because V is a state function the final result is valid however the solution

is in fact prepared.

Illustration 5.1

A polynomial fit to measurements of the total volume of a water/ethanol mi

xture at 25oC that contains 1.00 kg of water.

V = 1002.93 + 54.6664 x – .36394 x2 + 0.028256 x3

VE = (V/nE) = (v/x)

dv/dx = 54.66664 – 2(0.36394) x

+ 0.084768 x2

Molar volumes are always positive

but partial molar volume quantities

need not be.

Exp. The limiting partial molar volume

of MgSO4 in water is – 1.4 cm3 mol-1.

The salts break up the open structure

of water as ions become hydrated.

(b) Partial molar Gibbs energies

The chemical potential energy is defined as the partial molar Gibbs energy.

J = (G/nJ)p,T,n’

The chemical potential is the slope

of the Gibbs energy against the

amount of the component J

The total Gibbs energy of a mixture is

G = nA A + nB B.

The fundamental equation of chemical

thermodynamics relates the change

in Gibbs energy to changes in pressure,

temperature, and composition:

dG = Vdp – SdT + AdnA + BdnB + .

At constant pressure and temperature

dG = AdnA + BdnB + .

dG = dwadd, max

dwadd,max = AdnA + BdnB + .

The additional work (non-expansion, electrical work) can arise from the changing com

position of a system (or electrochemical cell).

(c) The wider significance of the chemical potential:

G = U + pV – TS ; U = – pV + TS + G

dU = – pdV – v dp + SdT + TdS + Vdp – SdT + AdnA + BdnB + .

= – pdV + TdS + AdnA + BdnB + .

At constant S and V; dU = AdnA + BdnB + .

J = (U/nJ)S,V,n’

J = (H/nJ)S,p,n’ ; J = (U/nJ)V,T,n’

(d) The Gibbs-Duhem equation:

At constant pressure and temperature, the total Gibbs energy of a binary mixture is

G = nA A + nB B

dG = AdnA + BdnB + nAdA + nBdB

dG = AdnA + BdnB

nAdA + nBdB = 0

The Gibbs–Duhem equation is J nJdμJ = 0.

In a system in which the compositions have mutual interaction or are mutually excha

ngeable, the chemical potential of one component can not change independently of the

chemical potentials of the other components.

In a binary mixture,

dB = – (nA/nB) dA

The same line of reasoning applies to all partial molar quantities.

Exp. Where the Vwater increases, the Vethanol decreases.

In practice, the Gibbs-Duhem equation is used to determine the partial molar proper

ties of one component in a binary mixture from the second component.

Example 5.1 Using the Gibbs-Duhem equation:

The experimental values of the partial molar volume of K2SO4(aq) at 298 K are fou

nd to fit the expression: vB = 32.280 + 18.216 x1/2

x is the numerical value of the morality of K2SO4 (x =b/b).

Use the Gibbs-Duhem equation to derive an equation for molar volume of

water in the solution. vwater = 18.079 cm3 mol-1

nA dvA + nB dvB = 0 dvA = – (nA/nB) dvB

vA = vA* – (nA/nB) dvB ;

vA*: vA/cm3 mol-1 is the numerical value of the molar volume of pure A.

dvB/dx = 9.108 x–1/2 vA = vA* – 9.108 o (nA/nB) x–1/2 dx

nB/nA = nB/(1 kg)/MA = nBMA/1kg = bMA = x bMA

vA = vA* – 9.108 MAb o x1/2 dx = vA*– 2/3 {9.108 MAb(b/b)3/2}

b/b

b/b

Example 5.1 Using the Gibbs-Duhem equation:

vA /(cm3 mol-1) = 18.079 – 0.1094 (b/b)3/2

5.2 The thermodynamics of mixing:

At constant temperature and pressure,

systems tend toward lower Gibbs energy.

A spontaneous mixing process is that of

two gases introduced into the same container.

The mixing is spontaneous, so it must

correspond to a decrease in G.

(a) The Gibbs energy of mixing of perfect gases:

= Gm = + RT ln p/p

: standard chemical potential, the of the

pure gas at 1.0 bar.

For simpler notation, the p/p is replaced by p

p: the numerical value of p in bars.

= + RT ln p

The Gibbs energy of the total system:

Gi = nAA + nBB = nA (A + RT ln p) + nB (B

+ RT ln p)

After mixing the partial pressure of the gases are pA and pB with pA + pB = p.

Gf = nA (A + RT ln pA) + nB (B

+ RT ln pB)

Gmix = nA RT ln (pA/p) + nB RT ln (pB/p)

Use the relation between partial pressure and mole fraction; pJ/p = J

Gmix = nRT (A ln A + B ln B)

Because J < 1, Gmix < 0

Gmix is negative for all compositions.

Exp. 5.2 Calculate a Gibbs energy of mixing

Gi = (3.0 mol) ((H2) + RT ln 3p)

+ (1.0 mol) ((N2) + RT ln p)

Gf = (3.0 mol) ((H2) + RT ln 3/2 p)

+ (1.0 mol) ((N2) + RT ln ½ p)

Gmix = (3.0 mol) RT ln (3/2 p/3p)

+ (1.0 mol) RT ln (½ p/p)

= – 6.9 kJ

Gmix = the mixing itself and the changes in pressure of the two gases.

When 3.0 mol H2 mixes with 1.0 mol N2 at same pressure, with the volumes of the

vessels adjusted accordingly, Gmix = – 5.6 kJ

(b) Other thermodynamic mixing functions:

(G/T)p,n = – S

The entropy of mixing of two perfect gases

is given by

ΔmixS = (mixG/T)p,nA,nB

= –nR(xA ln xA + xB ln xB) > 0

For perfect gases, the enthalpy of mixing

ΔmixH = 0.

The enthalpy of mixing is zero because there

are no interactions between molecules forming

in the gas mixture.

5.3 The chemical potentials of liquids:

We should denoted quantities to pure substances by a superscript *.

A* is the chemical potential of the liquid A*(l).

A* = A0 + RT ln pA* (relative pressure pA* = pA/p0)

If another substance (solute) is

also present in the liquid,

the chemical potential is changed to A.

A = A0 + RT ln pA (pA = pA/p0)

A0 = A* – RT ln pA*

A = A* – RT ln pA* + RT ln pA

= A*+ RT ln pA/pA*

In a series of experiments on mixtures of closely related liquids, the partial vapor

pressure of each component to its vapor pressure of pure liquid, pA/pA* is

approximately equal to the mole fraction of A in the liquid mixture.

Raoult’s law: pA = ApA*

Mixtures that obey the law throughout the composition range from pure A to pure B

are called ideal solution.

A = A* + RT ln A

Molecular interpretation 5.1

The presence of a second component reduces

the rate at which A molecules leave the surface

of the liquid but does not inhibit the return rate.

rate of vaporization = k A

rate of condensation = k’pA

At equilibrium, k A = k’pA

pA = (k/k’) A

For pure liquid, A = 1 pA* = (k/k’)

Some solutions depart significantly from Raoult’s law.

The law is therefore a good approximation for the properties of the solvent if the

solution is dilute.

(b) Ideal-dilute solution:

William Henry: for real solutions at low concentrations, although the vapor pressure

is proportional to its mole fraction, the

constant of proportionality is not the

vapor pressure of the pure substance.

pB = BKB

Mixtures for which the solute obeys

Henry’s law and the solvent obeys

Raoult’s law, are called ideal-dilute

solutions.

In a dilute solution:

The solvent molecules are in an environment very much like the one they have in

the pure liquid.

In contrast, the solute molecules are surrounded by the solvent molecules, which

is entirely different from their environment when pure.

Unless the solvent and solute molecules

happen to very similar.

Raoult’s law.

Example. Investigate the validity of Raoult’s and Henry’s laws:

Rault’s law: pJ = JpJ*

Henry’s law: pJ = J KJ

KJ is an empirical constant chosen

so that the plot of the vapor pressure

of J against its molar fraction is tangent

to the experimental curve at J = 0.

K(propane) = 23.3 kPa

K(chloroform) = 22.0 kPa

In practice, Henry’s law is expressed in terms of mobility, b, of the solute.

pB = bB KB

respiration: oxygen, diving, mountaineering, anesthetics.

I5.1 Gas solubility and breathing:

We inhale about 500 cm3 of air with each breath we take.

As the diaphragm (橫隔膜 ) is depressed and chest expands a decrease in

pressure of 100 Pa related to atmospheric pressure.

The total volume in the lungs is about 6 dm3 (L).

Some air remains in the lungs at all time to prevent the collapse of the alveoli (肺

泡 ).

Alveolar gas is in fact a mixture of newly inhaled air and air about to be exhaled.

pO2 in arterial blood is about 40 Torr pO2 in freshly inhaled air is about 104 Torr.

Arterial blood remains in the capillary passing through the wall of the alveolus for ab

out 0.75 s, but such is the steepness of the pressure gradient that it becomes fully sa

turated with oxygen in about 0.25 s.

If the lungs collect fluid, then the respiratory membrane thickens, diffusion is greatly

slow, and body tissue begin to suffer from oxygen starvation.

CO2 moves in the opposite direction across the respiratory tissue, but the pCO2 gr

adient is much less, corresponding to about 5 Torr in blood and 40 Torr in air at eq

uilibrium.

Because the CO2 is much more soluble in the alveolar fluid than oxygen is, equal a

mount of oxygen and carbon dioxide are exchanged in each breath.

A hyperbaric oxygen chamber, high pO2, is used to treat certain types of disease,

such as CO poisoning, disease caused by anaerobic bacteria.

In scuba diving, one unfortunate consequence of breathing air at high pressure is

that N2 is much more soluble in fatty tissue than in water, so it tends to dissolve in ce

ntral nervous system, bone marrow, and fat reserves.

nitrogen narcosis (氮醉 ).

The Properties of Solutions:

5.4 Liquid mixtures:

(a) Ideal solution

The Gibbs energy of mixing of two liquids to form an ideal solution is calculated

as in exactly the same way as for two gases.

Before mixing: Gi = nAA* + nBB*

After mixing: Gf = nA {A* + RTln A} + nB {B* + RTln B}

mixG = nRT {A RTln A + B RTln B} ; n = nA + nB

mixS = nR {A RTln A + B RTln B}

mixH = mixG + TmixS = 0

The mixV = 0 (G/p)T = V ( mixG/p)T = mixV = 0

mixG is independent of pressure.

In a perfect gas there are no forces acting between molecules

In an ideal solutions, there are interactions, but the average energy of A–B interactio

ns in the mixture is the same as the average energy of A–A and B–B interactions in t

he pure liquids.

Real solution:

No only may there be enthalpy and volume changes when liquids mix, but there ma

y also be an additional contribution to the entropy arising from the way in which the m

olecules of one type might cluster together instead of mingling freely with the others.

Large and positive mixH or if mixS << 0, then the mixG might be positive for mixing.

separation is spontaneous and the liquid may be immiscible.

When they are miscible only over a certain range compositions, the liquid might

be partially miscible.

(b) Excess functions and regular solutions:

An excess function (XE) is the difference

between the observed thermodynamic

function of mixing and the function for

an ideal solution.

SE = mixS – mixSideal

A regular solution: as one in which two

kinds of molecules are distributed randomly but

have different energies of interactions with each

other.

HE 0 but SE = 0.

Suppose the HE depends on compositions:

HE = nRTAB

HE = nRTAB

is a dimensionless parameter that is a

measure of the energy AB interactions

related to that of the AA and BB interactions.

If < 0, mixing is exothermic and

solute-solvent interactions are

more favourable than solvent-solvent

and solute-solute interactions.

If > 0, mixing is endothermic.

For a regular solution:

mixG = nRT {AlnA + BlnB + AB}

mixG = nRT {AlnA + BlnB + AB}

The important feature is that > 2

the graph shows two minima separated

by a maximum.

At >2, the system will separate

spontaneously into two phases with

compositions corresponding to the

two minima.

5.5 Colligative properties: (眾數性質 )

A colligative property is a property that depends only on the number of solute

particles present, not their identity.

the elevation of boiling point, the depression of freezing point, osmotic pressure.

Throughout the following:

(1). The solute is not volatile.

(2). The solute does not dissolve in the solid solvent.

(a) The common feature of colligative properties:

For an ideal-dilute solution, the reduction of the chemical potential of the solvent is

from A* for pure solvent to A* + RT lnA when a solute is present.

There is no direct influence of the solute on the chemical potential of the solvent v

apor and the solid solvent because the solute appears in neither the vapor nor the s

olid.

A reduction in chemical potential of solvent

boiling point is raised

and freezing point is lowered.

Molecular interpretation 5.2

The lowering of the chemical potential is not the energy of interaction of the solute

and solvent particles, because it occurs even in an ideal solution.

If it is not an enthalpy effect,

it must be an entropy effect.

S = kB ln

The vapor pressure reflects the tendency

of the solution towards greater entropy.

When a solute is present, there is an

additional contribution to the entropy of

the liquid, even in an ideal solution.

S(solution gas) < S(solvent gas)

lower vapor pressure

and higher boiling point.

(b) The elevation of boiling point:

The heterogeneous equilibrium between

the solvent vapor and the solvent in solution

A*(g) = A*(l) + RT ln A

The presence of a solute at a mole fraction

B causes an increase in normal boiling

point from T* to T* + T,

T = KB ; K = RT*2/ vapH

Justification 5.1

ln A = (A*(g) – A*(l))/RT = vapG/RT

Gibbs-Helmholtz equation: ((G/T)/T)p = – H/T2

d lnA /dT = 1/R d (vapG/T)/dT = – vapH/RT2

0 d lnA = – (1/R) T* vapH/T2 dT

ln A = ln(1 – B ) = vapH/R (1/T – 1/T*)

When B << 1, ln(1 – B) – B

B = vapH/R (1/T* – 1/T)

1/T* – 1/T = (T – T*)/TT* T/T*2

lnA T

For practical applications, we note that the molar fraction of B is proportional to its

molality, b, in the solution.

T = Kb b

(c) The depression of freezing point:

The heterogeneous equilibrium between

pure solid solvent and solution with

solute present at a mole fraction B:

A*(s) = A*(l) + RT ln A

T = K’B ; K’ = RT*2/ fusH

T = Kf b

Cryoscopy (Greek "freeze-viewing") :

A technique for determining the molecular weight of a solute by dissolving a known

quantity of it in a solvent and recording the amount by which the freezing point of

the solvent drops.

(d) Solubility:

Solubility is not strictly a colligative

property (identity of the solute).

Saturation is a state of equilibrium, with the

undissolved solute in equilibrium with

the dissolved solute.

In solution B* = B*(l) + RT ln B

B*(s) = B*(l) + RT ln B

RT ln B = {B*(s) – B*(l)}/RT

= – fusG/RT

0 d lnB = – (1/R) Tf fusH/T2 dT

ln B = fusH/R (1/Tf – 1/T)

The solubility of B decreases exponentially as the temperature is lowered from

its melting point.

High melting point, large enthalpies

low solubility.

This predict fails to predict that solutes

will have different solubilities in different

solvents, for no solvent properties appear

in the expression.

(e) Osmosis:

Osmosis is the spontaneous passage of a pure solvent into a solution separated

from it by a semipermeable membrane, a membrane permeable to the solvent but

not to the solute.

The osmotic pressure, , is the pressure that must be applied to the solution to

stop the influx of solvent.

transport of fluid through cell membrane,

dialysis (透析 ), osmometry (the determination

of molar mass of macromolecules).

Equilibrium is reached when the hydrostatic

pressure of the column of solution match the

osmotic pressure.

The chemical potential of the solvent is lowered by the solute, but it is restored to

its “pure” value by the application of pressure.

The van ’t Hoff equation for the osmotic pressure is Π = [B]RT

[B] = the molar concentration of the solute.

Justification 5.3

On the pure solvent side: A*(p)

One the solution side:

A(A, p + Π) = *A(p + Π) + RT lnA

*A(p + Π) = *A(p) + p Vm dp

A(A, p + Π) =

*A(p) + p Vm dp + RT lnA

– RT lnA = p Vm dp Π = [B]RT

Because the effect is so readily measurable and large, osometry is commonly use

d to measure molar mass of macromolecules (exp. Protein…).

As these huge molecules dissolve to produce solutions that are far from ideal,

Π = [J]RT {1+ B[J] + …………}

B: osmotic viral coefficient.

Exp. 5.4 Osmotic pressure of PVC:

c (g dm–3) vs. h/cm

[J] = c/M; osmotic pressure = gh

h/c = RT/ gM (1 + Bc/M + …….)

= RT/ gM + (RTB/ gM) c

To find M, h/c vs. c , and expect

a straight line with intercept RT/ gM at c = 0.

1.2 x 102 kg mol-1 = 120 kDa (1 Da = 1g mol-1)

I5.2 Osmosis in physiology and Biochemistry:

Cell membranes are semipermeable and allow water, small molecules, and

hydrated ions to pass, while blocking the passage of biopolymers.

The influx of water also keeps the cell swollen, whereas dehydration causes the

cell to shrink.

To maintain the integrity of the cells, solution injected into the blood must be

isotonic with the blood. the same osmotic pressure.

Dialysis: a common technique for the removal of impurities from solution of

biological macromolecules and for the study of binding of small molecules to

macromolecules.

In a purification experiments, a solution of macromolecules and impurities (ions

and small proteins) is place in a bag made of semipermeable membrane, and the

bag is immersed in a solvent.

In a binding experiment, a solution contains a macromolecules ([M]) and smaller

molecules A ( the total concentration of A = [A]in) in a bag.

The total concentration A is the sum of [A]in = [A]free and [A]bound.

At equilibrium, the chemical potential of free A in macromolecule solution

is equal to that in the solution on the other side.

[A]free = [A]out

(the activity coefficient of A is the same in both solutions)

The [A]bound = [A]in – [A]out

The average number of A molecules bound to M molecules, v, is then the ratio

v = [A]bound/[M] = ([A]in – [A]out)/M

If there are N identical and independent binding sites on each macromolecule, e

ach macromolecule behaves like N smaller macromolecules, M’, with same value of

K for each site.

The bound and unbound A molecule to

each binding site in the macromolecule

are in equilibrium, M’ + A M’A

K = [M’A]/[M’]free[A]free

[M’A]/[M’]free = (v/N) / (1 – v/N)

K = v/N/{(1 – v/N) [A]out}

v/[A]out = KN – Kv (Scatchard equation)

If a straight line is not obtained we can

conclude that the binding sites are not

equivalent or independent.

Activities

Account the deviations from ideal behavior.

It is important to be aware of the different definitions of standard states and activities.

5.6 The solvent activity

The general form of the chemical potential of a real or ideal solvent is:

A = A* + RT ln (pA/pA*)

For the ideal solution: A = A* + RT ln A

The solvent activity of the real solution is related to its chemical potential by

A = A* + RT ln aA

aA is the activity of A, a kind of ‘effective’ mole fraction.

The activity is defined as aA = pA/pA*.

Illustration 5.3 The vapour pressure of 0.500 M KNO3 at 100oC is 99.95 kPa, so the

activity of the solution at this temperature is

aA = 99.95 kPa/ 101.325 kPa = 0.9864

Because all solvents obey the Raoult’s law at A 1

aA A as A 1

A convenient way of expressing this convergence is to introduce the activity

coefficient, aA = A A A 1 as A 1

A = A* + RT ln A + RT ln A

5.7 The solute activity:

For solutes, they approach ideal-dilute (Henry’s law) behavior as B 0.

(a) Ideal-dilute solutions: (Henry’s law: pB = KBB)

The chemical potential of solute, B:

B = B* + RT ln (pB/pB*) = B* + RT ln (KB/pB*) + RT ln B

a new standard chemical potential B0 = B* + RT ln (KB/pB*)

B = B0 + RT ln B

For ideal solution: KB = pB* ; B0 = B*

(b) Real solution:

We now permit deviations from ideal-dilute solution:

B = B0 + RT ln aB ; aB = pB/KB

It is sensible to introduce an activity coefficient through

aB = BB

Because the solute obeys Henry’s law: [solute] 0

aB B and B 1 as B 0.

Deviations of the solute form ideality disappear as zero concentration is

approached.

Example 5.5 Measuring activity

c (chloroform); pc/kPa ; pA (acetone)/kPa

When chloroform is solvent: aC = pC/pC*

If it is solute: aC = pc/Kc ; C = aC/C

Raoult’s law Henry’s law

(c) Activities in terms of molarities:

In chemistry, compositions are often expressed as molalities, b, in place of mole

fractions.

B = B0 + RT ln bB

The chemical potential of the solute has its standard value of o when molality of

B is equal to bo (1.0 mol kg-1).

as bB 0 , B –

The solute becomes increasingly stabilized.

In practical consequence, it is very difficult to remove the last trace of a solute

from a solution.

aB = B (bB/b0) where B 1 as bB 0

For the chemical potential of a real solution at any molality:

= 0 + RT ln a

(d) The biological standard state:

The conventional standard state of hydrogen ions (pH = 0; [H+] = 1.0 M) is not

appropriate to normal biological conditions.

In biochemistry, the biological standard state at pH = 7.0 is common to be

adapted.

The corresponding standard thermodynamic functions as G; H; ; S.

The biological standard state (pH = 7) is related to the thermodynamic standard

state by H+ = H+

0 + RTln aH+ = H+0 – 7RT ln 10.

At 298 K, H+0 – H+

40 kJ mol-1.

5.8 The activities of regular solutions

The original of deviations from Raoult’s law and the its relation to activity

coefficient.

The Gibbs energy of mixing to form a nonideal solution is

mixG = nRT {A RTln aA + B RTln aB}

If each activity is replaced by

mixG = nRT {A RTln A + B RTln B + A ln A + B ln B}

mixG = nRT {AlnA + BlnB + AB}

mixG = nRT {AlnA + BlnB + AB(A + B)}

mixG = nRT {AlnA + BlnB + AB2 + BA

2}

ln A = B2 ; ln B = A

2 (Margules equations)

A 1 as B 0 and B 1 as A 0

The activity of A:

aA = A A = A exp(B2) = A exp{(1 – A)2}

pA = pA* [A exp{(1 – A)2}]

> 0 (endothermic mixing, unfavourable

solute-solvent interaction), give vapour

pressure higher than ideal ( = 0 ).

All the curves approach linearity and coincide

with Raoult’s law as A 1.

When A << 1 ; pA = pA* (exp) A

pA = KAA (Henry’s law)

K = pA* exp

5.9 The activities of ions in solution

Interactions between ions are so strong that the approximation of replacing activit

ies by molalities is valid only in very dilute solutions (< 10-3 m in total ion concentratio

n) and in precise work activities themselves must be used.

(a) Mean activity coefficients

The chemical potential of a univalent cation M+ is denoted as + and that of a

univalent anion X– is denoted as –, the Gibbs energy of the ions in the electrically

neutral solution is the sum of these partial molar quantities.

Gm = +ideal + –

ideal

For a real solution,

Gm = + + – = +ideal + RT ln + + –

ideal + RT ln –

= Gmideal + RT ln

There is no experimental way of separating the product +– into contributions from

cations and anions.

For a 1,1-electrolyte, we introduce the mean activity coefficient as geometric m

ean of the individual coefficients:

= (+–)½

The individual chemical potentials of the ions as

+ = +ideal + RT ln ; – = –

ideal + RT ln

The nonideality is shared equally.

This approach to the case of a compound MpXq ( p cations + q anions):

The molar Gibbs energy of ions is the sum of their partial molar Gibbs energies:

Gm = p + + q – = Gmideal + p RTln + + q RT ln–

= (+p–

q)1/s ; s = p + q

i = iideal + RT ln G = p + + q –

Both types of ion now share equal responsibility for the nonideality.

(b) The Debye-Hückel limiting law:

The long range and strength of the Columbic interaction between ions means

that it is likely to be primarily responsible for the departures form ideality in ionic

solutions and to dominate all other contributions to nonideality.

The Debye–Hückel theory (1923):

Oppositely charged ions attract one another.

As a result, anions more likely to be found near

cations in solution, and vice versa.

The time-average, spherical haze around the

central ion, in which counter ions outnumber ions

of the same charge as the central ions, has

a net charge equal in magnitude but opposite

in sign to that on central ion, and is called its ionic atmosphere.

The energy, and chemical potential of any given central ion is lowered as a result

of its electrostatic interaction with its ionic atmosphere.

The stabilization of ions by their interaction with their ionic atmospheres is part of

the explanation why chemists commonly use dilute solutions, in which stabilization is

less important, to achieve precipitation of ions from electrolyte solutions.

The Debye–Hückel limiting law is

log = – |z+z-| A I1/2 ; where I is the ionic strength,

I = ½ i zi2(bi/b0). (dimensionless)

at 25oC, A = 0.509 for an aqueous solution.

For a solution consisting of two types of ion at molalities b+ and b–

I = ½ (b+z+2 + b–z–

2)/b0

The mean activity coefficient of 5.0 x 10-3 mol kg-1 KCl(aq) at 25oC.

I = ½ (b+ + b–)/b0 = b/b0 (b+ = b– = b) ;

log γ± = – 0.509 x (5.0 x 10-3)1/2 = – 0.036 = 0.92.

The ionic strength emphasizes the charges of the ions because the charge

numbers occur as their square.

Ionic solutions of moderate molalities may have activity coefficients that different

from the values given by the eq. 5.69, yet all solutions are expected to conform as b

0.

The agreement at very low molalities

(less than 1 mmol kg-1, depending on

charge type) is impressive, and convincing

evidence in support of the model.

(c) The extended Debye-Hückel law

The extended Debye–Hückel law is

ln ± = – |z+z–|AI1/2/(1 + BI1/2) + CI. .

The equation accounts for some

activity coefficients over a moderate

range of dilute solution (up to about

0.1 mol kg-2); nevertheless it remains

very poor near 1.0 mol kg-1. .