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## Material Information- Title:
- Application of fluctuation solution theory to strong electrolyte solutions
- Creator:
- Cabezas, Heriberto, 1952- (
*Dissertant*) O'Connel, John P. (*Thesis advisor*) Hooper, Charles F. (*Reviewer*) Westermann-Clark, Gerald B. (*Reviewer*) - Place of Publication:
- Gainesville, Fla.
- Publisher:
- University of Florida
- Publication Date:
- 1985
- Copyright Date:
- 1985
- Language:
- English
- Physical Description:
- xii, 193 leaves : ill. ; 28 cm.
## Subjects- Subjects / Keywords:
- Chemicals ( jstor )
Correlations ( jstor ) Diameters ( jstor ) Electrolytes ( jstor ) Experimental data ( jstor ) Ions ( jstor ) Parametric models ( jstor ) Solvents ( jstor ) Teeth ( jstor ) Water tables ( jstor ) Chemical Engineering thesis Ph. D Dissertations, Academic -- Chemical Engineering -- UF Electrolyte solutions ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt ) thesis ( marcgt )
## Notes- Abstract:
- Fluctuation solution theory relates derivatives of the thermodynamic properties to spatial integrals of the direct correlation functions. This formalism has been used as the basis for a model of aqueous strong electrolyte solutions which gives both volumetric properties and activities. The main thrust of the work has been the construction of a microscopic model for the direct correlation functions. This model contains the correlations due to the hard core repulsion, long range field interactions, and short range forces. The hard core correlations are modeled with a hard sphere expression derived from the Percus-Yevick theory. The long range field correlations are accounted for by using asymptotic potentials of mean force and the hypernetted chain equation. The short range correlations which include hydration and hydrogen bonding are modeled with a density expansion of the direct correlation function. The model requires six parameters for each ion and two for water. The ionic parameters are valid for all solution and those for water are universal. The model has been used to calculate derivative properties for six 1:1 electrolytes in water at 25c, 1 ATM, the calculated properties have been compared to experimentally determined values in order to confirm the adequacy of the model.
- Thesis:
- Thesis (Ph. D.)--University of Florida, 1985.
- Bibliography:
- Bibliography: leaves 188-192.
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Heriberto Cabezas, Jr.
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- University of Florida
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- University of Florida
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- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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14971957 ( OCLC ) AEH8290 ( NOTIS )
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APPLICATION OF FLUCTUATION SOLUTION THEORY TO STRONG ELECTROLYTE SOLUTIONS By HERIBERTO CABEZAS, JR. A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1985 To Flor Maria ACKNOWLEDGMENTS I would like to express my sincere gratitude to Professor J.P. O'Connell, a man of wisdom and knowledge, for his guidance and encouragement during the course of this work. I also wish to thank Drs. G.B. Westermann-Clark and C.F. Hooper, Jr. for serving on the supervisory committee and for making very pertinent suggestions regarding this work. It is a pleasure to thank Mrs. Smerage for her excellent typing and patience and Mrs. Piercey for her help with the figures. Finally, I am grateful to the Chemical Engineering Department of the University of Florida for financial sup- port and for providing the kind of intellectual environment in which this work could take place. I am also grateful to the National Science Foundation for providing the financial support that made this work possible. TABLE OF CONTENTS Page ACKNOWLEDGMENTS.................................. iii KEY TO SYMBOLS...................................... vi ABSTRACT.......................................... xi CHAPTERS 1 INTRODUCTION......... ...... .................. 1 2 FLUCTUATION THEORY FOR STRONG ELECTROLYTE SOLUTIONS .................................. 11 Introduction................................ 11 Thermodynamic Property Derivatives and Direct Correlation Function Integrals... 12 Direct Correlation Function Integrals from Solution Properties..................... 23 Summary ................ .................. .29 3 A MODEL FOR DIRECT CORRELATION FUNCTION INTEGRALS IN STRONG ELECTROLYTE SOLUTIONS.. 33 Introduction................................... 33 Philosophy of the Model.................... 33 Statistical Mechanical Basis ............... 37 Expression for Salt-Salt DCFI.............. 51 Expression for Salt-Solvent DCFI............ 63 Expression for Solvent-Solvent DCFI........ 69 Summary...................................... 74 4 APPLICATION OF THE MODEL TO AQUEOUS STRONG ELECTROLYTES ......... ............... ....... 77 Introduction................................ 77 Calculation of Solution Properties from the Model. .............................. 78 Model Parameters from Experimental Data.... 90 Comparison of Calculated Properties with Experimental Properties.................. 104 Discussion. ............................... .105 Conclusions ............................... 113 5 CONCLUSIONS AND RECOMMENDATIONS............ 144 APPENDICES A HARD SPHERE DIRECT CORRELATION FUNCTION INTEGRAL FROM VARIOUS MODELS................ 148 B RELATION OF McMILLAN-MAYER THEORY TO KIRKWOOD-BUFF THEORY...................... 152 C RELATION OF DENSITY EXPANSION OF THE DIRECT CORRELATION FUNCTION TO VIRIAL EQUATION OF STATE: ALTERNATE MIXING RULES. ...................................... 172 D EXPONENTIAL INTEGRALS........................ 180 E MODEL PARAMETERS........................... 184 REFERENCES ........................................ 188 BIOGRAPHICAL SKETCH............................... 193 KEY TO SYMBOLS ai = hard sphere diameter of species i. a. = distance of closest approach of species i and j. 13 1/2 B = K/II/2 B. = sum of all bridge diagrams, second virial 13 coefficient. C = mixture third virial coefficient. C.. = direct correlation function integral for species 13 i and j; two-body factor in third virial coefficient. C =direct correlation function integral for components a and 3. C.i = third virial coefficient for i, j, k. ijk AC.. = short range direct correlation function 13 integral. c.. = direct correlation function. 1J Ac.. = short range direct correlation function. 13 D = dielectric constant of solvent or solvent mixture. D = pure solvent dielectric constant. E = exponential integral or order n. e = electronic charge. AF = spatial integral of Af... -u../kT f. = e -1 = Mayer bond functions. 13 Af. = f.. f.HS fLR = differences of microscopic 1] 13 jj ij two-body coefficient. g = pair distribution function. 1 n 2 I Z. p = ionic strength i=l K2 k N N. N o N Oct n n o P INT qi r,r. r. -1 vii 2 = -8T I = Debye-Huckel inverse length. DkT =Boltzmann's constant. = total number of moles of all species. = total number of moles of species i. = total number of moles of all components. = total number of moles of component a. = number of different species, integer greater than one. = number of different components. = pressure. = internal partition function. = separation between species i and j. = position vector of i. 6 1/2 = (2 3k3T3) = Debye-Huckel limiting law D k T efficient. = temperature. = pair potential. = total system volume. = partial molar volume of species i. S Y T u. 13 V V. i Vo = partial molar volume of component a. W.. potential of mean force. 13 X. = N/N = mole fraction of species i. N X = mole fraction of component a on a oa N species basis. Z = dimensionless parameter in exponential integral. Z. = valence of ion i. 1 a = Euler's constant, empirical universal constant for ion-solvent correlations. i = activity coefficient of species i. a = activity coefficient of component a. 6.. = Kroniker delta. 13 8 = Eulerian angle between a charge and a dipole. 81i,'li = Eulerian angles of dipole of solvent molecule i. < = isothermal compressibility. Ki = isothermal compressibility of pure solvent (1). A. = ideal gas partition function. . = chemical potential of species i. l = dipole moment of solvent. Via = number of species i in component a. V = total number of species in component a. n K S = -6 p. a. = reduced density. K 6. 1 1 i=l S = P., osmotic pressure. N p density of all species. p = vector of species densities. viii N. P = = density of species i. N oca po = density of component a. .ijk = spatial integral of .ijk" Aijk = spatial integral of Ai .j 1ik ijk ijk = microscopic three-body coefficient. HS Aijk = ijk ijk = difference of microscopic three- body coefficients. = orientation dependence of dipole-dipole interaction. Q = f dw. = integral over orientation coordinates. 1 . = angular orientation coordinates of i. 1 1 n 2 = Z v. Z.. Y 2 i= y 1 Superscripts F = Final. FLL = Friedman's limiting law. HNC = hypernetted chain. HS = hard sphere. KB = Kirkwood-Buff. LR = long range or field type correlations or interactions, Lewis-Randall. MM = McMillan-Mayer. P = Pure component. PY = Percus-Yevick. R SAT TB 0,O0 solvent. component. Special Symbol < > = integration over orientation. w = Reference. = Saturated. = Three body. = infinite dilution in salt. Subscripts = species. = components. jj, 0 = . . Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy APPLICATION OF FLUCTUATION SOLUTION THEORY TO STRONG ELECTROLYTE SOLUTIONS By Heriberto Cabezas, Jr. August, 1985 Chairman: Dr. J.P. O'Connell Major Department: Chemical Engineering Fluctuation solution theory relates derivatives of the thermodynamic properties to spatial integrals of the direct correlation functions. This formalism has been used as the basis for a model of aqueous strong electrolyte solutions which gives both volumetric properties and activities. The main thrust of the work has been the construction of a microscopic model for the direct correlation func- tions. This model contains the correlations due to the hard core repulsion, long range field interactions, and short range forces. The hard core correlations are modelled with a hard sphere expression derived from the Percus-Yevick theory. The long range field correlations are accounted for by using asymptotic potentials of mean force and the hyper- netted chain equation. The short range correlations which include hydration and hydrogen bonding are modelled with a density expansion of the direct correlation function. The model requires six parameters for each ion and two for water. The ionic parameters are valid for all solutions and those for water are universal. The model has been used to calculate derivative prop- erties for six 1:1 electrolytes in water at 25C, 1 ATM. The calculated properties have been compared to experimentally determined values in order to confirm the adequacy of the model. xii CHAPTER 1 INTRODUCTION Aqueous electrolytes are present in many natural and artificial chemical systems. For example, the chemical processes of life occur in an aqueous electrolyte medium. All natural waters contain salts in concentrations ranging from very low for fresh water to near saturation for geo- thermal brines. Industrially, electrolytes are used in azeotropic distillation, electrical storage batteries and fuel cells, liquid-liquid separations, drilling muds, and many other processes. Since a quantitative description of the properties of these systems is required for under- standing, design, and simulation, the ability to predict and correlate the solution properties of electrolytes is both scientifically and technologically important. In attempting to fill this need, many models of aqueous salt solutions have been developed. Essentially all describe only activities of the components but ignore the volumetric properties. Several extensive reviews of electrolyte solu- tion models are available in the literature (Pytkowicz, 1979; Mauer, 1983; Renon, 1981). To be concise, the various models have been classified here into three general cate- gories and a few examples of each briefly discussed. First, there are models based on relatively rigorous statistical mechanical results which can be called "theoretical." Second, there are those composed of a mixture of rigorous theory and empirical corrections which can be named "semi- empirical." Third, there are those models which directly correlate experimental data and are thus termed "empirical." Neither this classification nor the following list pretends to be either unique or all-inclusive. Among the "theoretical" models, the earliest and still the most widely accepted is the theory of Debye and Huckel (1923) which gives the rigorous relation at very low salt concentration (the limiting law) for salt activity coeffi- cients but fails at higher salt concentration. This theory has been amply treated in the literature (Davidson, 1962; Harned and Owen, 1958). The Debye-Huckel theory considers an electrolyte solution as a collection of charged hard spherical ions embedded in a dielectric solvent which is continuous and devoid of structure. This is the physical picture generally called the "Primitive Model." The correct formalism for the application of modern statistical mechani- cal techniques to the "Primitive Model" is given by the McMillan-Mayer theory (1945). A major method developed for this formalism is a resumed hypernetted chain approxima- tion to the direct correlation function. This, together with the Ornstein-Zernike equation (1914), forms a solvable integral equation for the primitive model ion-ion distribution function which has been used to calculate the properties of electrolytes up to 1 M salt concentration (Rasaiah and Friedman, 1968; Friedman and Ramanathan, 1970; Rasaiah, 1969). This method requires tedious numerical calculations to obtain the properties. A simpler and more generalizable approach is the Mean Spherical Approximation (MSA) which has been applied to both primitive (Blum, 1980; Triolo, Grigera, and Blum, 1976; Watanasiri, Brule, and Lee, 1982) and nonprimitive (Vericat and Blum, 1980; Perez-Hernandez and Blum, 1981; Planche and Renon, 1981) electrolyte models. The MSA method essentially consists of solving the Ornstein-Zernike (1914) equation for the distribution functions subject to the boundary conditions that the total correlation function is minus one inside the hard core and that the direct correlation function equals the pair potential outside the hard core. This is equivalent to the Percus-Yevick method for rigid nonionic systems (Lebowitz, 1964). The MSA generally gives good thermodynamic properties if these are calculated from the "Energy Equation" (Blum, 1980). It does not yield good correlation functions and further suffers from the need to numerically solve complex nonlinear relations for the value of the shielding parameter at each set of conditions. This last problem grows progressively worse as the sophisti- cation of the model increases. Due to their complexity none of the modern "theoretical" models is widely used in engineering practice. The most successful of the semiempirical models is that due to Pitzer and coworkers (Pitzer, 1973; Pitzer and Mayorga, 1973; Pitzer and Mayorga, 1974; Pitzer, 1974; Pitzer and Silvester, 1976). Model parameters for activity coefficients have been evaluated for a large number of aqueous salt solutions, but volumetric properties and multi- solvent systems have not been treated. To construct the model, Pitzer adopted the "Primitive Model" and inserted the Debye-Huckel radial distribution function for ions into the osmotic virial expansion from the McMillan-Mayer formalism. This latter is analogous to using the "Pressure Equation" of statistical mechanics (Pitzer, 1977). The resulting expression contains the correct limiting law. He then added empirical second and third virial coefficients which are salt and solvent specific. Although Pitzer's model correlates aqueous activity coefficients superbly, it does not add to the fundamental understanding of these solutions; further, its extension to multisolvent systems would pose some serious problems associated with the mixture dielectric constant as has been recently pointed out (Sander, Fredenslund, and Rasumussen, 1984). Another semiempirical approach uses the NRTL model for solutions of nonelectrolytes (Renon and Prausnitz, 1968) adapted for short range ion and solvent interactions (Cruz and Renon, 1978; Chen, Britt, Boston, and Evans, 1979) in nonprimitive models of electro- lyte solutions. Cruz and Renon separate the Gibbs energy into three additive terms: an elecrostatic term from the Debye-Huckel theory, a Debye-McAulay contribution to correct for the change in solvent dielectric constant due to the ions, and an NRTL term for all the short range intermolecular forces. Chen et al. adopted a Debye-Huckel contribution and an NRTL term for the Gibbs energy but no Debye-McAulay term. More recently, the UNIQUAC model for nonelectrolytes has been modified for short range intermolecular forces in electrolyte solutions (Sander, Fredenslund, and Rasmussen, 1984). The resulting UNIQUAC expression has been added to an empirically modified Pitzer-Debye-Huckel type electro- static term to form the complete Gibbs energy model. Although the two NRTL and the UNIQUAC models correlate activity coefficient data reasonably well even in multi- solvent systems, they have to be regarded as mainly empirical. First, their resolution of the Gibbs energy into additive contributions from each different kind of interaction is not rigorous. Second, the problems associated with the mixture dielectric constant are resolved in an empirical and somewhat arbitrary fashion. As a result, such models add little to our understanding of these systems and may not be reliable for extension and extrapolation. Of the various empirical methods developed, two have been chosen to be discussed here because they represent distinct approaches. First, there is the method of Meissner (1980) which is a correlation for the salt activity coeffi- cient in terms of a family of curves that are functions of the ionic strength and a single parameter which can be selected from a single data point. This method has been extended to multicomponent electrolyte solutions and is useful over a wide range of salt concentration (0.1-20 MOLAL), though it is not very accurate. Second, there is the method of Hala (1969) which is more conventional in that it consists of a purely empirical model for the Gibbs energy of the solution. This method is an excellent correlational tool, but it is not predictive. It has four parameters per salt-solvent pair. The existence of so many models to correlate and predict the thermodynamic behavior of electrolyte solutions is indicative of the complexity of these systems and, perhaps, the relatively poor state of the art. As examples of the physical complexity of electrolyte solutions, the composition behavior of the salt activity coefficient (Figure 1) and of the species (ions and solvents) density (Figure 2) is presented. Figure 1 shows the large deviation from ideal solution behavior (y= 1) even at very low salt concentration for all salts. Second, it indicates that salts of the same charge type show similar behavior at low salt concentration but are widely different at higher salt concentration. In Figure 2, the difference in the salt composition behavior of the species density is obvious even for relatively similar salts, i.e., the solution seems to expand for KBr while it seems to contract for all other salts. The activity coefficient data were taken from the compilation by Hamer and Wu (1972). For NaCl and NaBr the density data of Gibson and Loeffler (1948) were used. For LiCI, LiBr, and KBr the density data were taken from the International Critical Tables. For KC1 the density data of Romankiw and Chou (1983) were used. In the hope of improving the situation for obtaining properties of solutions, a new model of strong aqueous electrolyte solutions is presented here. This model has been carefully constructed so that it overcomes a number of the deficiencies of previous methods. For example, this model is simple enough for economical engineering calculations, yet sufficiently sophisticated to rigorously include all the different interactions (ion-ion, ion-solvent, solvent-solvent) and the principal physical effects (electro- static, hard core repulsion, hydration, etc.) that contribute to each interaction. The model is also extendable to multi- salt and multisolvent systems in a straightforward fashion. Finally, it addresses both activity and volumeric properties. In the chapters that follow, a detailed development of the new model is presented. Chapter 2 has the general relations between solution properties and correlation 8 functions. Chapter 3 has the full development of the new model. Chapter 4 shows the application of the model to solutions of aqueous strong electrolytes and the calculation of solution properties. Chapter 5 has suggestions for further work and conclusions. 1.4 1.2 1.0 0.8 0.6 0.4 0.0 Figure 1. 0.5 1.0 1.5 2.0 2.5 1 (Molality) 2 Salt Activity Coefficient in Water at 250C, 1 ATM. Data of Hamer and Wu (1972). 0.06 0.59 0.58 0.57 0.56 0.55 0.54 0.00 0.02 0.04 0.06 X05 Figure 2. 0.08 0.10 0.12 Species Density in Aqueous Electrolytes at 250C, 1 ATM. For data sources see text. CHAPTER 2 FLUCTUATION THEORY FOR STRONG ELECTROLYTE SOLUTIONS Introduction There are three general relations among the thermodynamic properties of a solution and statistical mechanical correlation functions. The first two are the so-called "Energy Equation" and "Pressure Equation" which are obtained from the canonical ensemble with the assumption of pairwise additivity of inter- molecular forces. These equations relate the configurational internal energy and the pressure respectively to spatial integrals involving the intermolecular pair potential and the radial distribution function (Reed and Gubbins, 1973; McQuarrie, 1976). The third relation is the so-called "Com- pressibility Equation" which is derived in the grand canonical ensemble without the need to assume pairwise additivity of intermolecular forces. This equation relates concentration derivatives of the chemical potential to spatial integrals of the total correlation function (Kirkwood and Buff, 1951) and to spatial integrals of the direct correlation function (O'Connell, 1971; O'Connell, 1981). This last method is generally known as Fluctuation Solution Theory. Fluctuation solution theory has been applied to the case of a general reacting system (Perry, 1980; Perry and O'Connell, 1984), and the formalism has also been adapted to treat strong electrolyte solutions which are considered as systems where the reaction has gone to completion (Perry, Cabezas, and O'Connell, 1985). The main body of this chapter consists of a derivation of the general fluctuation solution theory. Although the final results are identical to those previously obtained by Perry (1980), the development is more intuitive and mathematically simpler, though less general. The remainder of the chapter illustrates the calculation of direct correlation function integrals (DCFI) from solution properties and sets theoretically rigorous infinite dilution limits on the DCFI's. Thermodynamic Property Derivatives and Direct Correlation Function Integrals A general multicomponent electrolyte solution, contain- ing n species (ions and solvents) formed from no components (salts and solvents) by the dissociation of the salts into ions, is not composed of truly independent species due to the stoichiometric relations among ions originating from the same salt. It is, therefore, not possible to change the number of ions of one kind independently of all the other ions. However, the independence of ions has been assumed traditionally for theoretical derivations, and it will lead us to the correct results by a relatively simple mathematical route. Thus, with the assumption that any two species i and j are independent of all other species, Fluctuation Solution Theory gives the following well known result (O'Connell, 1971; O'Connell, 1981): 1 ^i 6.. c. 1 (2-1) RT 9N. N. N T,V,Nkj where = the chemical potential per mole of species i. N. = the number of moles of species i. N = the total number of moles of all species. 6.. = the Kroniker delta. 13 2 C. = 4Tp J 13 1=3 the direct correlation function. N p = = molecular density of all species. V The microscopic direct correlation function 13 W is an angle averaged direct correlation function defined by ij> c1 dw. dw. (2-2) 'J j2 ij 1 J Q where Q2 = dw. d . In order to arrive at the first and simplest of the desired relations, we define the activity coefficient for species i on the mole fraction scale as (T,P) = MP(T) + RT In X.Y.(T,P) (2-3) where = the reference chemical potential. 1 Ni Xi = N = mole fraction of species i. Yi = the activity coefficient of species i. P = the vector of species mole densities. By differentiating equation (2-3) with respect to the number of moles of species j, we obtain 1 i RT 3N. T,V,Nk ny+ i1 1 (2-4) 3N. N. N 3 T,V,Nk j which upon insertion in equation (2-1) gives 1ny. 1 C.. 1 = N1i (2-5) 3N. N 3 T,V,Nkj and when multiplied by the system volume on both sides of the equation, 81ny. 1 C.. 1 1- (2-6) p Tp p J TPkfj N-r where p = n-= molar density of species i. J V By performing a sum over all species i and j on equation (2-6) S n n lny. v v- v iaej B Pj a B i=l j=l j T iT'Pk~j n n 1-C.. S1 D1 (2-7) a i=l j=l where vi = number of species i in component a. Sa = total number of species in component a. By noting the definition of the mean activity coefficient of a component a, 1 n Iny, = V. Iinyi (2-8) a v a i-i i a i=l and also assuming that species j is formed from an arbitrary component 8 so that Pj = VjB Po (2-9) one then arrives at the first relation 1 alny 1 aOB T,Po (2-10) 1 n n 1-C . i=1 j=l a6 which upon identification of n n 1-C.. 1 C = (2-11) i=l j=l a3S assumes the simpler form alny PV p= v (1-C ) (2-12) pv T Equation (2-12) relates the DCFI to the derivative of the activity coefficient of any component a (salt or solvent) with respect to the molar density of any component 3 (salt or solvent) at constant volume, temperature, and mole number of all components other than 8. Because most experiments and many practical calculations are performed at constant pressure rather than constant volume, it is of interest to derive a relation between the activity coefficient derivatives at constant pressure and direct correlation function integrals. First, a change of variables is executed. i i 3N. N. 3 T,V,Nkj 3 T,P,Nk/j yi N .P (2-13) aN. T,N T,V,Nk^j and the following identifications are made, i | p = V (2-14) T,N aP av v. P T,N = r (2-15) aN. @N. VK 3 TV'Nkj V TPNkj T,PNkj Then equations (2-1), (2-14), and (2-15) are inserted into equation (2-13) to obtain lli 6. C.. V.V. 1 = -1 13 1 3 (2-16) RT 3N. N. N VKTRT 3 T,P,Nk 1 T where = partial molar volume 1 8N. 1 T,P,Nkfi volume of species i. V = the system volume. -1 av KT V P = isothermal compressibility. T,N To develop a relation in terms of activity coefficients, the chemical potential is written as in equation (2-3) and the proper constant pressure derivative is taken. 1 i lny RT aN. = Nj + 3 TP,Nk.j TP,Nkj 6. 13 1 (2-17) N N From equations (2-16) and (2-17), 18 81ny. 1 -e. V.V. 1 = L- 1 3 aN N VK RT (2-18) T,P,Nk#. By multiplying equation (2-18) by the system volume and rearranging, one finds N alny. 1 C V. V. 1 1 pKTRT aN pKTRT TRT K TRT (2-19) T,P,Nkj upon which a double summation over species i and j is performed to obtain N 1 n n alny PK RT v v L IiaVji 1N. T Ta i=l j=l 1 T,P,Nkfj 1 1 n n 1 v. (1-C..) - KT a 8 i=l j=1 n n 12 1. v V V. (2-20) (K RT)2 V v i=l j=l a j8 i j T a B Equation (2-20) must be simplified so that all the properties appear as component rather than species quantities. This is done with the aid of equations (2-8), (2-11), and the assumption that species j is formed from an arbitrary component 8 so Nj = vjNo (2-21) Additionally, the partial molar volume of a component a or B is expressed as a sum of the species partial molar volumes. -V 1 aV V- N- N (2-22) DN NiB No 1 iS Of3 T,P,Nk5i T,P,N V n Vo8 N =i i Vi (2-23) S T,P,N oY$S Equation (2-20) is now transformed to Nva alny PK RT 8N T o5 T,P,N v V (1-C ) V V (2-24) a s aS oa oS (2-24) PKTRT K RT KTRT To make further progress, the relationship of the Vo bulk modulus of the solution (p TRT) and the group, -T, to the direct correlation function integrals must be found. First, the compressibility equation is derived from the basic fluctuation theory result of equation (2-1) starting with the Gibbs-Duhem equation for an isothermal but nonisobaric process. n N N.dP. = VdP (2-25) i=l 1 1 Upon differentiation of equation (2-25) with respect to the mole number of an arbitrary component j, SN. = V P i=1 i T'V j aN T,V,N dj T,V,N kj and by insertion of the equation (2-1), n RT Z (6 X.C..) = V i=l ij 13 N. ST,V,Nkj n 1 P 1- 1 x. C. RT ap T Fk 1 13 RT j i= T'Pk$j (2-26) (2-27) (2-28) Equation (2-28) is the general multicomponent compres- sibility equation expressed in terms of species quantities. This relation is now transformed to one in terms of com- ponents by performing a summation over species j and use of equations (2-9) and (2-29). n o X = 1 Vi X o S B=lI 1 n ap RT X Vj T,PR~i (2-29) n n = 1 X.v C. i=1 j= 1 (2-30) 1 DP RT 3p oa T,p n o n n 1 XI v. Xi v .' C.. @=1 i=1 j=1 (2-31) which by use of equation (2-11) becomes n 1 @P 1 = v Xo Ca (2-32) RT Dp oa 0 0 T,p Equation (2-32) is the multicomponent compressibility equation expressed in terms of components. The density derivative of pressure is related to the partial molar volume as -V P aV o (2-33) ap V 8N o T,N oa T T,p o T,P,N which when inserted in equation (2-32) gives one of the desired relations. n V o oa v v X (1-C ) (2-34) KTRT a O~ X In order to relate the bulk modulus to direct correla- tion function integrals, the total volume is related to the partial molar volumes. n0 n V= ON = V No (2-35) VN oa oa oa a=l oa T,P,N a=1 T,P,N Dividing equation (2-35) by the mole number of species (N) yields 1 V n S= X (2-36) p N oa oca a=l and when applied to equation (2-34) aP/RTI D IT,N n n o o RT= 1 VX X (1-Ca) OK RT l a 6 oa O ( T a=1 8=1 which is the second necessary relation. Substitution of equation (2-34) and equation (2-37) into equation (2-24) gives Nva 1nya pKTRT 8No T,P,N YO = v a R n n o o SV vX X [(l-C )(l-C ) (1-C )(1-C )] y=l 6=1 Y 6X y 06 y6 ay a5 (2-38) which is further transformed by substituting for the bulk modulus Nv ny a a N T,P,Noy3 n n o o v v I vI v v X Y6[(1-C Y)(1-C a)-(1-C Y)(1-C6 )] y=l 6=1 o 06 n n o o S V X oyX 06(1-C y6 y=l 6=1 (2-39) Equation (2-39) relates changes in the activity coefficient of any component a with changes in the mole number of any component 8 where the process occurs at constant pressure (2-37) and temperature to sums of direct correlation function integrals for components. In summary, it should be noted that of the various relations developed in this section,only a few are of prac- tical importance in relation to this work. These are listed at the beginning of the next section. Direct Correlation Function Integrals from Solution Properties The previous section consists of a relatively simple but lengthy derivation of several basic relations between solution properties and direct correlation function integrals. The relations that are of most importance to this work are listed below. 1ainy PV D v v (1-C ) (2-12) V n oco oa (2-34) TRT- V V X (1-C ) (2-34) KTRT a =1 o aB n n SP/RT 1 o o ~ P/RT 1 v0 vX X (1-C- 1 (2-37) P T,N TRT a=l =iTxaB=1 Nva 1nya PKTRT 8No0 T,P,N, Y 8 n n o o a 8 6=1 Y 6 oy 0 Y=l 6=1 [(l-Cy )(1-Ca ) (1-Cay)(l-C68)] N N v. v. (1-C..) S-C = ia jB 1 i=l j=l a Y (2-38) (2-11) Useful bounds on the value of the direct correlation function integrals as the system approaches infinite dilution in all components except one (usually the solvent) can be deduced from the preceding relations. Thus, by taking the limit of pure solvent (component 1), one obtains ---0M V oa = (1-C )0 v K RT al a1 S (1-C11 ) P K RT ol 1 (2-40) (2-41) N 31ny ol alnya V 3N 1 No IT,P,N Y N 31ny ol a V 3N 0 3 ~o T,P,N Y -(1 V K 1KRT (1-Cal) (L-C ) (1-C(1-C ) (l- 11 #B 2-42a) (2-42b) m LIM where (1-C ) = (1-C ) and where equations (2-42) represent a constant pressure limit on the DCFI. A corresponding constant volume limit can be obtained from equation (2-12). For a binary system consisting of one solvent (1) and one salt (2), the fluctuation relations become 91ny y n2 22 (2-43) ap02 T 01 Po2 T'Pol o2 = (1-C ) + V2X (1-C ) (2-44) SRT = 2Xol 12 2 o2 22 T aP/RT 1 2 p pKRT ol (1-C )+ 3P T,N PKTRT 01 11 2 2 2olXo2(-12) + (-22 (2-45) Nv2 31ny2 pK RT 8No2 T,P,No 2X [(1-C 1 )(1-C22) (1-C12)2] (2-46) 2 ol 11 22 12 v+2 (1-C 1+) + V-2 (1-C 1-) 1 -C12 = (2-47) 12 v 1 C22 2 2 v (1-C )+2+2v (1-C )+ (1-C__ ) +2 ++ +2-2 +- -2 (2-48) and the respective infinite dilution limits are V \K2RT 1-C21 V 2Kl1RT = (C21 1 P p K RT ol 1 N1 lny2 "2 a2N o2IT (l-Cll) (2-49) (2-50) (2-51a) p 2 Po (V ) 22 V2 K RT 2 1 Nol V2 a1ny2 aN o l 02 T,P,N o1 -( 1-C122 (1-C ) (1-C12 22 (1-C11) (2-51b) The significance of equations (2-51) can be further understood by realizing that any correct model for the ,P,N ol activity coefficient of an electrolyte must approach the Debye-Huckel Limiting Law at very low salt concentration. Thus, the mean activity coefficient of a salt on the mole fraction scale is given by this law as 1 n 2 1/2 lny = S ( Z) (2-52) ny2 i=l 2 6 1/2 where S =( 332T ) S Dk T Debye-Huckel limiting law coefficient. e = the electronic charge. D1 = pure solvent dielectric constant. k = Boltzmann's constant. T = temperature. Z. = valence of ion i. 1 2 I = ZiP = ionic strength. i=l and when the proper derivative is taken, N P -1/2 0ol 2 y ol 2 3N2 4v2 T,P,N 2 n n 2 2 v1 1 i2 Ziz2 (2-53) i=l j=l Insertion of equation (2-53) into equation (2-51a) gives P -1/2 Sp I n n Y ol- 2 + 2 i2j22 1 4v2 i=1 j=1 P 2 Po (V O) 21 02 = (1-C2) (2-54) v2 1RT which approaches negative infinity as the salt concentration approaches zero. In order to construct a model capable of correlating and predicting the solution properties of electrolytes, it is helpful to calculate the experimental behavior of the DCFI's from solution properties. To that purpose, equations (2-43), (2-44), and (2-45) have been inverted so that the three DCFI's can be calculated from 1 2 1 C11 = [1 X V p] + 11 2 OKRTo2 o2 XolP TRT 2 X 2 iny2 ol V N 2 (2-55) 2 2 DN ol T,P,No Vo2 2 1-C =1[-X V p1- 12 V2XolKTRT -o2 o2 Xo2 N n 2 (2-56) X N 0ol o2 T T,P,Nol 2 p(Vo ) N in 2 1 C = -- (2-57) 22 2 vK aN 2 T T,P,Nol Figures 3-5 show the results of equations (2-55) and (2-57) for six different salts at 1 ATM and 25C. The compressibility data used were those of Gibson and Loeffler (1948) for NaCL and NaBR. For LiCL, LiBR, KCL and KBR the compressibilities of Allam (1963) were used. The activity coefficient data were taken from the compilation by Hamer and Wu (1972). The density data of Gibson and Loeffler (1948) were again used for NaCL and NaBR. For LiCL, LiBR, and KBR the density data were taken from the International Critical Tables. The newer density data of Romankiw and Chou (1983) were used for KCL. The pure water data were those of Fine and Millero (1973). The infinite dilution partial molar volumes were also from Millero (1972). Summary The present chapter has introduced the basic relations of interest, has shown how they have been used to calculate the experimental behavior of the DCFI's, and has given some bounds on the values of the DCFI's. The next chapter introduces a model for correlating the observed experimental behavior of the DCFI's. 30 25 20 15 1 -C22 10 5 0 0.00 05 02 0.00 0.02 0.04 0.06 0.08 0.10 X02 Figure 3. Salt (2)-Salt (2) DCFI in Aqueous Electrolyte Solutions at 250C, 1 ATM. For data sources see text. 0.12 24 20 1-C2 16 12 8 0.00 0.02 0.04 0.06 0.08 0.10 0.12 X05 Figure 4. Salt (2)-Water (1) DCFI in Aqueous Electrolyte Solutions at 250C, 1 ATM. For data sources see text. 28 26 24 1-C 12 22 20 18 0.00 0.02 0.04 0.06 0.08 0.10 0.12 X05 Figure 5. Water (1)-Water (1) DCFI in Aqueous Electrolyte Solutions at 250C, 1 ATM. For data sources see text. CHAPTER 3 A MODEL FOR DIRECT CORRELATION FUNCTION INTEGRALS IN STRONG ELECTROLYTE SOLUTIONS Introduction In order for the formalism introduced in the previous chapter to be of practical value, a model to express direct correlation function integrals in terms of measurable quantities (p, T, x) must be constructed. The present chapter describes such a model. First, a general physical picture of electrolyte solutions and its relation to micro- scopic direct correlation functions is discussed. Second, a rigorous statistical mechanical basis is laid for the microscopic direct correlation functions and their spatial integrals. Third, equations are given for each type of pair correlations in the system (ion-ioin, ion-solvent, solvent-solvent). Lastly, a summary is presented of the model parameters and estimated sensitivity of results to their values. Philosophy of the Model The complex thermodynamic behavior of liquid electro- lytes is the observable result of the very complex interac- tions between the species in solution, i.e., the ions and 34 solvent molecules. In the absence of a complete understand- ing of all these forces, models use simpler or, at least tractable, interactions which may have the essential charac- teristics of the real forces. In addition, some semiempiri- cal terms are used to account for those interactions that cannot be simply approximated. Thus the interactions between the ions at long distances are modeled as those of charges in a dielectric medium containing a diffuse atmosphere of charges. At very short range, however, the dominant interaction becomes a hard sphere-like repulsion. There exist rigorous statistical mechanical methods to treat these two types of interactions, but these two are not adequate to correlate and predict the solution behavior with sufficient accuracy. Interactions that are important at intermediate ion-ion ranges must be incorporated. Unfortunately, these intermediate range forces cannot be simplistically approximated because they involve strong many-body effects such as dielectric satura- tion, ion-pairing, polarization, etc., which are not well understood. In the present model the ionic and hard sphere interactions are treated theoretically while the rest are included in a semiempirical fashion. The interactions between ions and solvent molecules at large separation can be treated as those of charges and multipoles in a dielectric medium containing an ionic atmosphere. In general, quadrupoles and higher order multipoles are not included, because their contribution is expected to be numerically insignificant in an aqueous system. The short range interactions are treated as hard sphere repulsion. Intermediate range forces for the ion- solvent case are very important because they include solva- tion which makes a larger contribution than the long range charge-multipole forces. Solvation of the ion by the solvent is intimately related to the partial molar volume of the salt and must be incorporated if there is to be any hope of correlating and predicting the volumetric behavior of the solution. As for ions, the long and short range intera- tions are treated theoretically while the intermediate range forces are incorporated semiempirically. The forces between solvent molecules at long range can be considered to be those of dipoles in a dielectric medium which has an ionic atmosphere. Higher order multi- poles may again be neglected because their contributions are less important and can be covered in other ways. The short range forces are again treated as hard sphere repul- sions. The intermediate range interactions for the solvent- solvent case are dominated by association type forces such as hydrogen bonding which make a larger contribution than the long range dipole-dipole term. As above, the long and short range interactions are treated theoretically while the effects of the intermediate range forces are included semiempirically. In summary, there are three distinct classes of inter- action: ion-ion, ion-solvent, solvent-solvent. Each class has unique contributions from long-range, field-type forces, short-range, repulsive forces, and intermediate range forces. Traditionally, models have been written for the excess Gibbs or Helmholtz energy of the system by adding contribu- tions from some of the above forces in an ad hoc and, gener- ally, nonrigorous fashion. The fact that free energy contributions do not naturally separate into the types of forces and that experimental values for each cannot be separately determined has caused many of these models to be complex and/or inconsistent. Further, they do not yield volumetric properties along with the activities. Within the framework of Fluctuation Solution Theory, the contributions of the pair correlations to the thermo- dynamic properties can be rigorously added. Thus, there are terms from the salt-salt, salt-solvent, and solvent- solvent DCFI's, as shown in Chapter 2. Further, the experi- mental behavior of each of the three DCFI types can be separately calculated from solution data as seen in the previous chapter. It is then possible to construct separate and accurate models for each one of the DCFI's. These models can later be manipulated to yield thermodynamic properties. As may be inferred from the above discussion, each of the three types of DCFI's contains long range, short range, and intermediate range interactions. These can be theoretically separated into a simple additive form as will be shown in the next section of this chapter. It is important to note that the separation is first developed at the level of microscopic direct correlation functions which are later integrated to obtain the DCFI's. Although our particular additive separation of the micro- scopic direct correlation function is not fully rigorous, we believe it is more reasonable than a similar resolution of the radial distribution function into an additive form (Planche and Renon, 1981). In fact, the radial distribution function can naturally be resolved only into a multiplicative rather than an additive form. The intermolecular potential and, consequently, the potential of mean force can be approximately decomposed into additive contributions from interactions of different characteristic range, but this potential appears in an exponential in the radial distribu- tion function. Thus, resolving the radial distribution function into additive contributions is quite inappropriate. Statistical Mechanical Basis The above philosophy is a qualitatiave concept which must be expressed in quantitative terms. To this end, we now establish a rigorous statistical mechanical basis for a model of microscopic direct correlation functions. First, consider the diagrammatic expansion of the direct correlation function (Reichl, 1980; Croxton, 1975) for species i and j, u.. c..(T,p,r ,r ., .) = g. 1 Zn g + B.. (3-1) 13 -j -1- n g ij kT 13 -W.ij ./kT where gij = e = radial distribution function. W = potential of mean force. ui. = pair potential. B.. = sum of all bridge diagrams also known as elementary clusters. Although equation (3-1) is an exact expression for the direct correlation function, it is of little practical value because the bridge diagrams cannot be summed analyti- cally. This series is Sn n B.. f.f f .f.f. dr drd d Bl 2'! Okp fikfk fZjfifkj dkd wkd k=l =1 + (3-2) for a system consisting of n species. -u. ij./kT where f.. = e 1 = Mayer bond function. To obtain the hypernetted chain (HNC) approximation (Rowlinson, 1965) all of the bridge diagrams are neglected (B.. = 0). This introduces an error which is second order 1j in density and ignores some four body contributions. It is, therefore, exact up to the order of a third virial coefficient. Thus, the HNC direct correlation function is cHNC ij = gi ij - 1 ng. - 1ij kT From the definition of the radial distribution function, W.. Z.n g..= 11 1] kT (3-3) (3-4) W.. W . giJ -1 1 kT ) 2 13kT 72T kT 3 1 W. 3! kT which on insertion into equation (3-3) gives, 2 3 HNC u W.. W.. Cij T 1 3! 1 ) + ij IT 2! kT 3! kT " To apply equation (3-6) requires at least approximate expressions for the potential of mean force in terms of measurable variables. Such expressions, valid in the limit (3-5) (3-6) of zero salt concentration and large separation between the two interacting species, are available for ion-ion interactions from the Debye-Huckel theory and for ion-dipole and dipole-dipole interactions from more recent work (H'ye and Stell, 1978; Chan, Mitchell, and Ninham, 1979) which yields results identical to those of Debye and Huckel for ionic activities. Thus, the long range direct correlation function is based on these potentials of mean force, WLR. Then, our HNC approximation is LIM HNC LR I-*o c.. c.. (3-7) r ij _0_00 3 13 12 LR 2 LR 3 LR 1 W1 W cL. + 1 -+ (3-8) 13 kT 2! kT 3! kT The potentials of mean force, however, are unphysical inside the hard core of the molecules and must be set equal to zero. LR W R = 0 r.. < a.. i] 1] ji (3-9) LR LR W.. = W.. r.. > a.. ij ij] 3 13 1 where aij = (aii + a..) = distance of closest approach of species i and j At contact and inside the core of the molecules, the direct correlation function is dominated by a very strong repulsion which is modelled as a hard sphere interaction. To obtain the appropriate expressions for the hard sphere direct correlation functions, the Percus-Yevick theory (1958) was used since it has been shown to give a compres- sibility equation of state which is in good agreement with simulation results for hard spheres (Reed and Gubbins, 1973). The Percus-Yevick (PY) microscopic direct correlation function for hard spheres is zero outside the core. Thus, HS PY-HS c.. = c.. (3-10) HS uS/kT PY-HS HS (1 ijkT cij = gij (1 e ) (3-11) u. = 0 r.. > a.. 13 13 where HS u.. = r.. < a.. 1] 1] 13 Although the PY microscopic direct correlation function is formally used in the development that follows, it was not actually employed in obtaining the final expressions for the DCFI's. Rather, the expression for the hard sphere chemical potential as derived from Percus-Yevick theory through the compressibility equation was used together with equation (2-1) to obtain the desired relation (see Appendix A). Although the more exact Carnahan-Starling (Carnahan and Starling, 1969; Mansoori, Carnahan, Starling, and Leland, 1971) expression could have been used, it is somewhat more complex and relatively little improvement in accuracy would be expected. At this point, we have established a viable, albeit traditional, theory for the behavior of the direct correla- tion function as r.. m and at r.. < a... However, many 1] I] J1 interactions which are important in aqueous electrolyte systems such as hydration of ions by water, hydrogen bonding between water molecules, and ion pairing are strongest at r.. just outside the core. Further, this is that kind of interaction for which liquid state theory is not well developed. Therefore, we attempt here to develop a method for interpolation of the direct correlation function between long and short range. Because generally available theory offers little guidance, the method can at best be semiempiri- cal. For this purpose, the Rusbrooke-Scoins expansion of direct correlation function (Reichl, 1980; Croxton, 1975) for species i and j in a system of n kinds of species is now introduced. cij(T,p,r,rj ,wi'O.) = fij(T) + n + Z pk ijk(T) + (3-12) k=l where (T) = J f. fi fj dr ijk ij ik jk d-k k Since equation (3-12) represents the entire direct LR HS correlation function including cij and cij, these two must be subtracted to obtain the interpolating function. There- fore, the complete model for the microscopic direct correla- tion function for species i and j in a system of n species is c.. = c.H + c. + Ac.. (3-13) ] i i] 1] HS LR where cij is defined by equation (3-11), c by equation (3-8), and HS LR Ac.. = c c.. c. (3-14) 13 1] 13 3 which is approximated by the Rushbrooke-Scoins expansion as HS- LR n Ac.. = (fi.. f f ) + I (p ] i] ij k= k ijk HS o LR P H P o L (3-15) k ijk k ijk (3-15) LIM o LR LIM where = r.. P k ijk 13 k ijk I o The series in equation (3-12) is truncated at the first order term in density to be consistent with the HNC theory and because inclusion of the more complex higher order terms was empirically unnecessary. For the sake of simplicity in notation equation (3-15) is expressed as Ac = Af.. + n po LR Skl k ijk ijk3-16) where Af.. = f.. fHS fLR 13 i] ij ij HS A No attempt was made in this work to analytically calcu- late the coefficients in equation (3-16); rather, their spatial integrals were fitted to data. The importance of equation (3-16), however, is in providing a theoretical framework for describing the properties for a class of molecu- lar interactions which are not well understood. Thus, the first term represents the contribution of pairing or repulsion in the case of ion pairs, solvation in the case of ion-solvent pairs, and hydrogen bonding in the case of solvent pairs. The second term represents the effect of a third body (k) on the direct correlation between species i and j. If one or two of the three are solvent and the rest ions, then this term is dominated by hydration. If all three species are ions, then this term is dominated by ion association or repulsion. The physical significance of these terms will be discussed further below. As pointed out in Chapter 2, solution properties are related to spatial integrals of the direct correlation function. In order to relate this model to thermodynamic properties, equation (3-13) is integrated over angles first and separated later. Thus C (T,p) = 47'p f S o 0 ij 13 Cj(T,P) = CHS + CLR + C. (3-19) ij 13 1j 13 HS where C is obtained directly from the chemical potential i3 as shown in Appendix A. Thus, CLR is defined by ij LR 47p LR 2 C1 = f Ci kT o i] ) ij iJ 2 p LR 2 LR2 2dp LR 3 2 +2p> <(w ) 2> r2 dr.-. 2 f <(w ) > r dr. kT o j 13ij 3kT 0o ij J + (3-20) Lastly, AC.j is defined by formally integrating equation (3-16). n LR ACij = pAFij(T) + p (kAO(T) pk(T)) (3-21) k=l ijk ijk where AF(T) = 47 J ij o ] W 3 13 A(T) = 47 f ijk o ijk 13 LR 2 LR > 2 (T) = 47 ijk o Equations (3-19), (3-20), (3-21), and the expression for C. from Appendix A are the general forms of the model for species direct correlation function integrals. To obtain practical expressions one needs merely to introduce the appropriate pair potential and potential of mean force into equation (3-20) and perform the indicated integration as illustrated in the sections that follow. Since the coefficients in equation (3-21) are fitted to data rather than evaluated analytically, it is of importance to develop mixing rules to reduce the amount of data necessary to model multicomponent systems. The aim here is to predict all the coefficients from quantities associated with no more than two different species so that only binary or common-ion solution data would be required. For aqueous electrolytes, the situation can be improved due to the relative simplicity of ion-ion interactions which can be generally scaled with the ionic charge (Kusalik and Patey, 1983). Thus, two and three ion coefficients are expressed from quantities related to a single ion. If i, j, and k are ions, then AF (T) 1 (AFi + AF ) (3-22) ij 2 ii ji AQ(T) (A + Aj + AD ) (3-23) 3 iii jjj kkk ijk LR L(T) (L + .. + LR ) (3-24) ijk 3 11i 33 kkk If one or two of the species i, j, and k are solvents while the remainder are ions, then the mixing rule must be expressed from quantities involving each of the species and water. The reason for this is that ion-solvent inter- actions cannot possibly be predicted from solvent-solvent and ion-ion interactions separately. Therefore, if i is an ion and j a solvent, then AF..(T) = AF.. (3-25) If i and j are ions while k is a solvent, then AO(T) = (A ii + A k) (3-26) ijk 2 LR (T) ( + LR (3-27) i 2 iik jjk ijk If i is an ion and j and k are solvents, then AM(T) = A-jk (3-28) ijk LR LR $(T) = D(T) (3-29) ijk ijk Lastly, if i, j, and k are all solvents, then AF(T) = AF. (3-30) ij 13 AI(T) = Ai. (3-31) ijk ijk LR LR ((T) = ijk (3-32) ijk It should be noted that these additive mixing rules are not the only possible ones. In fact, theory would suggest that geometric mean type mixing rules might be more appropriate. Geometric mean rules, however, only work for positive quantities which turned out not to be the case with our empirically fitted coefficients. This situation is further discussed in Appendix C. The last point that needs to be addressed here is the extension of the model to multisolvent systems. First, the extension of the expression for c. is well known. LR Second, the extension of equation (3-20) for cLR requires potentials of mean force applicable to the system. Assuming all solvents are dipolar requires only knowing the dipole moment of each of the solvent molecules and the dielectric constant of the solvent mixture. Neither of these are expected to present a problem in general. Third, the exten- sion of equation (3-21) for Ac.. involves a few more coeffi- cients and slightly different mixing rules for some three body terms. Thus, while equations (3-22) to (3-27) would remain the same for all solvents, equations (3-28) and (3-29) where i is an ion and j, k solvents would be altered to ijk D(T) = (LR. + LR (3-34) ijk 2 ii3 ikk ijk which reduce to the previous result only when j and k are equal. Here, any nonadditive interaction between j and k has been tacitly ignored because the difference in the interactions between different solvents is likely to be less important to direct correlation function integrals than that from the much stronger ion-solvent interactions. This assumption is based on previous investigation of solvent-solvent interactions which are dominated by angle independent forces (Brelvi, 1973; Mathias, 1978; Telotte, 1985; Campanella, 1983; Gubbins and O'Connell, 1974; Brelvi and O'Connell, 1975). Finally, equations (3-30), (3-31), and (3-31) where i, j, and k are solvents would become AF (T) (AF + AF ) (3-35) ij 2 ii Fj 1 AM (T) = (At. + A + ) (3-36) ijk 111 331 kkk LR LR 1 LR LR LR f(T) = i ii + + k) (3-37) ijk 3 111i 333 kkk ijk The above mixing rules for an aqueous system (single solvent) have been tested against data for a number of salts and may be regarded as established. The rules for a multisolvent system, however, have not been tested. They can only be seen as physically reasonable in the light of previous experience but still tentative. The next two sections deal with the application of the theory developed here to specific interaction in order to construct practical expressions. Expression for Salt-Salt DCFI The salt-salt direct correlation function integral (C a) can be expressed as a stoichiometric sum of ion-ion DCFI's (c. ) given by equation (2-11). n n v. v (1-C..) 1 C = 1 la 18 (2-11) i=l j=l ae It is, thus, only necessary to develop general and practical expressions for the ion-ion DCFI's and insert these into equation (2-11) to obtain a general expression for the salt-salt DCFI. The basic model for ion-ion DCFI's is represented by equation (3-19). The expression for CHS ij has been developed in Appendix A and that for AC.. is given by equation (3-21). This section is then chiefly concerned with performing the integration in equation (3-20) to LR obtain an expression for CLR The pair potential between two ions is given by Z.Z .e2 LR z _ u. L3 (3-38) 13 r Here the potential of mean force is approximated by a gener- alized form given by the Debye-Huckel theory. Z.e K(a. .-r. ij) "LR = i e r > a.. (3-39a) ij kTr.. D(1+Ka..) l] LR W. = 0 r < a.. (3-39b) ij 13 where K2 4ne2 n 2 K DkT zi Pi = Debye-Huckel i=l inverse length. D = the dielectric constant of the solvent or mixture of solvents. Insertion of equations (3-38) and (3-39) into equation (3-20) gives 2 LR 4TpZ Z e 2 C f r.i d r.. ij kT i 2 2 4 2rpZ Z .e 2Ka.. + 1 1 e 13 J (DkT) (1+Ka..) a 336 3Ka.. 2JrpZ.Z .e 3Ka i e ] f 3 3 3(DkT) (1+Ka..) a.. 1J 1J -2Kr.. e 13 dr.. - -3Kr. 17 e i rj e dr.. + r.. 13 i] The first term of equation (3-40) contains a divergent integral. However, when it is introduced into equation (2-11) which relates it to thermodynamic properties, electro- neutrality makes the coefficients of the integrals sums to exactly zero. (3-40) 2 n n SkTi 1 Z vjZ f r.. dr.. = 0 (3-41) v V kT i=la i j=1 1 n where v. Z. = o i=l ic 1 The second term of equation (3-40) is integrable and contains the implications for DCFI's of the Debye-Huckel limiting and extended laws (see equation 2-54). Then, 27rpZ2 2e 2Ka.. m -2Kr.. i e 13 e dr.. (DkT)2(1+Ka..)2 a. 1J 2 2 -1/2 Z.Z.S pI = 1 'Y (3-42) 1/2 2 (l+a..IB 1/2) where 2e6 1/2 where S = ( 2 ) Y D3k3T3 K = B I1/2 Y e2 1/2 B =K I /2 DkT 1 2 1 2 Zi Pi i=l The third term of equation (3-40) is also integrable but more complex. The integral is the first order member of a class of functions known as the exponential integrals. These cannot be evaluated explicitly but a number of asymptotic expansions and numerical approximations are available (see Appendix D). It is convenient to express the integral in dimensionless form. Letting X = r/ai. then -3Kr. e 13 a idr.. r.. 13 -(3Kai)x = e- dx = E (3Ka..) 1 x 1 l] (3-43) where E1(3Ka ij) = the first exponential integral The third term in equation (3-40) becomes i ] 27rpZ.Z .e 3(DkT) (l+Ka..) 3Ka.. -3Kr.. e 13 r..- dr.. a.. 13 a.3 3a..B II/2 3 3 2 3a. y12 Z.Z Spe Pe S(l+a. .B E (3a..B 11/2) 3 1/2 13 (l+a..iB I 1] Y (3-44) which contains the implications for DCFI's to a higher order limiting law for unsymmetric electrolytes (Friedman, 1962). Because of electroneutrality, this term, when inserted into equation (2-11), is always very small for symmetric electrolytes, and it approaches zero as the con- centration of salt decreases. For unsymmetric electrolytes, however, the sum over the ionic charges is not small and this term actually diverges logarithmically as the salt concentration decreases. To further explore the relation of (3-44) to Friedman's limiting law and to elucidate the low salt concentration behavior, the exponential integral (E1) can be expanded for low values of the ionic strength (I 0). E (3ai.B I1/2) = n(3aiB I1/2) a + O(I1/2) (3-45) where a = 0.5772 = Euler's constant. This expansion is valid only at extremely low ionic strength. Equation (3-44) then becomes ji S2p PE (3aB 11/2- 3 Y olEl(3ai ) Y z33 Z 1 3 ol 2 n + + n 3a + 0(11/2) (3-46) where InI diverges as I 0 while a + In 3a..B are all 13 Y constant. The contribution of Friedman's limiting law to the activity coefficient of a salt (a) is 1 n .3 2 V x ia i FLL 1 _i=l 2 FLLn y1 i= Z S2 IknI (3-47) a 3 n 2 Y i=l La 1 and by taking the first derivative with respect to the mole number of a salt 8 at I o, FTL 2 P Sn y Sypo n n 3 2-1L 01 3 ..zz N 23v v X 3 3 oNo ITPN 3vaB i=l j=la Z T,P,N 1 1 (- An I + -) (3-48) Z 2 Rearrangement of equation (2-24) gives N any pV V0 N -C oa oB (3-49) v @N 0CaB$ v K TRT S oBS a c aB T,P,N 4 If equations (3-48) and (3-49) are compared, it is clear that the contribution of Friedman's form of the limit- ing law to the salt-salt direct correlation function integral is 2 P cFLL S ol n n1 CF LLX z z (- Zn I + ) (3-50) U 3vv ia j i j ]2 2 S 3vaeB i=1 j=1 Comparing equations (2-11) and (3-50) gives the ion-ion DCFI. 3 3 CFLL S2 P 1 + 1) (3-51) ij 3 y ol 2 2- Substitution of equation (3-46) in equation (3-40) gives the expression for the limiting contribution of the third term in equation (3-40) to the ion-ion DCFI. 3 3 SS2 P ( nI + + In 3a.. B) (3-52) 3 y o 2 13 Equations (3-51) and (3-52) have essentially the same behavior as I 0 since they differ only by a small constant which is negligible compared to ZnI as I 0. Therefore, equation (3-44) contains the higher order limiting law. LR The general expression for C is -1/2 2 2 S pl12 n n n i. v Z.Z. cLR 4v v 1/2 2 4t B i=l j=l (l+a..B I ) 13 Y 1/2 S3a. .B I / Sp n n i. Z3Ze E (3ai..B 3v V 1 1/2 3 3a i=l j=l (1 + a..B I2) 11 Y (3-53) The expression for CHS is aB HS 1 n n HS CS 1 V V. V. C (3-54) a ji=l j=l Lastly, the expression for ACB is n n AcB = v I Via. AF. + a B i=l j=l aB n n n + v (p A po ) (3-55) Va i=l j=l k=l a j k ijk kjk Equations (3-53), (3-54), and (3-55) form the complete model for the salt-salt DCFI. S = CHS + CLR + AC (3-56) CaB CaB aB Since the limits of DCFI's as salt concentration approaches zero are well defined, it is advantageous to use equations (3-53) to (3-55) to model the deviations from this limit. To this purpose, the infinite dilution limit of the salt-salt DCFI is now explored. From equation (2-42a) N n p V N1 N y = (1-Co) s S(1T-C ) ola (2-42a) S N a v vT,P, K RT S oS T~~Noy a6 it is seen that the constant temperature and pressure limit has divergent terms associated with the activity coefficient, a first constant related to the partial molar volume, and a second constant associated with the activity coefficient and which is not so well defined. This second constant is loosely related to a term linear in salt density which often appears in empirical expressions for the salt activity coefficient (Guggenheim and Turgen, 1955; Guggenheim and Stokes, 1969). In the present model the divergent terms are contained in equation (3-53). The first constant can be calculated directly from infinite dilution partial molar volumes and solvent quantities. The second constant must be fitted to data using terms from equation (3-55) which have only ion-ion and long range ion-water correlations. This reflects the fact that triple ion direct correlations are zero at infinite dilution and any contributing short range ion-solvent correlations would generally be contained in the first constant. Thus, 0 LIM LR TB C XL1 (C aB- A ) (3-57) a X +91 aB -ca aB where TB n n n where VAC = v. v jB ,ijk SB i=l j=l k=l p- o_ o p V V n n 1-c oa oa B p (F (1-C) KRT v v v (AF Sa i=l j=1 P LR + P 1l ) (3-58) 01 ijl Finally, the general expression for the salt-salt DCFI model including the infinite dilution limit is m LR HS HSo' 1-C = (-C ) (C -HS C aB cB aaB ca6 ) TB TBm (A AC ) (3-59) where cHS = LIM HS where C C aB Xol l aB TBC LIM TB AC AC aB X oll aB P Pol n n = 01 y y v PA V v I I itX jp o1 ijl Va i=l j=l Although equation (3-56) can be used in place of equation (3-59), it was felt that the latter was more appropriate for calculations at constant temperature and pressure. Therefore, equation (3-59) was used in the com- parisons and correlations in this work. In calculations where pressure varies, equation (3-56) would be more convenient since it would eliminate the need to obtain partial molar volumes as a function of pressure. For illustrative purposes, equation (3-59) will now be written for a binary system consisting of a solvent (1) and a salt (2) which dissociates to formvy cations and v anions. n = 1 + v + V = V + v2 = + - (1-c22)LR HS 22 = 22 22 (C22 SHSc 22 TB TB- - (ACTB ACT ) 22 22 (3-60) where TB TB- C P (AC22 AC ) -2 2 + 2v+v _(plAl+- 2 [v (p AD + p AD + p AD ) + + 1 1++ + +++ -++ + p+A++- + p-A(_+ ) + 2 + V (PlAl--_ + p+AI+__ + pA___ )] + P 2 (P p ) 2 9 01 [v 2A + 2v v A+ + v2A1 ] 2 + ++ + 1+- 1-- V2 (3-61) R S -1/2 2 4 cLR= ++ + 22 2 1/2 2 4v (l+a BI ) 2 ++ y 22 24 2v _Z2Z2 v2 Z + ( BI1/22 + (1/2)2 (l+a+ BI ) (l+a BI ) +- -- y S3a++B I1/2 2 6 + y + + (1+a++B I/) ++ y 2v 3a B 1/2 3 3 3a+- B 2+ Z_ -+ E (3a+ B 1/2) 1 /7 E (3a B 1 /2) 1 -- 2 6 3a B II/ 2V -- y v_Z_e (l+a B 11/2)3 2 P 2 SPol(Vo2) p 2 1 -C22 [V (aF SC22 =2 -2 + ++ v2 1RT 2 + 2v+v_ (AF_ (CHS HSm) 1 22 22 2 2 + 2V v_ (CHS + +- P LR - P l++) + o1 1++ P LR 2 P LR - Po (l ) + v (AF Po l ) S1+- -- 01 1-- [v2 (CHS ) + + ++ ++ CHS) + v2 (HS HSm) +- + __ 2 S3 --v 3v2 (3-62) (3-63) (3-64) (1 B1/2 3 (l+a BI ) +- Y E (3a__ By1/2 ) Expression for Salt-Solvent DCFI The development in this section parallels that of the previous one. Thus, a general expression for the solvent-ion DCFI is derived and then inserted in equation (2-11) to yield the salt-solvent DCFI relation. Although any type of interaction can, in principle, be included, it was assumed here that ion-solvent interac- tions are dominated by dipole-charge forces at large separa- tion, and no other interactions were included. The pair potential for an ion (i) and a dipolar solvent (1) is LR l e u = cos 6 (3-65) ril where pl = the dipole moment of solvent 1 in Debyes. S= the Eulerian angle between dipole and charge. The potential of mean force is approximated by a func- tional form inspired by some recent applications of the mean spherical approximation (Chan, Mitchell, and Ninham, 1979) and of perturbation theory (Hoye and Stell, 1978) to nonprimitive electrolyte models. wL lK(ail-ril LR i 1 e il il 2 (cos 0) e r > ail (3-66a) kTri LR Wi = 0 r ail (3-66b) iiil -1 where a is a universal constant that we have set equal to 4.4 empirically. Since equations (3-65) and (3-66) are functions of orientation, it is necessary to first perform the integration over angles as indicated in equation (3-20). = u LR dw dwl (3-67)il 2 i1 i 1 where dwo = sin e.di.d4. 1 1 1 1 S= f dw. = f sin 6.de. f2 7di = 47 1 0 1 1 0 i When the integral in equation (3-67) is evaluated, it is found that LR = 0 (3-68)il W The second term in equation (3-20) has LR 2 1 I (W)LR2 <(W ) > (W) dw. dw (3-69) il Q 2 il 1 After the integral in equation (3-69) is evaluated, it gives L2 2 2K(ail-ril) 2 Z ile )e <(W i) > = T 4 (3-70) ilDkT 4 r.l The third term in equation (3-20) contains LR 3 1 )3 <(W ) > ( dw d ii 2 il 1 which also equals zero. LR 3 <(W ) > = 0 Therefore, for ion-dipole pairs there is only one term in equation (3-20). i 2 Tr c 2 2Ka -2Kril LR 2I 1I ) 2 il e e il 3 P DkT ) e f - aij il 1] 1 dril (3-71) (3-72) (3-73) The integral in equation (3-73) is also an exponential integral (E2) which is expressed in dimensionless form as before. C -2Kr il -(2Ka )X S1 il E (2Kail ) 2 e dr = 1 f e dX = ail 2 i1 a.i 1 a ail ril Equation (3-73) then becomes 2Ka 2 i1 iR -2T lZ el e E2(2Kail 11 3 DkT ail (3-74) (3-75) Then, the general expression for the salt (a) and solvent (1) DCFI is S2a iB I1/2 2 11 LR 27p ea 2 2 n E. 1/2 Cl = ( ) E (2a il ) al 3v DkT i=l ail2 il y a 1=1 11 (3-76) The expression for CHS is given by n HS 1 HS Cl v Via Cil a i=l and the relation for ACal is n n n AC v= -P- v. AF. + p vi al a il a il V a il k la a l=c a i=l k=l (Pk Ailk (3-77) (3-78) o LR - p ik) k ilk Again, equations (3-76), (3-77), (3-78), and (3-79) form the complete general model for the salt-solvent DCFI. SCHS LR C = CHS + C + AC al al al al (3-79) As previously discussed, it is convenient, particularly for isobaric calculations, to use the model only for 67 deviations from infinite dilution. (For nonisobaric calcu- lations, equation (3-79) would be more appropriate.) The infinite dilution limit of Cal is given by equation (2-40) LR and that of CLR is (see Appendix D) acl P 2 P2eep 2 n y Z2 LRO LIM LR 2ol ( ea 2n i (3-80) 1C C = ) P 1 a1 (3-80) al X ol+1 l 3v DkT 1 ail ol a i=l a HS while the infinite dilution limit of C H is formally HSm LIM HS (3-81) C1 =X1 Cl (3-81) Lastly, the infinite dilution limit of AC is P A LIM ol n P LR al X o1 al i (iFil Pol ill ol0 i=a P Pol n + 1 o' pnolA l (3-82) iV Vi P1 ill a i=l The complete general relation for the salt-solvent DCFI including the infinite dilution limit is 00 V oc HS HS o LR LRm 1- IT- RT Cl- Cal) (C Cal (ACal ACal) (3-83) Finally, equation (3-83) will now be written for a binary system consisting of solvent (1) and salt (2) with v+ cations and v_ anions. V o 02 (HS 21 v K2 RT 21 HS LR - C2 ) (C2 21 21 (AC21 AC21) where HS _HS- 1 HS 21 21 v2 +(C+1 "2 -HS + (HS HSm) +1 ) -1 -1 ) 2 CLR CLR 2- r ( ) e ) 21 21 3v DkT 2 2 2 + + 1 [ a+1 2a +B I1/21/ (pe Y E2 (2a+1B 2 1y \) z2 -p +T ol a_l AC C = [v (AF+ 1 C21 + +1 2 2a 1B BI/2 (pe E2(2a lBI2) p)] (3-86) P LR P LR - i ) + _(AF Pol )] 01 +11 -1 01 -1 (p p ) + + ol + p+ p + V2 + +1 1+1 + P+A++ + -+ V2 SLR- 21 (3-84) (3-85) + v_(plA11_ + P+A+ + P_A __ )] P 2 (Pol [v+ &1+1 + v _l_ (3-87) Expression for Solvent-Solvent DCFI The solvent-solvent direct correlation function integral has the simplest relation since the solvent does not dissociate so the species and component integrals are the same. As previously noted, any type of interaction can gen- erally be included in this theory, but it was assumed that solvent-solvent interactions at large separation are domi- nated only by dipole-dipole forces. The solvent (1)-solvent (1) pair potential is LR 1 1 u 3 1 (3-88) rl r11 where ( = 2 cos 611 cos e12 sin 011 sin 012 cos (11 012 611' 11 = Eulerian angles of solvent molecule number 1. 612' 12 = Eulerian angles of solvent molecule number 2. The potential of mean force is approximated by a function inspired by previously mentioned work (Chan, Mitchell, and Ninham, 1979; Hzye and Stell, 1978). K(a -r) LR 1 1 e 1 W = kr D W11 3 D kTr11 11 LR W I 01 11 r > al r < al 11 (3-89a) (3-89b) Again, equations (3-88) and (3-89) are inserted into equation (3-20) and the required integration over angles performed. SLR = L l d11 11 Q 2 11 d11 12 dli= sin 1i deli d li S= f d6l = f sin 6li dli o 2Tr f di = 47 0 The integration of equation (3-90) gives LR 011 W The second term in equation (3-20) has LR 2 1 LR 2 <(W ) > f (W ) dw dw <11 ( 2 (W11 d11 d12 which yields upon evaluation, LR 2 1 i l11 2 <(W ) > (D -) 11 w 3 DkT 2K(al-rl) 2K(11-11 e 6 11 (3-93) where (3-90) (3-91) (3-92) The third term in equation (3-20) has LR 3 1 LR 3 (W ) > (W ) d <11 2 11 1 12 which becomes upon integration LR <(W )> = o 11 (3-94) (3-95) Thus, for dipolar solvents only one term of equation (3-20) remains after the angle integration. R = 2 e2Kal -2Krl 1 3 (DkT e r 4 all rll dr11 (3-96) The integral in equation (3-96) is also an exponential integral (E4) which can be expressed in dimensionless form. -2Krll -(2Kall)X S11 -(2Ka11 )X E (2Ka ) e 1 e 4 11 d 4 dr l = -- -4 dX = 3 al r11 all 1 X all (3-97) Equation (3-96) is then transformed 2alB Il/2 LR 4 ( 1 (1 1)1 C11 3 DkT 3 all E (2a B 11/2) 4 11 (3-98) which is the general expression for the solvent-solvent DCFI. Since the solvent does not dissociate, there is no HS summation over species in C11. However, AC11 does have a sum over third bodies. n CFo R ( 9LR AC11 = pA1 + (Pkllk llk (3-99) k=1 Equations (3-98), (3-99), and (3-100) form the complete general model for the solvent-solvent DCFI. C = CHS + + AC1 (3-100) 11 11 11 11 Again, the infinite dilution limit of C11 is introduced so that for isobaric calculations the model need only account for deviations from the infinite dilution value. Also, equation (3-100) would be more practical for nonisobaric cases. The infinite dilution limit of C11 is the bulk modulus of the pure solvent given by equation (2-41). The infinite dilution limit of CLR is given by 11 P LRm LIM cLR 4pol r T D 2 CLR- XLIM CII l ( (3-101) 11 X 01 11 3 fDkT )-0 11 and that for C S is formally 11 HS- LIM HS C =1 Xol1 CI (3-102) 11 X -1 11 (3-102) ol The infinite dilution limit for AC11 is given by AC LIM ACP (AF P LR P 2 A S11= Xol 11 01ol 11 ol 111 ll (3-103) Finally, the complete general expression for the solvent-solvent DCFI including the infinite dilution limit is 1 HS HS" LR LR" 1 C (Cl l ) (C C ) polKRT --11 11 11 11 (ACll ll) (3-104) Again, the application of equation (3-104) to a binary system consisting of solvent (1) and a salt (2) with v + cations and V_ anions is shown. However, for the solvent- solvent DCFI all of the terms except AC11 appear similar to the general case since they have no summations over species. Thus, only AC11 is illustrated below. oo P P LR AC ACI = (P p )(AF Po + 11 11 01 11 P ll + P(P1 111 + P+ A)11 + pA-11) - (p P)2 A (3-105) 01o 111 Summary A general statistical mechanical model of the direct correlation function has been presented. In principle it is applicable to any system, but it has been specialized here to treat strong electrolyte solutions. The next chapter shows the application of this model to six aqueous strong electrolyte binary solutions. As a preview to the calcula- tions, the relative magnitude of the three contributions HS LR to the DCFI (C C R AC ) will now be discussed, the model parameters will be listed, and the sensitivity of solution properties to parameter value considered. The salt-salt DCFI is dominated at very low salt con- LR centration by C n which contains the long ranged electro- LR static interactions. However, the magnitude of C 8 decreases very fast as the salt concentration increases so that above 2M or so in salt density the dominant term becomes C HS This reflects the increasing shielding of electrostatic forces by more ions that more frequently repel each other. AC makes a contribution that is generally not dominant in either regime but is always numerically significant above 0.5M. HS The salt-solvent DCFI is always dominated by C H with LR Cl making a small but not negligible contribution. Due to the relative strength of the short ranged hydration interactions, ACal makes the largest contribution after CHS Cl* "al" The solvent-solvent DCFI is also dominated by CHS over the entire range of salt concentration up to about LR 6M. C11 makes a negligible contribution reflecting the relative weakness of long range dipole-dipole interactions. HS Again, the largest term after CI is AC which contains the short ranged hydrogen bonding between solvent molecules and the hydration related effect of an ion on two solvent molecules at short range. The parameters of the model are species specific and universal. It is, therefore, necessary to build only a relatively small set of parameter values to predict the behavior of a large number of systems. Thus, a hard sphere diameter (a. ) for each species is required for CHS and 11 a LR CaB (where a, B can be salts or solvents). To avoid con- fusion, the parameters for ACa will be those of a system with one solvent (1), one salt (2), and many ions (i, j). Then, AC involves AF PLR which is ion independent, 11 11 ol 111 A11 which is usually neglected, and Al for each ion. 111 ill AC has AF POaLR i l and A. AC22 includes 1A2 i 01o lil 1il' 1i22 AF. p A and AoP.... This totals to two solvent 11 ol iin, ii, 111 specific parameters if AO11 is neglected and six parameters for each ion (note that A lii = A.il and AD. = Al ) lil ill 111 ii three of which involve solvent-ion pairs. Properties predicted with the model are most sensitive HS to the value of the hard sphere diameters because the C is a very strong function of the diameters. But it is 76 not as sensitive as is the case with other models. This is due to the fact that the two body coefficients AF.. 13 are fitted to infinite dilution quantities that include HSm C so there is a degree of compensation for changes in the diameters. The sensitivity of the results to the value of the coefficients in ACa is generally small since they make a small contribution to the DCFI's. CHAPTER 4 APPLICATION OF THE MODEL TO AQUEOUS STRONG ELECTROLYTES Introduction In Chapter 2, the formal relations between DCFI's and thermodynamic properties were introduced. In Chapter 3, a model expressing the DCFI's in terms of measurable variables was constructed. In the present chapter we illustrate the use of the formal relations and the model in the calcula- tion of thermodynamic properties. We also explore the scheme used to fit model parameters; further we compare calculated values to experimental ones for the salt-salt, salt-solvent, and solvent-solvent DCFI's and for the bulk modulus, partial molar volume, and salt activity coeffi- cient. Finally, a discussion of the above results and a few conclusions are presented. The use of Fluctuation Theory in general fluid phase equilibria problems has been treated in detail by O'Connell (1981). The specific case of liquids containing super- critical components has been addressed by Mathias and O'Connell (1981) and Mathias (1978). The present treatment generally follows these developments, but there are important differences for the present case of electrolytes. Calculation of Solution Properties from the Model The formal relations between solution properties and DCFI's are given by equations (2-12), (2-34), (2-37), and (2-38) for a system consisting of no components, salts (a,B) and one solvent (1). a2enyc pv ac ap~ ~ ooY = V v (1-C ) V n V o OT = Ea Vx (1-Ca ST =1C n n aP/RT o o PT = E Z v VX X (1-C Tp a=l =l a oa oB aB a=1 B-l T,X NVa aQtny PK TRT a@i oT TIPN0 n n o o v v V Y v vvX X 0 a y=1 6=1 6 oy 06 [(1-C )(1-C a) (1-Cy )(1-C )] *Y (2-12) (2-34) (2-37) (2-38) ae P/RT 1 where = - ap I PK RT ST,N T In order to evaluate the change in solution density with pressure while the composition and temperature are constant, one needs to integrate equation (2-37) from a known reference density (pR) at the reference pressure (pR) at the temperature and composition (mole fraction) of the F system up to the desired density (p ) at the system pressure (P). n n F R o o P (T,P,X) v V X X (1-C )dp (4-1) RT E =i =l oa o R R T,N ( p R(T,R ,X) Equation (4-1) represents an implicit equation for the F unknown density (p ) which can only be solved numerically with realistic models. It should be appreciated that equation (4-1) cannot be applied to an isobaric change because that would imply that pressure, as well as temperature and composition, were F R held constant. Then p would be the same as p so the state of the system would not vary at all. To evaluate the change in solution density isothermally with varying composition, a different approach is required. To develop the necessary relations we start by considering that in Fluctuation Theory the pressure is treated as the 80 dependent variable, a function of temperature, density, and mole fraction. P = P (T,p,X) (4-2) Taking the total differential of pressure gives aP aP dP dp+ dT ap T,TN Tp, T,N p,N n o z aX a=2 oa dX oa T,pX oyfa If the change is isothermal and if we divide by RT, 1 dP T 1 aP 1- 3- dp RT 3p T T,N 1 + - RT C= o p S ax :2 oc dX TP oa T,pX oyfa By making some identifications we obtain n o B=I DP ax oY T,V,X / oy#a aN ax a Ny4 a VKT VK T DN 0 o3 oa N oy/8 N -v X Inserting equations (4-5) and (4-6) into equation (4-4) gives dP 1 aP dp + RTT RT ap T T a TN n 0 ct-2 n - SK RT 8=1 T dX 6 -vX B oa 6eS Boa (4-7) (4-3) (4-4) (4-5) (4-6) We next insert equations (2-34) and (2-37) into equation (4-7). n n 1 dP = [= V V X (1-C ) dp + RT T c=l 8=1 a oaoB 0 n n n + 0 V X (1-C pdXo (4-8) a=2 8=1 y=l Y 6 -vX c G B-uoc Equation (4-8) permits us to evaluate the change in solution density with both pressure and composition along an isotherm. This equation is also applicable to an isobaric and isothermal process where the solution density changes as a function of composition only. To obtain the density (p ) of a given solution at a known temperature, pressure, and composition (X ), we -o isothermally integrate equation (4-8) from the known reference density (p ) at a system temperature and a conven- iently chosen reference pressure (P R) and composition (X ) -o F F up to the desired density and composition (p and X ). It -o is suggested that for aqueous electrolytes the reference density be chosen to be that of pure saturated water at the system temperature. 82 F F SAT n n P (T,P,X ) P-P o 0 -o R = I v B SAT X X (1-Ca )) dp + RT SAT oc oaS Ra=l =1 Poi (T) T F n n n X o o o oa pdX + I I oy(-C) (4-9) a=2 8=1 y=l Y oy a6 BXa T In evaluating the integrals of equation (4-9) each integral involves variables appearing in other integrals. To explicitly find pF requires further manipulations discussed below. The activity coefficient on the mole fraction scale for any component (a) can be obtained by integration of equation (3-12) from the reference molar density (P R) to the molar density of each component (PF ) at constant temperature. F Y n p 1-C n v R dP oS (4-10) R R R p a T 7 8=1 8 Po T a P05 where Po5 = Xo8P Equation (4-10) is applicable to any isothermal change, isobaric or not, and the reference state composition where YaR = 1 need not be that chosen. However, for aqueous electrolytes it is natural to choose pure saturated liquid water at the system temperature as was done for equation (4-9). F F P (T,P,X)I n P (T,P,X) ZnYc = al dol + Z f 1-Ca dp pSATT T 8=2 o p OT ol (4-11) Equation (4-11) can be used for either isobaric or nonisobaric changes. In equations (4-9) and (4-11) one can use the DCFI model represented by equations (3-59), (3-83), and (3-104) for isobaric integration. But, for nonisobaric integra- tions with equations (4-1), (4-9), and (4-11) the DCFI model of equations (3-56), (3-79), and (3-100) will be more applicable because the pressure behavior of the DCFI infinite dilution limits, some of which involve salt partial molar volumes, is not generally available. The composition behavior of component activity coeffi- cients on the mole fraction scale at constant temperature and pressure could also be obtained from equation (2-38) with composition expressed as mole fractions. Thus, we express the differential of the activity coefficient of a component (a) as no 8nya dnya = z X dX (4-12) ST,P B=2 a TPX TIIoyX$ ,nya n eny o 8 ny aN = 3o o N(4-13) 06 T,P, Xo l T,P,Noa 0 N OY 8 oy/c OY7c oo N = N (4-14) ax 6 -v x N OY4 oyf - Inserting equations (4-14) and (2-38) into (4-13) and then putting the resulting expression into equation (4-12) gives n n n n o o PK RT o o diny = I dX I Z v TV T,P 6=2 =1 B O 0 Y=l 6=1 XoyXo (1-C 6)(-Ca) (-Cy) (1-C6)] (4-15) To obtain the activity coefficient, equation (4-15) is isothermally and isobarically integrated from the reference to the desired state. F n n n n X o o o o o pK RT S= I R do 6 v IX iV 6 a=2 B=l y=l C=1 X a 6 oX oB Y OB T,P Xo Xo[(1-C 6)(1-C ) (1-C )(1-C6) ] (4-16) n n e o o where = v v X Xo (1-C ) SRT = 8=1 B a o 6 = Kroniker Delta Equation (4-16) can only be used for isothermal, isobaric changes and thus either the DCFI model of equations (3-59), (3-83), and (3-104) or that of equations (3-56), (3-79), and (3-100) may be used. Equations (4-1), (4-9), (4-11), and (4-16) express integration of the DCFI model formally. However, these cannot be explicitly evaluated because of the multiple variables involved in the integrals. To actually evaluate these integrals requires a change of variables as discussed by Mathias (1979) and O'Connell (1981). Rather than give their formal equations, we now give the above relations with explicit expressions for the present DCFI model. Those parts that are analytically integrable have been evaluated while simplified integrals are given for the others. The DCFI model used is that of equations (3-56), (3-79), and (3-100) which does not contain the DCFI infinite dilution limits. This form of the model yields simpler expressions which can be applied to both isobaric and nonisobaric changes. We start by rewriting equation (4-1) as _pR pPY-HS(pX) F PY-HSpRX) RT RT RT F R - (p -p ) n n n n o 0 1 LR S 0 0 v v X X X J C L (t)dt - = oa 0 oy a U=l =l y=1 o F F RR n p p -p p 2 . i=l F F FR R R p P p -p p p 3 n j=1 n i=l FF P,F RR P,R P P P -pp p ol ol n where p(t) = I(pR + y=l1 X.X. AF.. - 1 3 13 n n SI X.X X Aj + j=1 k=l 1 3 k n n S z x.x. ,LR i=1 j=1 ijl (4-17) n F R o (P F-P R)t)= I v p (t) oy oy Y Y Equation (4-9) can also be changed to SAT PY-HS F F PY-HS R R P-P TT) P ( ,X ) P -(p ,X) 1 -o 0 RT RT RT n n n LR 0 FF RR a C t) v v (X p -X p ) fp (t)p (t) p dt - = 8 = oY -oy o o o p(t)p(t) a=l 8=1 y=1 o n n n n 1 oo o o 1 1-C (t) 2 ll yl =1 Y o P (t) (t) F (t) dt - a=2 6=1 Y=I 6=1 0 oy poa(t) a 6 -v p( rB 8 p(t) n F FFF R RRR S(X. X p p -X X. pp ) AF. - j=l 1 3 1 3 n n F F FFFF R R RRRR S(X. /X. X pp-X. X. A4. + j=1 k= i j P P -Xi j X P p p ) A ijk j=l k=l n n F F F F P,F R R R R P,R LR (X. X. p p p -X Xj p p ) p j= p ol P ijl j=1 n 1 n 2 2 i=w n 1 3 i= n 1 3. 1+ where R P,R SAT S=P = T) R ol ol R X = 1 X R =0 a 1 oU n o X. = v. X 1i L )i Xot a=l RR FF R R p (t) = X p + (X p X p )t oa oa oa oa x FF R R X p -X p F (t) op(t) a p(t) n 0 FF R R p o (t) (X p -XR p ) oa p oB O t= ) 2 p(t) (4-18) 88 The expression for PY-HS is given by equation (A-l) and LR that for C by equations (3-53), (3-76), and (3-98). In a similar fashion we transform equation (4-11) to 1 nya = a n i V. i=l a F Pi i - n ) (p R p i n o 6=2 F n Ij= j=1 n 1 1 a 1=1 PY-HS TpF PY-HS pR i T, ) pi TT, ) (1 - RT RT LR 1 C LR(t) F R) cl d ol 1o p(t) 1 CLRt) F - ) a dt + n - Sp(t) o p SF ia j R - pj) AF. n j=l n j=l n F F I .. (PijP k=l la 3 k F P,F S. (p .p i cc ol0 n o Pi = p v iaX a=1 R R h j ijk R P,R LR j- ol )ijl n o = v. P l ia oa a=1 and the other quantities have been defined above. n a i= 1 n 2v + n 2v i a i=1 where (4-19) |