MICHAEL J. PIKAL
3124 mixtures of simple molecules. If only thermodynamic contributions are considered for analyzing the data on n-alkanes, it appears that [qlV works better than [ q ] ~ . The reason for this is not clear. For longer chains (n > 25-30), continuum hydrodynamics is quite adequate, if dimensions are calculated according to the rotational isomeric states theory, as shown in Figure 1, and continued incorporation of thermodynamic contributions leads to the underestimation of [q]. One is thus led to believe that after a
certain value of n, the thermodynamic contributions decrease rapidly in significance and can be effectively neglected. This may be explained by solvent entrapment in the polymer domain so that solvent-solute contacts are no longer important in viscous energy dissipation.
Acknowledgments. This work was supported by grants from the National Institutes of Health and the Alfred P. Sloan Foundation.
Theory of the Onsager Transport Coefficients I,, and R,,for Electrolyte Solutions by Michael J. Pika1 Department of Chemistry, University of Tennessee, Knoxville, Tennessee 57916
(Received April 18, 197’1)
Publication costs borne completely by T h e Journal of Physical Chemistry
A simple limiting-law expression for the Z,j in multicomponent electrolyte systems is presented. An extended theory of the Zij and Rii coefficients is developed for symmetrical binary electrolytes. A term representing the effect of ion-pair formation appears in the extended theory as a natural consequence of the electrostatic interactions. The mobility of an ion pair is not an adjustable parameter but is determined by the theory. The theory includes terms of order c’/a, c log c, c, and c8/2. The extended theory is in good agreement with experiment. The physical interpretation of the lij and R t j coefficieiits in multicomponent and binary systems is discussed in terms of electrolyte theory. Certain aspects of the current “intuitive” interpretations given l , j and Rij are found to be inconsistent with electrolyte theory.
I. Introduction The phenomenological theory of irreversible thermodynamics may be applied to transport phenomena in electrolytes, resulting in a general description of ionic transport in terms of the Onsager “conductivity” coescients lij.1-3 An alternate, but equivalent, description in terms of the “friction” coefficients, Rij,may also be used. Ratios of the form Zij/c (c = concentration) are mobilities,2 and products of the form c,R,, have been interpreted as friction coefficient^.^ From the point of view of irreversible thermodynamics the 1,j or Ri,are more fundamental than the conventional transport properties, such as equivalent conductance, transference number, and diffusion coefficient.2 Therefore, systematic studies of the laj or R,, should lead to new insights into the nature of transport phenomena in electrolytes. I n this paper, we consider several extensions of earlier theoretical on the concentration dependence of the Id, and Rtj coefficients. Before proceeding with a The Journal of Physical Chemistry, Vol. 76, No. 20, 1971
discussion of specific objectives, it is useful to present the phenomenological relations of irreversible thermodynamics which define the ltj and Rtj coefficients. The irreversible thermodynamic description of the isothermal vector transport properties of an electrolyte solution may be given in the most general linear (one dimensional) form n
JI =
k=l
ltKXk
(i, k = 1, 2 . . ., n)
(1)
(1) D.G.Miller, J . Phys. Chem., 64, 1598 (1960). (2) D.G.LMiller,ibid., 70,2639 (1966). (3) D.G. Miller, ibid., 71, 616 (1967). (4) H.8. Dunsmore, S. K . Jalota, and R. Paterson, J . Chem. Soc., A , 1061 (1969). (5) L. Onsager and R. M. Fuoss, J. Phys. Chem., 36, 2689 (1932). (6) L. Onsager, Ann. N . Y . Acad. Sci., 46, 241 (1946). (7) L.Onsager and S. K . Kim, J . Phys. Chem., 61, 216 (1957). (8) J. Rastas, Acta Polytech. Scand., Chem. M e t . Ser., No. 50, 37 (1966). (9) H.Schanert, J . Phys. Chem., 73, 62 (1969).
THEORY OF
THE
Jt =
x
-
(2)
ciVto
+
-~ d/&’ = -d/hC ZkF-d6
(3) dY dY dY The numerical subscripts refer to ionic components 1, 2, . . . , n . The ionic conductivity coeficients l,, are defined by eq 1 in terms of linear relations between solvent-$xed flows J , and thermodynamic forces X k in mole units. Equation 2 expresses the solvent-fixed flows in terms of the concentration of component i (in moles per cubic centimeter), denoted by C,, and the velocity of ion component i (in centimeters per second) relative to the solvent, denoted by Vio. The thermodynamic force Xk is given by the negative of the gradient of the electrochemical potential of ion k , denoted by p k f . I n eq 3, XI, is related to the chemical part of the electrochemical potential, ptc, and the electrical potential, 4, in volts. The symbol F denotes the Faraday constant in coulombs per equivalent, 2, is the signed valence of ionic component i, and y is the distance in centimeters. For this choice of forces and flows, it is expected from statistical mechanical and other considerations10 that the Onsager reciprocal relations will hold. k -
3125
ONSAGER TRANSPORT COEFF~CIENTS
1ij
=
hi
(4)
The R i j (i, j = 1, 2, . . . , n) or “friction coefficients” are defined by2
Since the matrix of the l t j coefficients is nonsingular, the R f jmatrix (i,j = 1, 2, . . . , n ) is simply the inverse of the l f j matrix. The R f j coefficients also satisfy the Onsager reciprocal relations, R t j = R,,. It is conventional2 to define additional “friction coefficients’’ Roo and Rto (i = 1, 2, . . ., n) by the relation
The general purpose of this paper is to present some theoretical guidelines for the interpretation of l i j and R , in electrolyte systems. We have three main objectives: (1) to present a simple limiting-law expression for the Zij in multicomponent systems; (2) to extend the Onsager-Fuoss6 limiting-law theory to include terms of higher order in concentration; and (3) to point out certain aspects of the “intuitive” interpretations of lij and Rtj which are not consistent with electrolyte theory.
11. Onsager-Fuoss Limiting Laws A . Friction Coeficients. The theoretical description of ionic transport phenomena is usually given in terms of three effects: (1) the interaction of the moving ion with the surrounding solvent, which is assumed to be independent of concentration and therefore is related to the limiting ionic equivalent conductance; (2) long-range hydrodynamic interactions between the ions, termed the electrophoretic effect; and (3) longrange electrostatic interactions between a given ion and its asymmetric ion atmosphere, called the relaxation effect. Theoretical expressions for the relaxation effect and the electrophoretic effect are given in the classic 1932 paper of Onsager and FUOSS.~ These results are given for ionic motion in a general (multicomponent) electrolyte solution to order c”’, where c is the concentration, and thus are “limiting laws.” The limiting-law equations are discussed further by Onsager and Kim.7 Some of the results contained in these studies6J reveal certain deficiencies in the “intuitive” interpretations of the friction coefficients and thus deserve emphasis. Denoting the friction coefficient in “microscopic” unitsI3 by ri3, the limiting-law equations may be written in the form
dnxrjt= pjajr + dG,rjtl+ d n i r j t f f (i,j
n
where the subscript o refers to the solvent. We now summarize the “intuitive” interpretations of I,, and Rij. The quantity lii/& is the “intrinsic mobility” of ion i while li9/Ei (i # j ) is an “interaction mobility,” that is, a measure of the degree of coupling between the motions of component i and j . z , * l There is disagreement concerning the interpretation of the Rij coefficient^.^^^^^^'^ The notion that R i j represents the “friction” of component i with component j was criticized by hliller.2 Some of Miller’s objections have been circumvented by Dunsmore, et al. I n the latter work,4 the quantity &Riiis interpreted as the sum of all friction coefficients between component i and other components in the system, while -C,Rij (i # j ) is interpreted as a friction coefficient representing the frictional interaction between components i and j .
= 1,
2, 3, . . ., n) (7)
where ni and nj refer to the concentration of i- and j type ions in (number of ions) per cubic centimeter, and = 0, i # j ; = 1). The friction coefficient for the interaction of ion j with the solvent pr is given in terms of the absolute ion mobility w j or the limiting ionic equivalent conductance X I o by l/p, =
wj
=
xjoc x
10-8
FelZ,l
I n eq 8, C is the speed of light in centimeters per second, F is Faraday’s constant in colulombs per equivalent, e (10) See footnote 18 of ref 2 for further comments and references. (11) M. J. Pika1 and D. G. Miller, J . Phys. Chem., in press. (12) R. W. Laity, J . Chem. Phys., 30, 682 (1959). (13) The term “microscopic units” refers to the units resulting from the use of units of (number of ions) cm-2 sec-1 for flow and units of ergs per ion for the force. The Journal of Physical Chemistry, Vol. 7 6 , No. 20,1971
MICHAELJ. PIKAL
3126 is the electronic charge in electrostatic units, and .Z5 is the signed valence of the j-type ion. The second and third terms in eq 7 are the result of long-range coulombic interactions, and both are proportional to the square root of the total ionic strength. The single prime denotes the electrophoretic effect, while the double prime denotes the relaxation effect. Onsager and Kim7 show that r c z l 1is always positive while r i l l 1 (i # j ) is always negative, regardless of the sign of 2, and Zl. Thus ril (i # j ) may not be interpreted simply as an interaction coeficient reflecting pairwise i-j coulombic interactions. Another interesting result may be expressed in the form7 n
n,rrlll = O j= 1
If the above equation is combined with eq 6, the result Rro= 0 is obtained. Thus, at least to the limiting-law approximation, the ion-solvent friction coefficients, Rto (i = 1,2, . . . , n ) ,do not depend on the relaxation effect. However, it is easily shown that the electrophoretic contribution, Riot, is zero only in the special case of equal ionic mobilities. B. Conductivity Coeficients. General Limiting Law. The limiting-law expressions for the conductivity coefficients, li,, have been given explicitly only for the binary case.8 Limiting-law expressions for la3 in the general multicomponent case may be obtained by a Straightforward combination of several theoretical equations given by Onsager and ~oworkers.~~’ Denoting the conductivity coefficients in “microscopic” unitsla by Czl, we find
where the single prime represents the electrophoretic effect
and the double prime represents the relaxation effect
The term Q i , is our notation for a unitless complex function of ionic mobility and ionic strength fractions, which in the Onsager and Kim notation’ may be written in the form Qt5
=z
2(1-
d&xt’xtP
(14)
The original paper7 should be consulted for a definition of the terms in eq 14. I n practice, Q(, must be evaluated by numerical methods’ for each multicomponent electrolyte system considered. By a straightforward extension of the arguments made in establishing the sign of r z j t l it , may be shown that & ’ I is always negative and C i j l r (i # j ) is always positive, regardless of the sign of Zz and 2,. However, as shown in eq 10, the electrophoretic term &,’is negative if ions i and j have charges of like sign, and Cv‘ is positive if ions i and j have charges of opposite sign. Thus, lij (i # j ) may not be interpreted simply in terms of pairwise i-j coulombic interaction^.^,^ Simplified Limiting Law. Since the function Qil cannot, in general, be expressed in simple form, the dependence of li, on ionic strength fraction and mobility is not obvious from the form of eq 11. It would be useful to obtain a simple (approximate) expression for ldl which would correctly represent the gross features of the ltl for dilute multicomponent systems. Our numerical calculations show that although la, is a sensitive function of ionic mobilities w t and wl, Qu is a rather insensitive function of the ionic mobilities. We note that, for equal ion mobilities, the function Qu may be expressed in simple form14
Thus, if the equal ion mobility approximation, eq 15, is used to evaluate Q I j , and the result substituted into eq 11, one might expect to obtain a good approximation for the relaxation effect i z j l ’ . Combining eq 9-11 with Q r j given by eq 15, and converting into “macroscopic” units t o obtain l t j , we find
10121tz/~t = 0.10740, A t 0
-
I n eq 10 and 11, 1 / is ~ the familiar radius of the ion atmosphere, p i is the ionic strength fraction of ionic component i
i=l
and i is an average ionic mobility defined by n
(14) This result (eq 15) follows directly from theoretical equations presented in ref 5 and 7.
The Journal of Physical Chemistry, Vol. 76,N o . $0,15’71
THEORY OF THE ONSAGER TRANSPORT COEFFICIENTS The terms involving Atj are due to the relaxation effect, with Ai, given by
3127
Table I: Ternary Zij:
Comparison of Limiting-Law Theories for H20-LiCI-KCI 1 = Lif; 2 = K+; 3 = C1-HzO-LiCl
where I is the total ionic strength, D is the dielectric constant of the solvent, k is Boltzmann’s constant, and T is the absolute temperature. The electrophoretic effect is given by the terms involving BO,where
B o -
Ft
cx
10-8
K 1 P 237rq
(19)
Equations 16 and 17 are valid approximations for the limiting laws in any electrolyte mixture and, of course, include binary systems as a special case. Our numerical calculations indicate that eq 16 and 17 are surprisingly good approximations for the complete (Le., exact) limiting laws even when the ionic mobilities differ considerably. For example, in the binary LaCl$-H20 system, where the ionic mobilities differ by more than a factor of 3, the approximate relaxation effects given in eq 16 and 17 are in error by about 20%. I n the CaC12-H20 system, the errors in the relaxation terms are only about 10%. The electrophoretic terms are identical in both the approximate and complete equations. I n Table I, the approximate and complete limiting laws are compared for the ternary system H20-LiC1(0.25 M)-KCl(0.2 M ) . The subscripts 1, 2, and 3 refer to Li+, K+, and C1-, respectively. Experimental Zij valuesa are also included for comparison. Although the ionic mobilities of Li+ and C1- differ by about a factor of 2, the approximate limiting law is found to be an excellent approximation for the complete limiting law. The total ionic strength (0.45) is much too high to expect quantitative agreement between experiment and a limiting-law theory. However, it is significant to note that the limiting laws are in qualitative agreement with experiment. I n particular, lZZ/C3 and Za3/c3 are correctly predicted by the limiting laws to be nearly Also, the equal and about twice the value of ZII/CI. limiting laws correctly predict the signs and relative magnitudes of the Zij (i # j ) coefficients. Since the charges on ions 1 and 2 are of like sign, the electrophoretic effect for 112 is negative. The negative sign and small magnitude of Z12 is simply the result of the negative electrophoretic effect being slightly larger than the positive relaxation effect. C . Experimental Tests of the Limiting Laws. The general limiting laws include binary systems as a special case. The validity of the limiting laws for the more common transport coefficients, such as equivalent conductance, has ample experimental support in binary systems. l 6 Since the common transport coefficients are combinations of the 1ij,2 the limiting laws for Ztj should also be valid, a t least in binary systems. This conclusion is supported by the work of Lorenz,l6 where
I+O
(theory)
10’211l/C1 10‘2/22/c: 10‘2lSS/C* 1 0 ‘ ~ h 2 / ~ 10’2ZlS/~S 10~~Z*S/1/CZ8
4.154 7.895 8.200 0.0 0.0 0.0
Complete L.L.
M)-KCl (0.2 M)Approximate ExperiL.L. ment
3.25 6.33 6.35 -0.34 1.12 1.17
3.28 6.35 6.38 -0.36 1.09 1.20
(0.25
3.60 7.29 7.17 -0.20 0.52 0.69
dilute binary lij data are compared with predictions of the limiting laws. Surprisingly, the Z12 data for 1-1 salts‘ remain in good agreement with the limiting law up to fairly high concentrations (c = 0.05).16 As expected, unsymmetrical electrolytes show the greatest deviations from the limiting 1aws.16 Our own li, and R t j calculations on additional binary systems2 support the generality of Lorenz’s observations and indicate, as expected, that the limiting laws for R,, are also valid. The limiting law for diffusion of a tracer ion in a binary matrix solution is another special case of the general limiting laws. I n tracer diffusion, the electrophoretic effect vanishes, and the concentration dependence of the intrinsic mobility of the tracer ion (Zii/ci)*17depends on concentration of the matrix solution through the relaxation eff ect.6 Experimental evidence18indicates the limiting laws for tracer diffusion are valid. A quantitative test of the limiting laws for general multicomponent systems is not possible due to lack of suitable data. However, since the limiting laws are apparently valid for the special cases of tracer diffusion and binary Zij, it may be inferred that the general limiting laws are also valid.
111. Extended Theory of Z,, and Ri, for Symmetrical Electrolytes The limiting laws offer only a first-order approximation for the concentration dependence of Zi, and Ri,, and in general, agreement with experiment is expected only at high dilutions. It is of interest to extend the (15) See, for example, H . S. Harned and B. B. Owen, “The Physical Chemistry of Electrolytic Solutions,” 3rd ed, Reinhold Publishing Corp., New York, N. Y., 1958. (16) P. B. Lorenz, J . Phys. Chem., 6 5 , 704 (1961). (17) The intrinsic mobility of the tracer ion ( L J c J * in Miller’s units2 is related to the tracer diffusion coefficient D,*in cmZ/sec by8
(18) It. Mills and E. W. Godbole, J. Amer. Chem. Soc., 82, 2395 (1960). The Journal of Physical Chemistrg, Vol. 76, N o . 10,1971
MICHAEL J. PIKAL
3128 theory for Zi, and Ri, to higher concentrations by evaluating the contribution of higher order terms in the relaxation and electrophoretic effects. I n particular, the contribution of ion-pair formation to the 1,j is of interest since this information is of interest in establishing a detailed physical interpretation of the l,, and Ri,.We now consider the development of an extended theory of lit and Rtjfor binary symmetrical electrolytes. It has been shown1~z~8 that values of Ztj in binary electrolytes (i.e., 111, 112, and lzz)may be calculated from a knowledge of equivalent conductance A, transference numbers ti, and thermodynamic diffusion coefficient L according to the relationship
Therefore, theoretical expressions for l,, may be derived by combining theories of equivalent conductance, transference, and diffusion. This procedure is satisfactory provided the theories for A, t,, and L employ the same physical model and are mutually consistent in their mathematical approximations. To develop theoretical equations for lzl, existing mutually consistent “extended” theories for conductanceIg and diffusionz0are combined using eq 20. The model common to both “extended” theories is the usual charged sphere in a continuum approximation which results in the “ion-atmosphere” description of electrolytes.6 Both the “extended” conductance theory19 and the “extended” diffusion theoryz0 are developed retaining the full exponential of the Boltzmann equation. As a result, the effect of electrostatic ion-pair formation appears as a natural consequence of coulombic interactions. Both extended theories result in Bjerrumzlion-pair formation.20 The recovery of the 1,, equations from the conductance and diffusion theories is complex algebraically, but is straightforward. The procedure may be summarized as follows. The theoretical equations for the transference number and the equivalent conductance are evaluated from the Kremp-Kraeft-Ebeling (MKE) conductance theory.lg I n the KKE theory all terms of higher order than c in concentration are dropped. Thus, when the theoretical product t,t,A is formed, all product terms of higher order than c are dropped. Next a slight modification of the KKE theory is made. The function denoted by QRE1(b)in the KKE theorylgis written in the form Q””’(b)
=
1.9341
+ 2- T ( b ) - 1 - 6&(b)
where Q(b) is the Bjerrum functionz1 (defined by eq 23) and T ( b ) is defined in the Appendix, eq A6. Since &(b) is directly proportional to the Bjerrum equilibrium constant for ion-pair formationlZ1KA, the term in the conductance equation proportional to Q(b) The Journal of Physical Chemistry, Vol. 76, N o . 20,1971
represents the effect of ion-pair formation. Since Q(b) is negative when b < 2, this interpretation of the &(b) term can be valid only when b > 2. If the degree of association is small, the association term in a conductance theory is directly proportional to KACfj2where f is the mean ionic activity coefficient. If terms of higher order t,han c are dropped, as in the KKE theory, f becomes unity. We thus (‘extend” the KME theory to include some terms of order c8’* by replacing &(b) in the KKE equations by fz&(b). Since this substitution is valid only when b is large, we adopt the convention! = 1when b is small (i.e., b 2). The diffusion theory defined by eq 51 and 53 of ref 20 is employed in the lij calculations. The negative exponential integral function E i ( 2 ~ a )which appears in the diffusion theory is approximated by
-
Ei(2~a% ) -0.5772
- ln 2 ~ a
Finally, the theoretical expressions for t&,A and L/c are combined according to eq 20. A . The Theoretical Equations. The lgj Coeficients. The resulting theoretical equations for the intrinsic mobilities llz/c may be written in the form
where b is the Bjerrum parameter, and a is the usual “effective ionic diameter” with
The functions Q(b) and are complex integral functionsz0-z2whose asymptotic expansions for large b are given by
I(b)
eo F=.
ga(1
+ 3/b + . . .)
-
( b >> 1)
(24)
When b is small ( i e . , b 2, as in aqueous 1-1 electrolytes) we take f = 1. When b is large, f denotes the (19) D. Kremp, W. D. Kraeft, and W. Ebeling, Ann. Phys. ( L e i w i g ) , 18, 246 (1966). (20) M . J. Pikal, J. P h y s . Chem., 75, 663 (1971). (21) (a) N. Bjerrum, Kgl. Danske Videnskab. Selslca!,., 7, No. 9 (1926); (b) see also R. M. Fuoss and F. Accascina, Electrolytic Conductance,” Interscience Publishers, New York, N. Y., 1959. (22) This function is defined in the Appendix, eq A l l .
3129
THEORY OF THE ONSAGER TRANSPORT COEFFICIENTS limiting-law approximation for the mean ionic activity coefficient fZ = e-PK (25) The symbols a and El’ represent “constants” arising from the relaxation effect, while PO and Ez’ represent “constants” arising from the electrophoretic effect. The function j ~ ( b arises ) from the relaxation effect, while j,(b,Ao) is a function coming from the electrophoretic effect. The notation used here is precisely defined in the Appendix. The theoretical result for the interaction mobility Zlz/c may be written in the form
where J&A0) is an “electrophoretic” function defined in the Appendix (eq A12). The Ri, Coeficients. The theoretical expressions for the RII, R12, and Rzzcoefficients are obtained by inversion of the ,Z, matrix.2 The various terms in cRll (i,j = 1,2) are grouped according to their concentration dependence to give equations similar in form t o eq 21 and 26. Terms of order cl/’, c In c, c, and cf” are retained. All higher order terms are dropped. Using the theoretical expressions for Rllr R12, and R22 and eq 6, theoretical expressions for the “solvent friction coeficients” Rto (i = 0,1,2) are derived. Since the “friction” coefficients Rlz, RIOland Rzo are normally preferred for interpretation of data,4 explicit equations will be presented only for these coefficients. The theoretical equations for Rlz and Ri0 (i = 1, 2) may be written in the form
The functions X1z(b,Ao) an Alo(b,Ao)are rather complex and are defined in the Appendix (eq A13 and A14). I n the theoretical equations for Zv and RII,the terms involving a and El‘ are due to the relaxation effect (ie., asymmetry of the ion atmosphere), while the terms involving Po and Et’ are due to electrophoretic effect (ie., hydrodynamic interactions due to the presence of an ion atmosphere). Although the terms proportional to cf” originate in the relaxation and electrophoretic calculations, it will be shown later that these terms may be interpreted in terms of ion-pair formation if b is large. It should be noted that while both the relaxation effect and the electrophoretic effect make significant contributions to all the concentration-dependent terms for li,/c and cR12, the relaxation effect does not affect Rio measurably. The only terms in Rto involving the relaxation effect is a very small relaxation-electrophoretic product term in yto, which depends on aPo* (Azo - Azo). This result is expected. As Miller shows,2 Rt0 may be expressed in terms of the transference number and the thermodynamic diffusion coefficient. Diffusion in a binary system is independent of the relaxation eff ect.6-20The transference number, as calculated from the KKE theory, shows only a slight dependence on the relaxation effect through the product OrPO(Xl0
-
AZO).
The theoretical expressions for 10-12cRtt and 10-12(co2/c)Ro, are similar to eq 28. The limiting values for 10-12cRtt and - 10-12coRto are identical. However, 10-12cRlI depends on both the relaxation effect and the electrophoretic effect. The quantity 10-12(c,2/c)Roo depends only on the electrophoretic effect, and its limiting value involves both h0 and A Z O . The Fuoss-Onsager-Skinner (FOS) conductance theoryzs is also developed using the full exponential of the Boltzmann equation. Theoretical expressions for lit and R t j based upon the FOS conductance theory are similar to the results given in eq 21-28. The limitinglaw terms are identical. The small differences between “c In c terms” are not experimentally detectable. Although differing in detail, the “c terms” are of similar form and magnitude. We note that the R,, coefficients are completely independent of the relaxation effect using the FOS theory. The most significant differences occur in the terms proportional to cf2, basically because the KKE theory predicts Bjerrum ion-pair formationlg while the FOS theory leads to “contact” ion-pair formationSza The KKE theory and the extended diffusion theoryz0 are consistent in that both theories lead to Bjerrum ion-pair formation.20 Thus, only the results based upon the KEE theory are explictly given here. However, most of our conclusions based upon eq 21-28 also follow from the corresponding equations based upon the FOS conductance theory. (23) R.M. Fuoss, L.Onsager, and J. F. Skinner, J . Phys. Chem., 69, 2581 (1906).
The Journal of Physical Chemistry, Vol. 76, No. 20, 1071
MICHAEL J. PIKAL
3130
Sch6nerte recently suggested that the concentration dependence of thee lij or R,, coefficients may be expressed in terms bf power series expansions involving concentrations of the components in the solution. Our results are in partial agreement with Schonert's suggestions. Terms of order cl" and (c"')~ are present. However, terms of order c In c are also present. A power series representation of the c In c terms converges slowly when c < 1. Thus, the concentration dependence of lij and R15in dilute electrolytes is best described with a series composed of terms of order cniz (n = 1, 2, 3, . . .) and cnlz In c (n = 2,3, . . .). Because of mathematical and physical approximations made during the development of the KKE conductance theorylg and the extended diffusion theory,20 eq 21, 26, 27, and 28 are limited to dilute solutions, i e . , systems for which Ka
