On the calculation of thermodynamic properties of electrolyte solutions fromKirkwood–Buff theoryRaghunath Behera Citation: The Journal of Chemical Physics 108, 3373 (1998); doi: 10.1063/1.475738 View online: http://dx.doi.org/10.1063/1.475738 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/108/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Restricted primitive model for electrolyte solutions in slit-like pores with grafted chains: Microscopic structure,thermodynamics of adsorption, and electric properties from a density functional approach J. Chem. Phys. 138, 204715 (2013); 10.1063/1.4807777 Primitive models of ions in solution from molecular descriptions: A perturbation approach J. Chem. Phys. 135, 234509 (2011); 10.1063/1.3668098 Kirkwood–Buff integrals for ideal solutions J. Chem. Phys. 132, 164501 (2010); 10.1063/1.3398466 Thermodynamics of electrolyte solutions in the modified mean spherical approximation J. Chem. Phys. 113, 292 (2000); 10.1063/1.481794 Pseudolattice theory of strong electrolyte solutions J. Chem. Phys. 107, 6415 (1997); 10.1063/1.474301

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NOTES

On the calculation of thermodynamic properties of electrolyte solutionsfrom Kirkwood–Buff theory

Raghunath Beheraa)

Department of Chemistry, Indian Institute of Technology, Kanpur 208016, Uttar Pradesh, India

~Received 14 October 1997; accepted 18 November 1997!

@S0021-9606~98!50308-X#

Thermodynamic properties of multicomponent systemscan be calculated from Kirkwood and Buff’s statistical me-chanical theory of solutions.1,2 For m component system,these relations are1

A5ABA, ~1!

where, A and B are symmetric square matrices of dimensionm with elements

Aab5~1/kBT!~]ma /]Nb!Ng5~V/kBT!~]ma /]rb!rg

,

Bab5radab1rarbGab5~kBT/V!~]ra /]mb!mg, ~2!

with Gab5*hab(r )d3r , ra5Na /V is the number density ofthe speciesa. The subscriptmg ~or Ng ,rg) means allm ’s~or N’s, r ’s! except the one indicated in the differentiation.ConstantT andV are assumed throughout. The elements ofA matrix are related to various thermodynamic properties.1

When each species is an independently variable compo-nent the matrixB is invertible and the solution of Eq.~1! canbe given byA5B21. But, for ionic solutions when the ionsare treated as components, the condition of electroneutralitymakes some ionic components linearly dependent makingBa singular matrix, and thus the solutionA5B21 does notexist. This problem of non-invertibility ofB for ionic solu-tions was first observed and handled by Friedman andRamanathan3 by treating the salts and not the ions as com-ponents. Later on, taking ions as components Kusalik andPatey,4 overcome the difficulty by definingk-dependent ana-logs of Kirkwood–Buff equations and taking the appropriatek→0 limits analytically.

We note that, whenB21 exists, then Eq.~1! has uniquesolution, viz.A5B21. However, whenB21 does not exist,we have to solve Eq.~1! and the solution is not unique. Theform of Eq. ~1! states5 that B is a generalized inverse~g-inverse! of A. Friedman2 describes the relationship ofAand B as complicated. The general solution of Eq.~1! isgiven by @Theorem 3.4.1 of Ref. 5#

A5C~DBC!r2D, ~3!

where,C(n3p! andD(q3m! are arbitrary matrices;Xr2 is

the reflexive g-inverse~a particular kind ofg-inverse! of X.

For different choices ofC andD, various kinds ofg-inversescan be obtained. Thus, there exists the problem of non-uniqueness of the solutions, as stated earlier. But as we willsee, this non-uniqueness of solution ofA does not affect thethermodynamic properties. We will show this by taking twospecificg-inverses~and hence two specific sets ofC andD!,viz. the reflexive generalized~RG! inverse and the Moore–Penrose~MP! inverse. The corresponding choices ofC andD are5 ~i! C5D5I ~the unit matrix! for RG inverse, and~ii !C5B* B, D5B* ~the conjugate transpose ofB! for MPinverse.

Now we find the RG and MP inverse ofB ~denoted asBr

2 andB†, respectively!. As an example, we take the mul-ticomponent system, same as taken by Kusalik and Patey,4

consisting of a non-ionic solvent~1! and a salt~s!, one mole ofwhich, in solution producesn1 moles of positive ion~1! andn2 moles of negative ion~2!. For this system, matrixB afterimposing condition of electroneutrality4 can be written as:

B5S B11 B11 tB11

B11 B12 tB12

tB11 tB12 t2B12

D , ~4!

where, Bi , j ~i , j 51, 1, 2! are defined in Eq.~2!, t5r2 /r1 . We calculate the RG and the MP inverse ofB byPartitioned and Rank Factorizing method, respectively@Chap. 11 of Ref. 5# as:

Br25

1

D rS B12 B11 0

2B11 B11 0

0 0 0D,

~5!

B†51

DmS t8B12 2B11 2tB11

2B11 B11/t8 tB11/t8

2tB11 tB11/t8 t2B11/t8D ,

where, t8511t2, D r5B11B122B112 and Dm5t8D r . We

note thatBr2 andB† are the solutions ofA matrix defined in

Eq. ~1!. Thus, it is now possible to calculate the variousthermodynamic properties either fromBr

2 or from B†. Wefound thermodynamic properties~partial molar volumes, iso-

JOURNAL OF CHEMICAL PHYSICS VOLUME 108, NUMBER 8 22 FEBRUARY 1998

LETTERS TO THE EDITORThe Letters to the Editor section is divided into three categories entitled Notes, Comments, and Errata. Letters to the Editor arelimited to one and three-fourths journal pages as described in the Announcement in the 1 January 1998 issue.

33730021-9606/98/108(8)/3373/2/$15.00 © 1998 American Institute of Physics

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thermal compressibility, activity and osmotic coefficients!calculated either fromBr

2 or from B†, are identical to thatobtained by Kusalik and Patey.4

Just now, we have calculated theAi j , noting thatA isBr

2 or B† of Eq. ~5! ~for i , j 51, 1, 2!, by the method ofg-inverse. Motivated by Friedman and Ramanathan,3 wenow calculate the same fromAab with a,b 5 1, s ~to avoidconfusion, we denoteAab5 Aab).

Considering solvent and ions as components, the totalderivative dm i can be written6 ~for i 5 1, 1, 2! as: dm i

5( j 51,1,2Ai j dr j . Multiplying the equations fori 5 1, 2by n1 and n2 , respectively and using the relationsn1m1

1n2m25ms andrs5r1 /n15r2 /n2 , we get the follow-ing relations~for a51, s!

dma5Aa1dr11@n1~]ma /]r1!r1 ,r2

1n2~]ma /]r2!r1 ,r1#drs . ~6!

If we take solvent and salt as components, we get~for a51,s)

dma5 Aa1dr11 Aasdrs . ~7!

Equating the corresponding coefficients of Eqs.~6! and~7!, we get the following three independent equations in sixunknowns (Aab are calculated in terms ofBi j in Appendix!.

A1s5 A1s , n1A111n2A215 As1 ,~8!

n12 A111n1n2~A121A21!1n2

2 A225 Ass.

These equations can be solved uniquely, ifm1 ~or dm1)can be expressible in terms ofm2 ~or dm2) uniquely. Butthere is no unique way we can splitms into m1 and m2 .Since we have only one equation, viz.ms5n1m11n2m2

and two unknownsm1 andm2 . For an arbitrary choice ofm2 we get the correspondingm1 satisfying the equation, andfor each such choice, we get a particularg-inverse. The spe-cific choices of m1 and m2 which leads to a specificg-inverse can be obtained, as the solutions of consistent butdependent set of equations (Rx5y) consisting of the relationn1m11n2m25ms and itsn times. And the general solu-tion is given by5 x5Sy1(I 2SR)z where,z is an arbitraryvector,S is anyg-inverse ofR. Depending on the choices ofS, we get different set ofm1 and m2 . Corresponding toRG and MP inverse we found, respectively~i! m1

5(1/n1)ms1(n2/n1)c, m25c ~a constant! and ~ii ! m1

5@n1 /(n12 1n2

2 )2]ms, m25@n2 /(n12 1n2

2 )2#ms . It is easyto see that for choices of~i! and~ii !, the expressions forAi j

~i , j 51, 1, 2! calculated from Eq.~8!, reduces respec-tively, to Br

2 andB† of Eq. ~5!.Thus, the method of generalized inverse can be used to

calculate thermodynamic properties of ionic solutions usingKirkwood–Buff theory. The results obtained are identical to

that obtained earlier.4 The problem of non-invertibility ofBmatrix as discussed in Refs. 3 and 4 arises because of elec-troneutrality i.e., because ions cannot be added separately tothe solvent. The absence of unique splitting ofms to obtainm1 and m2 arises from the same fact. It is interesting thatthis is related to the non-uniqueness ofA matrix (]m/]r)and that the thermodynamic properties are unaffected by thisnon-uniqueness.

ACKNOWLEDGMENTS

I would like to thank Professor P. Gupta-Bhaya and Pro-fessor P. C. Das for valuable discussions.

APPENDIX

Consider the solvent and salt as the components, wehavedra5(b51,sLabdmb whereLab5(]ra /]mb)mg

~for a

51, s). Using the relationms5n1m11n2m2 , the aboveequations can be rewritten as

dra5La1dm11n1Lasdm11n2Lasdm2 . ~A1!

Now, we consider the same system as three component one,the solvent and the ions being components, we can write6

dr i5( j 51,1,2Bi j dm j for i 5 1, 1, 2. Adding the equationsfor i 5 1, 2 and usingr11r25(n11n2)rs , we obtainthe following two equations~for a 5 1, s!

dra5 (j 51,1,2

l a jdm j , l a j5~]ra /]m j !mg. ~A2!

Equating the corresponding coefficients of Eqs.~A1! and~A2!, we getLab in terms ofl i j and henceBi j . Inversion ofL gives (]ma /]rb)rg

for a,b51, s and henceAab

5(V/kBT)(]ma /]rb)rg. The result is

A51

DS rs2G12 2r1rsG11

2r1rsG11 r1G118D , ~A3!

where

G118 511r1G11

and

D5r1rs2@G111r1~G11G122G11

2 !#

a!Electronic mail: behera@iitk.ernet.in1J. G. Kirkwood and F. P. Buff, J. Chem. Phys.19, 774 ~1951!.2H. L. Friedman,A course in Statistical Mechanics~Prentice-Hall, Engle-wood Cliffs, New Jersey, 1985!.

3H. L. Friedman and P. S. Ramanathan, J. Phys. Chem.74, 3756~1970!.4P. G. Kusalik and G. N. Patey, J. Chem. Phys.86, 5110~1987!.5C. R. Rao and S. K. Mitra,Generalised Inverse of Matrices and Its Ap-plications ~Wiley, New York, 1971!.

6H. Margenau and G. M. Murphy,The Mathematics of Physics and Chem-istry ~Van Nostrand, Princeton, 1943!, p. 4.

3374 J. Chem. Phys., Vol. 108, No. 8, 22 February 1998 Letters to the Editor

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