[equilibrium 2] the multiple equilibrium model of micelle formation

7/30/2019 [Equilibrium 2] the Multiple Equilibrium Model of Micelle Formation

1/14

The Multiple Equilibrium Model of Micelle FormationAuthor(s): J. M. Corkill, J. F. Goodman, T. Walker and J. WyerReviewed work(s):Source: Proceedings of the Royal Society of London. Series A, Mathematical and PhysicalSciences, Vol. 312, No. 1509 (Sep. 2, 1969), pp. 243-255Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/2416220 .

Accessed: 03/03/2013 20:29

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of

content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms

of scholarship. For more information about JSTOR, please contact [email protected].

.

The Royal Society is collaborating with JSTOR to digitize, preserve and extend access to Proceedings of the

Royal Society of London. Series A, Mathematical and Physical Sciences.

http://www.jstor.org

This content downloaded on Sun, 3 Mar 2013 20:29:54 PMAll use subject to JSTOR Terms and Conditions
http://www.jstor.org/action/showPublisher?publisherCode=rslhttp://www.jstor.org/stable/2416220?origin=JSTOR-pdfhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/stable/2416220?origin=JSTOR-pdfhttp://www.jstor.org/action/showPublisher?publisherCode=rsl


2/14

Proc. Roy. Soc. A. 312, 243-255 (1969)Printed in Great Britain

The multiple equilibrium model of micelle formationBY J. M. CORKILL, J. F. GOODMAN, T. WALKER and J. WYER

Proctor and Gamble Limited, Basic Research Department,Newcastle upon Tyne

(Communicated by D. Tabor, F.R.S.-Received 28 November 1968-Revised 14 January 1969)

In dilute aqueous solution surface-active molecules aggregate to form micelles. This processhas been described by a multiple equilibrium model which considers changes in the distribu-tion of micelle aggregation numbers with concentration. Analysis of the colligative propertiesof dilute aqueous solutions of non-ionic surface-active agents has enabled the concentrationof monomer to be determined as a function of the total solute concentration. The validityof the treatment has been confirmed by showing that these monomer concentrations are ingood agreement with those deduced from nuclear magnetic resonance experiments, whichdo not depend on the assumptions employed in the thermodynamic method. Only onecolligative property need be determined as a function of concentration in order to describeadequately the average aggregation numbers and free energies of micelle formation.

INTRODUCTIONThe abrupt change in the colligative and other solution properties of surface-activemolecules in dilute aqueous solution at a critical concentration, known as thecritical micelle concentration (c.m.c.), is attributed to the formation of molecularaggregates known as micelles (see McBain I950). The description of this processhas been approached from two viewpoints, as a single kinetic equilibrium (Jones &Bury I927; Hartley I936) and as a phase separation (Stainsby & Alexander I950).Both models are limited by an oversimplification in the specification of the micellarspecies. Hall &Pethica (i 967) have applied Hill's (i 963, I964) small system thermo-dynamics to overcome this objection and have obtained exact thermodynamicrelations for micellar systems. The multiple equilibrium model provides an equallyrigorous approach for the thermodynamic treatment of micellization. This modelis developed here in a manner analogous to treatments of association in liquidmixtures (Prigogine & Defay I954) to obtain relationships between the averagethermodynamic properties of a micellar system; some of which are similar to thosederived for protein aggregation (Steiner I952; Adams &Williams I964). The theoryis used to obtain the concentration of monomeric solute as a function of the totalsolute concentration from thermodynamic measurements, and the validity of thetreatment confirmed by determining this quantity using a spectroscopic methodwhich is independent of thermodynamic assumptions. The variation of the surfacetension of a micellar solution with solute concentration calculated on the basis ofthis treatment agrees with values determined experimentally. Finally, the changein the standard free energy of micelle formation with aggregation number has beencalculated. [ 243

http://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsp


3/14

244 J. M. Corkill, J. F. Goodman, T. Walker and J. WyerTHEORY

1. Colligativepropertiesof micellarsolutionsIt is well known that the colligative properties of ideal solutions are determined

by the mole fraction of the solvent, xl. In a system in which the solute exists inboth the monomeric and associated states, we may define a solute colligative molefraction xc, by = = I xi, (1.1)where x2 and xr are the mole fractions of the monomeric and rth associated speciesrespectively and the summation extends over all aggregated species. The totalsolute concentration, C (g/ml), is related to these mole fractions by

C = (M2/Vm)X2+ E nrXr), (1.2)where M2 is the solute (monomer) molecular weight, Vm s the molar volume of thesolution and nr is the aggregation number of the rth micellar species. For dilutesolutions, Vmwill be indistinguishable from the solvent molar volume, Vl. In thesubsequent analysis, we shall find it useful to employ a 'stoichiometric' solute molefraction, xt, rather than C, where

Xt = X2+ EfnrXr. (1.3)We can now show that for systems in which the solute species are in chemical

equilibrium it is possible to determine x2 from the dependence of xc upon xt (or C).If ar and t2 are the chemical potentials of the rth micellar species and the monomerrespectively, we have for the equilibrium between the monomeric and any associatedspecies nr/2 = 1br (1.4)We choose the hypothetical standard states in which the solvated species have thepartial enthalpies and heat capacities corresponding to dilute real solutions andfree energies and entropies corresponding to unit activity (Lewis & Randall I96I).For solutions sufficiently dilute to behave ideally, we have, from (1.4),

ln xr =-(#0-nr?)/RT +nrln x2 (1.5)where ,arand 4a?are the standard chemical potentials of the associated and mono-meric species respectively, R is the gas constant and T the absolute temperature.By differentiation of (1.5) at constant temperature and pressure we have

dlnx2 nrxr (1.6)We multiply each side of this equation by nrkwhere k is a positive integer and sumwith respect to xr to obtain

d E, nrkx k +1(1dlnX -ow xrn (1.7)di 2If we now differentiate x, (equation (1.1)) with respect to InX2,substitute for the



4/14

Micelle formation 245derivative of the sum from (1.7) (k = 0) and introduce the definition of xt from(1.3) we have dxc/xt = dlnx2. (1.8)Integration of (1.8), using an experimentally determined relation between xc andxt, enables the change in lnx2 to be calculated. By choosing for the lower limits ofintegration solutions sufficiently dilute for association to be negligible (xc -- xt),x2 may be determined as a function of xc and hence xt (or C). The colligative molefraction may be determined from any suitable colligative property; thus in idealdilute solution from (1.1) pi/P1 = xi = 1-Xe, (1.9)where pi and p? are the equilibrium vapour pressures (strictly fugacities) of thesolvent in the solution and pure liquid states respectively.

2. Light scatteringAlthough we are dealing with a multi-component system, the condition of

chemical equilibrium between the solute species means that there is only onecompositional variable. The solute specific refractive increment is independent ofthe concentration above the c.m.c. in these systems and so we may assume that thespecific refractive increments of all solute species are the same. For these con-ditions, it is permissible to employ the two component light scattering equation(Debye I947) - HCRTV

(2.1)where T1 is the excess turbidity of the solution due to the solute, H is an opticalconstant containing the specific solute refractive increment and ir the solutionosmotic pressure. For ideal dilute solutions

7V1/RT = -ln x e, (2.2)By differentiation of this expression with respect to C and substitution into (2.1)we have from (1.2), after rearrangement

HC 1dxc 1 dxcT1 14d0 M2dx~~~~~~~~~ (2.3)V,dC M2 &t .

This equation enables either xc or x2 to be determined from the experimentaldependence of HC/I1 upon xt (or C). By choosing the lower limits of integration tocorrespond to solutions with negligible association (IICM2/1l -- 1) integration withrespect to xt enables xc to be calculated. By multiplying each side of (2.3) bydlnxt and M2we obtain from (1.8)

HCM2dlnxt = dxc = dInX2. (2.4)T d Xt

Integration of this expression enables x2 to be determined.Since the same information may be obtained from light scattering or colliga-tive property measurements, the choice of method is determined by the relative



5/14

246 J. M. Corkill, J. F. Goodman, T. Walker and J. Wyerexperimentalaccuracy.It has been found that vapour pressuredepressionmeasure-ments are the more sensitive when the degreeof association is low, light scatteringwhen it is high.

3. AggregationnumbersThe average association numbers are a set of parameters characterizingthedistribution of the solute between the monomeric and associated species. Withrespect to all solute species we may define number (N.) and weight (Nw) averageassociation numbersbyN - X2 -rXr=-T (from (1.1), (1.3)) (3.1)

X2+ >Xr Xc

and N = x2+nr(nrxr) (3.2)X2 + E nrXr

By differentiating xt with respect to lnx2 and substituting from (1.7) (k = 1) weobtain dxt/dlnx2 = x2+En2xr. (3.3)From (1.8) we have dx,/d lnx2 = x2+ EnrXr, (3.4)and hence, by division, N, = dxt/dx,. (3.5)We note than Nw may be obtained from either the dependence of xt upon xc ordirectly from light scattering data (equation (2.3)).

Average association numbers may also be defined with respect to the associatedsolute species. Thus we have for the number ()and weight (w)averages= rxr XtX2 (3.6)

EXr Xc-X2

w= >2nr(nrxr) =N-Xtx2 (3.7)If we differentiate (3.6) with respect to lnx2 we obtain, with the aid of (1.7) for thederivative of the sums, after rearrangement,

d/d lnX2 = (w-) . (3.8)Since for a system in which there is more than one associated species w> ,it follows that must increase with concentration since d lnx2 has the samesign as dxt.

4. Nuclear magnetic resonance chemical shiftsIf the resonance corresponding to a particular chemical group within a moleculehas a chemical shift Vj or a particular environment of the molecule, and if there areseveral possible environments with rapid exchange between them, then the observedchemical shift, P, is given by _ E


6/14

Micelle formation 247summation extends over all sites. If we assume that in solution the unassociatedmolecules have a shift of v2 and all associated species have the same shift vm, then

v X2 v2 +(EnrXr)vm (4.2)Xt

Rearrangement of this equation leads tox2=v2 v Xt. (4.3)

Thus, if Pm) P2 can be determined, the dependence of v upon xt leads to x2 as afunction of xt.5. Surface tensions

For a plane interface at constant temperature and pressure the Gibbs adsorptionisotherm may be written in the form

-dy = Fj d,j, (5.1)where y is the interfacial tension, F- and ,ajthe surface excess and chemical potentialof the jth solute species and the summation extends over all solute species.

For an associating solution, we have from (1.4)-dy = (F2 + E Frfnr)d/a2. (5.2)

As the surface excess Fr involves nr Fr moles of solute, the term in brackets in (5.2)is the total surface exceess (Ft) in terms of monomeric solute. We can express d,a2in terms of d nxt since dlnxt 1 dlnxt

dln2 =R1Tdlnx2' (5.3)and hence by substitution from (3.3) and the introduction of Nw (equation (3.2))we have - dy RTFt

dlnxt Nw(54If Ft can be determined, then y can be calculated from the relationship betweenNw and xt.

6. Standard ree energiesof associationIn the development of the thermodynamic theory it has proved useful to ignore

the individual micelles and consider only their average properties. If the averagechemical potential of the micellar species is defined as the mean Gibbs freeenergy per mole of the associated species then since

= ,arXr/ EXr, (6.1)it follows that = + RT. (6.2)

If we consider a mixture of the individual micellar species in their respectivestandard states in proportions corresponding to the distributions in a real solution,



7/14

248 J. M. Corkill, J. F. Goodman, T. Walker and J. Wyerthe free energy change of this mixing process will be

>Xr \X2r/mG=RT E Xr-n( Exr)= RT[-ln (E xr)]. (6.3)

The standard chemical potential of the associated species in the mixed state()is given by Kv0>-Kito> iMG, (6.4)and thus from equations (6.2) to (6.4) the free energy change for association permole of monomer referred to the mixed standard state is given by

2 R>-T (lnx2n x)) (6.5)Thus is related to quantities which may be determined experimentally.The converse problem, that of calculating the standard free energies for particularmicelles entirely on the basis of experimental results may in principle be solved bythe method given below.

From (1.5), by summation with respect to the associated species, we havea)

EXr = E>f(nr)exp(-nrs), (6.6)0wheref(O),f(l) are zero and for n > 2 (integer)

f(nr) = exp (-[? - nr/,tO]/RT) (6.7)and s=-lnx2. (6.8)The sum Exr is (XI - x2) and hence, from the experimental data, this may be obtainedas a function of s, say g(s). Equation (6.6) then becomes

co(s) = >f(nr) expHnr 8). (6.9)0The right hand side of this equation is related to the Laplace transform and from

the Fourier integral theorem (Sneddon I95I), it can be shown that1 Ifd+ir1f(nr) = 2ni)j g(s)exp(nrs)ds (nr > 0, integral), (6.10)

where a is a constant such that all singularities of g(s) in the complex plane lie tothe left of the path of integration.

Equation (6.10) provides a formal solution to the problem of obtaining the freeenergies of formation of the individual micelles from experimental data. In principlef(nr) may be obtained by fitting a suitable function of s to the relationship between



8/14

Micelle formation 249s and g(s) obtained from experimental results and applying (6.10). The approxima-tion of replacing the series summation (6.9) by an integrationand using the normalLaplace inversion to obtain a continuous approximation to f(nr) leads to largeerrors (- 20 %)when s small (- 5) for the simple trial function we investi-gated (f(nr) cx I/n +,8In- + A where a, /3,Z and A are constants).

EXPERIMENTALMaterials

The zwitterionic surface-active agent 3(dimethyloctylammonio)-propane--sulphonate (C8H17N Me2(CH2)3SO ) was chosen for this study. The change in thenuclear magnetic resonance chemical shift between the monomericand micellarstates is much largerfor a fluorine nucleus than for a proton (Muller& BirkhahnI967), and to enable 19Fnuclearmagnetic resonance experiments to be conductedthe seven terminalprotons were replacedby fluorineatoms to give the heptafluorocompoundC3F7(CH2)5N+Me2(CH2)3SOBoth compoundswere preparedfrom thecorrespondingbromidesby the following synthetic route.

CH2CH2CH2I IHNMe2 0~~S02RBr- 2RNMe2 -RN+Me2(CH2)3SOj

The products were recrystallizedfrom acetone and ethanol and gave the correctelemental analyses.The bromide used to preparethe heptafluorocompoundwas synthesized from4-pentene-1-oland heptafluoropropyl odide as follows:C3F71

CH2=CH(CH2)30H - C3F7CH2-CHI(CH2)30HPBr3 Zn/HCIC3F7(CH2)5Br PBr3 C3F7(CH2)50H

Thermodynamicmeasurements(i) LightscatteringAqueoussolutions of the fluorinatedsurface-active agent were clarifiedby filtra-tion through 'Millipore' filters (100nm pore size) and the intensity of scatteredlight at 25?Cmeasuredat angles of 45, 90 and 1350 to the incident beam by meansof a Sofica model 4200 P.G.D. instrument. The disymmetry (Z45) of the scatteredlight was less than 1.05 in all cases and the turbidities,T, were thereforecalculatedfrom the scattering at 900 on the basis of a benzene calibration (Coumou I960).Correctionsfor depolarizationof the scattered light were found to be negligible.Refractive index incrementswere determinedwith a Rayleigh interferometer.



9/14

250 J. M. Corkill, J. F. Goodman, T. Walker and J. Wyer(ii) Vapourpressureosmometry

The difference n temperature between a drop of solution and a drop of solventcontained in a constant temperature enclosure saturated with solvent vapour isproportional to the vapour pressure depression of the solution (Huff, McBain &Brady I95I). Thermistor beads are used to measure the temperature differenceand the apparatus calibrated with aqueous sucrose solution of known activity(Robinson &Stokes 1959). The Mechrolab 301 and Hitachi-Perkin-Elmer instru-ments were used for these measurements.The latter is more sensitive and wasrequiredto detect the small activity change in micellar solutionsof the fluorinatedcompound.(iii) Surface tension

Surface tensions of aqueous solutions of the fluorinated solute were measuredusing a drop volume apparatus which was immersed in a water thermostat(25.00+ 0.05?C). The drops were formed in a Teflon tip because difficulty wasexperienced in wetting correctly the glass tips normally used. The tip was attachedto a micrometer syringe held in an atmosphere saturatedwith solvent vapour andthe surfacetension was calculatedby the proceduredescribedby Harkins & Brown(I 9I9).

(iv) Nuclearmagneticresonance n.m.r.) measuremnentsN.m.r. measurementsfor the 19Fnucleus were carried out using a Varian DA 60system at a frequencyof 56.4MHz at 25?C.The external reference solution (40%trifluoroaceticacid in water) was contained in a 1mm capillaryand centred in thesample tube with Teflon bushes. The chemical shift of the terminal CF3 group ofC3F7(CH2)5N+Me2(CH2)3SOith reference to trifluoroacetic acid was measuredby using a side band calibrationwith a Hewlett Packard 200CD audiofrequencyoscillatormonitoredon a Vennerfrequencycounter.Chemicalshifts were correctedfor bulk susceptibility effects, which were found to be small.

RESULTSThe turbidities, T,of aqueous solutions of C3F7(CH2)5N+Me2(CH2)3SO3nd thecolligativemolefraction(xc)fromosmometrymeasurementsareshown as afunctionof the total solute concentration n figures1and2respectively.Valuesof xcobtainedby integratingthe light scattering data using equation (3.5) are in good agreementwith the values obtained directly by osmometry (figure 2), thus verifying theequivalence of the two methods. The chemical shift of the terminal CF3 groupis plotted as a function of the reciprocalconcentration (x-y1)n figure 3, and thevariation of the surface tension with the logarithm of the solute concentration(lgxt) in figure 4. The surface tension decreases with increasing concentration



10/14

Micelle formation 2516-

2-!;4 -

0 5 10 0 2 4 6102 C/gml-1 103XtFIGURE1 FIGURE2

FIGURE 1. Light scattering, C3F7(CH2)5N+(CH3)2(CH2)3SO-olutions, turbidity againstconcentration.FIGURE 2. Colligative properties, C3F7(CH2)5N+(CH3)2(CH2)3SO-olutions, xc against xt,0, Osmometry; , integration of light scattering results (calculated by using (3.5)).

300 -

250-

0 0.5 1.0 1.510-3x-1

FIGURE 3. Chemical shifts of the CF3 group forC3F7(CH2)5N+(CH3)2(CH2)3SO- solutions against x-1.



11/14

252 J. M. Corkill, J. F. Goodman, T. Walker and J. Wyerat all concentrations, but the decline is relatively small in micellar solutions(Xt > 1.5 x 10-3). Precise turbidity data could not be obtained for aqueous solutionsof C8H17N+Me2(CH2)3S0- because of the small aggregation numbers, but thevariation of xc with xt from osmometry measurements is shown in figure 5. The

36 -

34 -

0 -0_Q -0cn 32 -3.0 -2.5

lgxtFIGURE 4. Surface tension of C3F7(CH2)5N+(CH3)2(CH2)3SO- solutions against

lg xt. 0, Experimental; - -, calculated (by using (5.4)).

8-

4 -~~~~~~~~~~~~~

0 10 20103 Xt

FIGURE 5. Vapour pressure osmometry, C8Hl7N+-(CH3)2(CH2)3SOolutions.0, x. against xt; , x2 against xt (calculated by using (1.8)).

absence of any clearly defined discontinuity illustrates the difficulty in defininga c.m.c. precisely for surface-active agents with low aggregation numbers. Theplot of xc against xt shows deviations from ideal behaviour at concentrations wellbelow the c.m.c. region. A verification of the interpretation of the deviations fromideal behaviour in terms of solute aggregation was obtained by examining thehomologous compound C6Hl3N+Me2(CH2)3SO which was found to behave almostideally over the same concentration range.



12/14

Micelle formation 253DISCUSSION

Multiple equilibriummodelThe large changein the chemicalshift for the 19Fnucleus in the monomeric andmicellarstates enables x2to be determinedas a function of xt by a method that isindependent of thermodynamic assumptions. By extrapolation of the linearportions in figure 3, V2 and vmwere found to be 241.0 and 315.0 Hz respectivelyrelative to trifluoroaceticacid. The effect of micelle size on vmis assumed to benegligible.Valuesofx2as a functionofxtcalculated from (4.3)arein good agreementwith those obtained by light scattering measurementsby means of (2.4), as shownin figure 6. We concludethat the multipleequilibriummodel adequately describesthe aggregationof surface-activemoleculesin dilute aqueoussolution.

2-0

0 2 4 610wxt

FIGURE 6. The concentration of monomeric detergent, C3F7(CH2)5N+(CH3)2(CH2)3SO3,X2 against x,. 0, From n.m.r. results (calculated by using (4.3)); , from light scatter-ing results (calculated by using (2.4)).

SurfacetensionIt has been shown for a number of surface-active agents that a plot of surfacetension against the logarithm of the concentrationis linear well below the c.m.c.(van Voorst Vader I960) which is consistent with the experimental observationmadeby direct radiotracermeasurements hat the adsorbedmonolayer s essentiallycomplete at concentrations well below the c.m.c. (Corkill,Goodman, Ogden &Tate I963). Assuming that Ft is defined by the linear region of the relation of yagainst lg xt below the c.m.c. (figure4), the variation of y with xt over the wholeconcentration range may be calculated by using (5.4) and values of NRobtainedfrom light scattering experiments. The agreementwith the experimentallydeter-*mined alues of y against lg xt is good, showingthat the abruptchangein y againstxt observed at the c.m.c. is simply a consequence of the abrupt change in NR(Elworthy &Mysels I966).

I7 Vol. 312. A.



13/14

254 J. M. Corkill, J. F. Goodman, T. Walker and J. WyerAggregation numbers and free energies of micellization

For C8Hl7N+Me2(CH2)3SO-, x2 does not change abruptly at any well definedconcentration (figure 5) and values of and wcalculated from (3.6) and (3.7)are small and increase markedly with increasing concentration (figure 7). The largedifferencle between and wat any particular concentration shows that themicelles are polydisperse with respect to size.

10 w3>5

0 5 - /

0 10 201O3xt

FIGURE7. Aggregation numbers for C8H17N+(CH3)2(CH2)3SO3, (n>, and (n> against xt(calculated by using (3.6) and (3.7)).

As s small in the concentration range in which it can be accurately determinedthe integral approximation to the series inversion formula cannot be applied toobtain the dependence of the individual standard free energies of micelle formation(,u? nr/O) upon n,. However, the average free energy of micelle formation, withrespect to the mixed standard states can be obtained. The value of orC8H17N+Me2(CH2)3SO-decreases sharply with increasing from -2RT at = 2.5 at small values of . For = 8, has the value - 4.5RT and ismuch less dependent upon .Thus as the micelle size increases the free energy forthe addition of a further monomer molecule tends to become constant. For thesesmall micelles a considerable alkyl chain-water interaction in the core of the micelleis to be expected. As the association number increases the extent of alkyl chain-water interaction in the core region will decrease. The limiting value, approachedat high association numbers, represents the difference between the transfer of thehydrophobic chain from an aqueous to a non-polar environment and the freeenergy term associated with the packing of the head groups.The variation of x2with xt for C3F7(CH2)5N+Me2(CH2)3SOTs small at the higherconcentration (figure 6) and the concept of a critical micelle concentration(x* = 1.50 x 10-3) is reasonable. w alculated from equation (3.7) increases from20 to 25 over the concentration range studied above the c.m.c. Values calculatedusing the Debye approximation (Debye I949), where the micelles are considered to



14/14

Micelle formation 255be in a solvent definedby the solution composition at the c.m.c., are the same asthose calculated from (3.7) within the experimentalerror. For this compound,thestandard free energy of micellizationwith respect to the mixed standard state iswell approximated by AGO RTlnx (Corkill,Goodman & Tate I968), and is- 6.50RT at 25?C. Conclusions

The use of the multiple equilibrium model in the treatment of the properties ofmicellar solutions has shown how the various colligative properties may be relatedto the distribution of the solute between the monomeric and associated states.The connexion between the colligative and other (spectroscopic, surface) propertiesof the system has been demonstrated. Although a formally exact relationshipenabling the free energy of formation of a particular species to be calculated fromthe colligative properties has been deduced, mathematical difficulties have pre-vented us from applying it. It is, however, possible to obtain the average freeenergy of association as a function of the average association number. The behaviourof this quantity is consistent with the currently accepted model of the micellarstate.

REFERENCESAdams, E. T., Jun. & Williams J. W. I964 J Am. Chem. Soc. 86, 3454.Corkill, J. M., Goodman, J. F., Ogden, C. P. & Tate, J. R. I963 Proc. Roy. Soc. A 273, 84.Corkill, J. M., Goodman, J. F. & Tate, J. R. I968 Symposium on hydrogen bonded solvent

systems (University of Newcastle upon Tyne) (eds. A. K. Covington and P. Jones).London: Taylor and Francis.Coumou, D. T. I960 J. Colloid Sci. 15, 408.Debye, P. 1947 J. Phys. Coll. Chem. 51, 18.Debye, P. 1949 J. Phys. Coll. Chem. 53, 1.Elworthy, P. H. & Mysels, K. J. I966 J. Colloid Sci. 21, 331.Hall, D. G. &Pethica, B. A. I967 Nonionic surfactants (ed. M. J. Schick), ch. 16.London: Arnold.Harkins, W. D. & Brown, F. E. I919 J. Am. Chem. Soc. 41, 499.Hartley, G. S. I936 Aqueous solutions of paraffinic chain salts. Paris: Hermann.Hill, T. L. I963, I964 Thermodynamics of small systems, vols. 1, 2. New York: Benjamin.Huff, H., McBain, J. W. & Brady, A. P. 1951 J. Phys. Chem. 55, 311.Jones, E. R. & Bury, C. R. 1927 Phil. Mag. 4, 841.Lewis, G. N. & Randall, M. I96I Thermodynamics, ch. 20. London: McGraw-Hill.McBain, J. W. 1950 Colloid science, ch. 17. New York: Heath.Muller, N. & Birkhahn, R. H. I967 J. Phys. Chem. 71, 957.Prigogine, I. & Defay, R. 1954 Chemical thermodynamics (trans. by D. H. Everett), ch. 26.London: Longmans, Green and Co.Robinson, R. A. & Stokes, R. H. 1959 Electrolyte solutions, app. 8.6. London: Butterworths.Sneddon, I. N. 195I Fourier transforms, ch. 1. New York: McGraw-Hill.Stainsby, G. & Alexander, A. E. 1950 Trans. Faraday Soc. 46, 587.Steiner, R. F. 1952 Arch. Biochem. Biophys. 39, 333.van Voorst Vader, F. I960 Trans. Faraday Soc. 56, 1067.

I 7-2

[equilibrium 2] the multiple equilibrium model of micelle formation

Documents