Application of Irreversible Thermodynamics to Electrolyte Solutions. I

ISOTHERMAL VECTOR TRANSPORT PROCESSES IN BINARY ELECTROLYTE SYSTEMS

2639

Application of Irreversible Thermodynamics to Electrolyte Solutions. I. Determination of Ionic Transport Coefficients Zijfor Isothermal Vector Transport Processes in Binary Electrolyte Systems112

by Donald G. Miller Lawrence Radiation Laboratory, University of California, Livermore, California

(Received February 14, 1966)

Irreversible thermodynamics is applied to binary electrolyte solutions undergoing isothermal vector transport processes. Rigorous expressions for the conductance A, diffusion coefficient (D)o,and cell and Hittorf transference numbers tia and tth are derived systematically in terms of the Onsager transport coefficients Zi,. Conversely, expressions for the 1 , are derived in terms of the experimental quantities t i , A, and Values of Lij calculated from experimental data are given for HCI, LEI, NaCI, KCl, CaCI2, BaCI2, and Lac13 at 25” from c = 0 to as high as 3 N . Interesting aspects of their relative sizes and concentration dependences are discussed. New insights into ionic properties are possible because the li, are more fundamental than usual transport quantities (which are composites of the Z’s). Curie’s theorem is used to show that no rigorous quantitative connection can exist between the viscosity and D or A. Friction coefficients are also presented and aspects of that formalism are discussed.

I. Introduction I n a very dilute aqueous electrolyte solution, the ions are so far apart that long-range coulombic forces are the principal interaction and specific effects are negligible. Consequently many ionic properties are additive, as was discovered by Kohlrausch and others in the 19th century. In this circumstance one may assume that, in a diffusing or conducting system, the flow of an ion would be proportional to gradients only of its own properties and would be uninfluenced by the presence of other distant ions. On this basis, one can derive such limiting expressions as the Nernst-Hartley (N-H) equation for the diffusion coefficient. I n concentrated solutions, however, specific ion effects become important; e.g., activity coefficients become specific above 0.1 m, ionic conductances are no longer additive, the N-H equation fails, etc. Consequently it is intuitively clear that the flow of an ion must be influenced by the other ions present and the gradients of their properties as well. This paper is primarily concerned with vector transport processes such as conductance, diffusion,

ion transference, etc. , at arbitrary concentrations. The important vector process, heat conduction, will be omitted here. The macroscopic framework necessary to cover these complex situations is the thermoIt gives a dynamics of irreversible proces~es.~-~ natural, completely general description of irreversible processes in terms of linear transport coefficients It,. Clearly these Zi, are more fundamental than the common transport coefficients for special cases, such as the equiv(1) This work was performed under the auspices of the U. S. Atomic Energy Commission. (2) Portions of this work were presented at the 94th AIME meeting in Chicago, Ill., Feb 1965, and at the 16th CITCE meeting in Budapest, Hungary, Sept 1965. (3) S. R. DeGroot and P. Mazur, “Non-Equilibrium Therm+ dynamics,” Interscience Publishers, Inc., New York, N. Y., 1962. (4) R. Haase, “Thermodynamik der Irreversiblen Prozesse,” Dietrich Steinkopff, Darmstadt, 1963. (5) K. G. Denbigh, “Thermodynamics of the Steady State,” John Wiley and Sons, Inc., New York, N. Y., 1951. (6) D. D. Fitts, “Non-Equilibrium Thermodynamics,” McGrawHill Book Co., Inc., New York, N. Y . , 1962. (7) J. Meixner and H. G. Reik in “Handbuoh der Physik,” S. Flugge, Ed., Vol. 111/2, Springer-Verlag, Berlin, 1959, pp 415-523. (8) D. G. Miller, Chem. Rev., 60, 15 (1960).

Volume 70, Number 8 August 1966

DONALD G. MILLER

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alent conductance or diffusion coefficient, which turn out to be combinations of the Zi,. This description fully takes into account the effect on the flow of one constituent of gradients in the properties of all the other constituents. Thus, any vector transport process, no matter how complex, can be completely characterized once these ,Z are known as functions of temperature, pressure, and concentration. Especially important are the cross coefficients l,,, i # j , which directly represent the ionic interactions. Their neglect is responsible for the failure of simplified electrolyte theories. Although the ,Z are fundamental and give new insights to electrolyte behavior, little is known about their values and concentration dependence. A previous brief communicationg gave equations for the I,, in terms of experimental quantities and preliminary ZE/C results for three chloride^,^" LorenzlO gave some Z12/Nvalues at high dilution and the theoretical limiting slopes for these and two others, and Stockmayer” considered weak electrolytes. Here c is the molarity and N is the normality The principal purposes of this paper are (a) to derive rigorously the equationse relating isothermal li, to experiinentally measured quantities for a single binary electrolyte in a neutral solvent and (b) to present Zi, calculated from critically reviewed data for seven chlorides in HzO from infinite dilution to as high as 3 N . .41so discussed are the interesting differences between salts, characteristics of Zi, behavior, the importance of the cross coefficient ZI2, the desirability of using this thermodynamic description of ionic transport processes, and the relation of 11 to vector transport properties. The relevant equations and numerical results are also presented for the “friction coefficient” formalism Equations and calculations for ternary isothermal systems will be given in paper 11. Comparison of experimental ,Z with the Onsager-Fuoss limiting law theory12 rind consideration of I,, for nonisothermal situations will follow in subsequent papers.

11. Thermodynamic Preliminaries According to irreversible thermodynamics, the entropy production per unit volume u for an isothermal system of a neutral solvent (subscript 0) and n solutes, either ions or nonelectrolytes, can be writtena+ It

Tu

=

CJiX,

i=O

(1)

where T is the absolute temperature, J i the massfixed flow of matter in moles/cm2 sec, and Xi the The Journal of Physical Chemistry

thermodynamic “force” in mole units. The small viscous terms will be mentioned in VIII. The one-dimensional case is considered for convenience and because this is the usual experimental situation. Consequently X imay be written as

xi

=

-

[;Ii + a41 ax -

2,s-

where negligible gravitational terms are omitted and where p l is the chemical part of the chemical potential in joules per mole, 2, is the signed valence of an ion (its absolute value is the number of equivalents per mole) and is zero for a nonelectrolyte, 5 is the value of the faraday in coulombs per equivalent, is the electrical potential in volts, and x is the distance in centimeters. Because the systems of interest are in mechanical equilibrium, eq 1 is valid no matter what frame of reference is chosen for the f l 0 ~ s . I ~Hence, J , may be replaced by (J,)* where the asterisk refers to an arbitrary reference frame. The X i are not independent because of the GibbsDuhem equation, which may be written

L i X *= 0 ~

i=O

(3)

where cl is the concentration of constituent i in moles per liter. If the solvent term Xo is eliminated for convenience, then eq 1becomes

Tu

=

2[(Jt)* - ?(Jo)* co 1X i

(4)

i-1

Any reference frame may be used to fix the flows. For pure diffusion a volume-fixed (VF) frame is desirable because the measured diffusion coefficients are based on that frame. I n this study, however, the solventfixed (SF) frame (i-e., (Jo)o= 0) is extremely convenient because transference numbers are always referred to it. Transformations exist for passing from one frame to another.14-16 Because only SF flows are em(9) D.G.Miller, J. Phys. Chem., 64,1598 (1960). (9a) NOTEADDED IN PROOF.Equivalent equations were worked out independently by J. W. Lorimer (private communication) and A. Katchalsky and P. Curran, “Noncquilibrium Thermodynamics in Biophysics,” Harvard University Press, Cambridge, Mass., 1965, pp 139-144. (10) P. B. Lorenz, J. Phys. Chem., 65, 704 (1961). (11) W. H.Stockmayer, J . Chem. Phys., 33, 1291 (1960). (12) L. Onsager and R. M. Fuoss, J. Phys. Chem., 36, 2689 (1932). (13) See ref 3, pp 43-45. (14) G. J. Hooyman, Physica, 22, 751 (1956). (15) J. G.Kirkwood, R. L. Baldwin, P. J. Dunlop, L. J. Costing, and G. Kegeles, J . Chem. Phys., 33, 1505 (1960). (16) L. A. Woolf, D. G. Miller, and L. J. Gosting, J. Am. C h . ~ o c . , 84, 317 (1962).

ISOTHERMAL VECTORTRANSPORT PROCESSES IN BINARYELECTROLYTE SYSTEMS

ployed in this paper, they will be denoted henceforth by J , (previously used in eq 1 for mass-fixed flows) instead of the usual but more cumbersome (J&. In terms of its n-independent SF flows and n-inde1 component system pendent forces, T u for the n becomes

+

n

Tu = C J f X f i- 1

(5)

Experiment shows that linear laws such as those of Ohm or Fick hold between the J 1and X iand also that each J , depends on all of the X's. Hence n

J,

=

i = 1, . . ., n

&X,

j-1

(6)

where I, are phenomenological or transport coeficients. The notation l,, is reserved for ionic transport coefficients whereas L,, will be used for electrolytes as a whole. Irreversible thermodynamics states that if the flows and forces are chosen from Tu and are independent,l6,l7 then the Onsager reciprocal relations (ORR) hold ;la-27 namely i , j = l,..., n

1,,=1,,

(7)

It is important to note that the li, of eq 6 and 7 are SF transport coefficients.28 Equations 5-7 represent a completely general macroscopic description of isothermal vector transport properties.

111. Binary System Consider an isothermal system consisting of a neutral solvent (such as HzO, neglecting the small ionizationZ9)and a binary electrolyte which ionizes as

C,,A,, +rlCzl

+ rALP

(8)

where C is the cation, A is the anion, subscripts 1 and 2 refer to the cation and anion, and rI are the stoichiometric coefficients for ionization. If 1.41, 1.42, and p12 are the chemical potentials in joules per mole of the cation, anion, and electrolyte as a whole, then by definition 1.412 =

rlpl

+rw~

(9)

Moreover conservation of charge requires that

+ rzzz = 0

~ Z I

(10)

The independent flows for this system are those of the two ions, since JO = 0 for the SF reference frame. Therefore the transport equations are

+ hzX2 Jz = 121x1 + J I = AiXi

1 2 2 x 2

( W (1W

2641

where X i is given by eq 2. These equations completely describe the isothermal vector transport properties in a binary electrolyte solution provided the li, are known as functions of T , the pressure P, and c o m p o ~ i t i o n . ~ -Knowledge ~~ of the li, is clearly (17) See ref 3, pp 64-69. (18) The ORR were demonstrated by Onsager'g using statistical mechanics for a special class of thermodynamic force and flow variables. The statement about arbitrarily choosing J and X from Tu was objected t o by Verschaffelt," (cf. Davieszl), and more recently in detail by Coleman and Truesdell.22 The argument is that arbitrary transformations of Ji and X i in TUmay destroy the ORR. However, transformations which destroy the ORR lead to flow variables which are unreasonable on physical or chemical grounds.*S Secondly, statistical mechanical derivations of the ORR for the usual choices of variables have been given by DeGroot and Mazur,*' Mori," and Kirkwood and Fitts.a Thirdly, it is possible to transform the Onsager variables into the usual ones3-* while retaining the validity of the ORR.2' Finally the ORR based on the usual flows and forces3-8 have been amply confirmed by experiment for a large number of phenomena.* The ORR are valid for SF flows and forces because these variables can be obtained from a set known to satisfy the ORR by experiment or statistical mechanics by means of symmetry-preserving (Meixner) transformations. (19) L. Onsager, Phys. Rev., 37, 405 (1931); 38, 2265 (1931). (20) J. E. Verschaffelt, Bull. Classe Sci., Acad. Roy. Belg., 37, 853 (1951). (21) R. 0. Davies, Physica, 18, 182 (1952). (22) B. D. Coleman and C. Truesdell, J . Chem. Phys., 33, 28 (1960). (23) C. J. Pings and E. B. Nebeker, Ind. Eng. Chem. Fundamentals, 4, 376 (1965). (24) 9. R. DeGroot and P. Mazur, Phys. Rev., 94, 218 (1954). (25) H. Mori, ibid., 112, 1829 (1958). (26) J. G. Kirkwood and D. D. Fitts, J. Chem. Phys., 33, 1317 (1960). (27) See ref 3, pp 69-77. (28) Such coefficients for a different frame of reference will have different numerical values (the J's will also be different and the X ' s will no longer be given by eq 2) but the ORR will still be valid.16 (29) Cj. ref 16, footnote 67. (30) I n a recent paper, Baranowski and Cukrowski31 have derived equations for a binary system based on a reference frame fixed on convection flow. With this frame there are three independent flows (solvent and ions 1 and 2) and nine Onsager coefficients L;j, of which six are independent. The six experimental quantities required are the three self-diffusion coefficients Di*, A, and the hydration numbers 81 and 82. From them Lij (i = 0, 1, 2) can be computed as well as tlh and (D)o. Although their method is attractive because o give more insight into ion-solvent interactions, there the L ~ would seem t o be certain difficulties: (A) The ai are t o be determined by Washburn's "inert" reference substance technique, with the effect of the reference32 eliminated by extrapolating t o zero added reference. However the extrapolated values of 8 ; differ with different so-called "inert" ~ubstances.3~~ 33 Consequently the resulting Lij will not have unique values. (B) There do not appear to be enough independent experimental quantities t o test the ORR even in a binary system. (C) Baranowski and Cukrowski note that their method cannot be applied to ternary or higher order systems because there are insufficient independent experimental quantities even when the ORR are assumed. (D) The Di* are actually properties of a ternary system. (E) Comparison with existing theories is difficult because they are based on the SF frame. Because of ambiguity in the numerical values of Lij and because the method cannot be extended beyond the binary system, it appears that their scheme is not too useful. On the other hand there exist sufficient independent experiments for the S F description t o obtain the l i j unambiguously for systems of any number of components. (31) B. Baranowski and A. S. Cukrowski, Z. Physik. Chem., 228, 292 (1965). (32) M. Spiro, J . Inorg. Nucl. C h a . , 2 5 , 902 (1963).

VoZume 70,Number 8 August 1966'

DONALD G. MILLER

2642

desirable because from them one can compute the conductance, transference number, diffusion coefficient, as well as the transport coefficientsfor any other process no matter how complex. To obtain numerical values of the four ,Z, in eq 11, four independent experimental quantities are needed. Four suitable ones based on special cases are the conductance A, diffusion coefficient D, one Hittorf transference number th, and one emf transference number tC. All four cases will be considered because equations for testing the ORR in ionic systems are desired for use elsewhere. However in the actual calculations of ,Z, for this paper, the ORR (eq 7) is assumed. Hence only three independent quantities are required; one of the transference experiments is superfluous.

IV. Analysis of Special Experiments We now systematically derive rigorous expressions for A, th, tc, and D in terms of the Zij. Constant pressure is assumed. A . Conductance. The conductance is measured by passing the current per cm2

I

= (2dl

+

22J2)F

(12)

through a solution of uniform composition, where I is in amp/cm2. Since the composition is uniform

1ooox

A=-

N

where N is the number of equivalents per liter (Le., rlzlc), and the 1000 is really 1000.027 cm3/l. Therefore A=-

+

100052

N

[z1%1

2122(Z12

+ + 121)

222Z22]

(20)

Although A is customarily measured in an apparatusfixed reference frame, it is shown in Appendix 1 to have the same value in any reference frame. B. Hittorf Transference. The Hittorf transference number tth is the fraction of the current carried by the ith ion relative to the solvent in a solution of uniform compo~ition.~~ Because the SF frame is being used we may immediately write

tl h

ZIJIF

=--

I

Jis

21

(ZJl

+

(21)

22J2)S

Because of uniform composition, X i reduces to eq 14, so that on substituting eq 11we obtain t:

+ + z122(Z1z + + 212z11

= 212z11

- 212Z11

2122z12 121)

+

2122ZlZ

a!

222z22

(22) and similarly

(13) where c is the molarity in moles per liter of the electrolyte as a whole. Therefore X,becomes

x,=

-2

&J (uniform concentration)

(5-

bX

(14)

Using eq 11,12, and 14, we obtain

I = A(-$) where

is the measured specific conductance in (ohm If a is defined by a!

= x/52

(17)

then comparison of eq 15 and 16 yields a!

=

+

t2

=

1. Although I,

A, and a are reference-frame independent, the tt are

not. C. Concentration Cells with Transference. Consider an electrochemical cell with transference (concentration cell) with two identical electrodes and a single electrolyte. The electrolyte, however, has different concentrations at each electrode and there is a liquid junction of varying composition between the electrode solutions, e.g., Ag/AgNOa(cl)iAgNOs(cz) Ag. I n such a system diffusion occurs, and, owing to the different ion mobilities, a charge separation is set up. I n about low9see (less in solvents of lower dielectric constant than H20),36the powerful coulombic effects speed up

I

However, by Ohm's law

. cm) -I.

It is easily verified that 11

+

+ Zzi) + zz2k21

[~i~Zii ~ i ~ z ( Z i z

(18)

The quantity usually reported is the equivalent conductance A in cm2/ohm equiv, defined as The Journal of Physical Chemistry

(33) L. G. Longsworth, J . Am. Chem. Soc., 69, 1288 (1947). (34) A rigorous general definition of the Hittorf transference number, valid for nonelectrolytes as well as ions and based on the actual experiment, is given by D. G. Miller, Am. J . Phys., 24, 595 (1956), especially pp 598-600. The J's used there are apparatus fixed, but, as shown in footnote 19 of ref 16, the numerators of eq 17 for ions and eq 18 for nonelectrolytes are precisely the solvenb fixed flows (since n;/n, in that notation is c,/co in this notation), and the denominators are reference-frame free (Appendix 1). (35) D.R. Hafemann, J . Phys. Chem., 69, 4226 (1965).

ISOTHERMAL VECTORTRANSPORT PROCESSES IN BINARY ELECTROLYTE SYSTEMS

the slow ions and slow down the fast ones, so that no current flows through the solution. Therefore

I

=

+ 2zJz)S = 0

(24)

( ~ 1 J l

I n this situation, the bclt/bx and bC$/bx are not zero. Upon substitution of eq 2 and 11 in eq 24, and solving for -Sh$/bx, we obtain an expression for the liquid junction or diffusion potential (21111

+

22121)-b P 1

+

L-z12z11

bX

+ + + +

21z2(112

(21112

22Z22)-

121)

222122

”‘1 bX

212Z11

+

212z11

+ + +

-

2122z21

2122(112

121)

2l211l

222122

+

1

Z122Z21

ct

(26) and similarly for he. Then eq 25 becomes the classic expression for the diffusion potentia13‘j

--bx

= tlC - bcrl 21

D. Di$usion. For diffusion to occur a concentration gradient is necessary, and thus bbcli/dxdoes not vanish. However, the current vanishes. Consequently eq 24 applies, and the cation and anion motions are coupled. I n a binary system this coupling requires the cations and anions to move together, the electrolyte thus diffusing as if it were a nonelectrolyte. This is shown formally by substituting eq 10 in eq 24, yielding

(25)

Let us define the cell or emf transference number of ion 1, tie, to be tlC =

2643

tZC b cr2 +bx 22 bx

Kone of 4, p l , and p2 is accessible by experiment. However if electrode reactions are included, various terms combine to give measurable quantities. The result depends on the choice of electrode. Assuming an electrode reversible to the anion (to be definite) and integrating from the anode through the cell to the cathode, one can derive the classical result

where E is the emf measured potentiometrically a t zero current, k is the gas constant in appropriate units, a! and p refer to the anode and cathode, respectively, and u12 is the activity coefficient of the electrolyte as a whole. For electrodes reversible to the cation, replace subscript 1 by 2 in tic, r1, and 21. If E is measured for cells with varying concentrations at the cathode and a fixed reference concentration at the anode and if u12is known from other studies, then an experimental value of tlc can be obtained from the derivative of eq 28

Although it has been assumed since Helmholtz3’ that tlC and t? are the same, comparison of eq 26 and 22 show that this is true if and only if the ORR Zl2 = kl is valid.8p19

Since J1 is the total flow of cations, Jl/rl is the flow of cations as if they were tied up as the compound C,,A,,. Similarly for J2/r2. Hence J is the SF flow of the electrolyte as a whole. This result can also be obtained by analysis of T5.as The diffusion coefficients for our binary system are defined by J =

* bc - (D)o-

bX

=

-L- b P l 2 bX

where to maintain consistent units, we define

Here ( D ) o is the SF diffusion coefficient in cm2/sec, L is the SF thermodynamic diffusion coefficient (denoted by M in ref 9 and w by Harned and Owen39)in moles2/joule cm sec, and the 1000 is again really 1000.027 cm3/l. Comparison of (D)oand L yields

+

where r = rl r2, and y is the molar activity coefficient. The distinction between (D)o and the usually reported volume-fixed diffusion coefficients (D),, is important but has not always been observed in discussions of electrolyte solutions. Although the same at infinite dilution, they diverge significantly in concentrated solutions.40 using partial One may calculate (D)o from (D),, ~

(36) The derivation of this result by classical equilibrium thermodynamics is incorrect because the system undergoes irreversible diffusion. (37) H.yon Helmholtz, Ann. P h y s i k , 3, 201 (1878). (38) D.G.Miller, J. Phys. Chem., 63, 570 (1959),Appendix 1. (39) H. S. Harned and B. B. Owen, “The Physical Chemistry of Electrolytic Solutions,” 3rd ed, Reinhold Publishing Corp., New York, N. Y.,1958. (40) R. P. Wendt and L. J. Gosting, J. Phys. Chem., 63, 1287 (1959).

Volume 70,Number 8 August 1886

DONALD G. MILLER

2644

molal volume data by an equation from ref 15 or 16 specialized to one cbmponent ; namely

where co is the molarity of the solvent, and VO and VI2 are the partial molal volumes of the solvent and electrolyte as a whole, respectively. Alternatively (D)o/(l cd In y/dc) can be calculated directly from ( D ) v and the commonly reported data for y as a function of m by means of the relation (derived in Appendix 2)

+

(D)o d In y 1+c-dc

-

(D) V

(35)

d In y l+mdm

where m is the molality in moles per kilogram of solvent and y is the molal activity coefficient. and L in terms of the To get expressions for lrj, we must get J in terms of bp12/bx. Equations 2, 11, and 30 yield J = Ji - = rl

-j[ll1g +

112-h

2

+

dX

rl

Recalling that neither 4 nor the p i are accessible by experiment, we eliminate b$/bx by eq 25, a consequence of I = 0. If eq 25 is substituted into eq 36, after some manipulation one finds the p i in the measurable combination of eq 9. The results are41 J

=

%(

111122

- 112121

+ z1z2(1n + +

r1r2 ~ 1 ~ 1 1 1

121)

ye bx

(37)

~ 2 ~ 1 2 2

by Laity42to get these quantities in terms of the Rt, description. Although many of the above results are not new, some of the earlier derivations or equations were ambiguous because of mixed or unspecified reference frames, etc.

V. The I , in Terms of Measured Quantities If we know the l i j as functions of composition, etc., we can calculate A, t;, t?, and (D)o. Conversely, we can solve eq 20, 22, 26, and 39 or their equivalents simultaneously to get expressions for the li5 in terms of these four quantities. The procedure is as follows. Solve for E12 in terms of ath and 111, and for 121 in terms of atlCand 111. Substitute these into a to get 122 in terms of 111, at?, and atl'; then substitute for 112, 121, and 122 in L, which yields an expression for lll. Working backward yields the others. The results are simply expressed in terms of CY and L by

- -

The a and L in eq 40 are both proportional to c. Consequently all Iff 0 as c 0. This is physically correct because J , must be zero at c = 0 no matter how large the bp,/bx are. Moreover, Ell and k2 divided by something proportional to c will be relatively constant because the resulting additive terms for the right side of eq 40 are not strongly concentration dependent. A natural divisor is the equivalent concentration N because it gives identical limiting values for the same ion with different valence-type electrolytes. However, we expect 112", to be concentration dependent because 22 is negative, resulting in partial cancellation. On dividing eq 40 by N and introducing (D)oand A, etc., by means of eq 19 and 33, we obtain

+

Theref ore

or

or using eq 33

;

pa(111122 112121)]

(39)

E. Comments. Equations 20,22,26, and 39 are the desired equations for the four usual special cases in terms of the li5 and, except for eq 26,8 were given in equivalent form in ref 9. The same ideas were used The Journal of Physical Chemistry

(42) where Btl is the Kronecker delta. This result is valid (41) An equation equivalent to eq 38 was worked out independently and given earlier by R. Haase, Trans. Faraday Soc., 49, 724 (1963), asterisked footnote, p 728. (42) R. W.Laity, J . Chem. Phys., 30, 682 (1959).

ISOTHERMAL YECTORTRANSPORT PROCESSES IN BINARY ELECTROLYTE SYSTEMS

for any neutrrl solvent and applies to both strong and weak’ electrolytes. If 5 is in coulonibs/equiv, R in joules/mole deg, and (D)oin cm2/sec, then the units of lij are mole2/joule cm The units of ltj,/N are thus a mixed lot, mole2 l./joule cni sec equiv. By eliminating the , has a neater looking but not factor 1000 C U I ~ / ~ .liJN necessarily more useful set of units: mole2 cm2/joule see equiv. Equation 41 mas derived to permit testing the ORR. However, assuming the ORR from the outset eliminates the distinction between t: pnd tih, and eq 5 of ref 9 is obtained. The physical interpretation of the l,, is quite direct. The Z,,/N is an “intrinsic” ion mobility, Le., the motion the ion would have if there were no interaction with the oppositely charged ion. Besides the large obstruction (or “lattice” exchange) contribution, it includes smaller effects such as the coulombic interaction of other i-type ions, solvation, and reference frame interactions with the solvent. The 112/N is an “interaction” mobility and directly the cat- represents ion-anion interaction. However, because it is an SF coefficient, there is some solvent contribution. Qualitative evidence that Z12/N is primarily coulombic is that in ternary systems the cation i-cationj term has the opposite sign to the cation-anion terms. illoreover 112/N being zero at infinite dilution is consistent with no interaction when the ions are infinitely separated. The 1,JN for the same ion in .different salts have a common limiting value at infinite dilution, also because of no interaction at c = 0. It is useful to obtain a rigorous expression for J iin terms of t, A, and D. Upon substituting eq 40 or 41 in eq 11 and eliminating superscripts on ti by the ORR, we obtain

where

Equation 43 can be rewritten in terms of the current

I as

Equation 45 represents the superposition of electrical and diffusion transport. Therefore the argument which

2645

14.0--

--

12.0--

*

- too--

--

-

NO

8o 6 O--

2 o--

I

0 0

I

La, : ! ] ; : ; : ; ; ; ; : ; ; ; ; 2 3 4 5 6 7 8 9 I O I1 12 13 14 15 16 17 18

$

Figure 1. 1012i,~/,v US. ~ ‘ 1 2 .

led to it constitutes a rigorous proof that superposition is valid. We also see that the general electrical transport term of an ion in the presence of concentration gradients and electric fields is the ati [ ] / z i term of eq 43. Although eq l l a and l l b are more fundamental in principle, eq 43 and 45 are equally general and may be useful in analyzing complex processes.

VI. Concentration Dependence of ZJN for Several Aqueous Electrolyte Solutions Equation 41 may be used to obtain the concentration dependence of 1,JN if tl, A, (D)oand activity coefficient derivatives are known at each concentration. Although volumetric data can be bypassed by using eq 35 to calculate (D)o/(l cd In yldc), densities are usually required to get c from m or vice versa. Salts for which all of these data are available simultaneously over any reasonable concentration range are scarce indeed. Most lacking are transference numbers for concentrations greater than 2 N . With some effort the necessary quantities have been collected and critically examined for almost all of the salts with sufficient data, namely, HC1, KCl, NaC1, LiC1, CaC12, BaC12, and LaCh in H20a t 25”. The calculated l i j / N are given in Tables I-VI1 together with the best values for t ~ A, , (D)v, and (1 md In r/dm).

+

+

(43) When the units given earlier for each quantity on the right side of eq 41 are substituted, both terms reduce t o mole*/joule cm sec. If is given in erg units, dividing the A term by lo7 keeps the units consistent.


DONALD G. MILLER

2646

c

v

a

The Journal of Phgcrical Chemistrg

ISOTHERMAL VECTOR TRANSPORT PROCESSES IN BINARY ELECTROLYTE SYSTEMS

Q

2647

3 - 3 3 3 3 3 3 3 3 3 3 3 3 3 3

m

d

d

00000000000~3dci&

..

Volume 70,Number 8 AuOust 1966

DONALD G. MILLER

2648

-

42

H

-

I

P

z Q)

The Journal of Physical Chemistry

0 - m 0 - m 0000310

099930 0 0 0 0 0 3 0

2649


Let us now consider the qualitative behavior of the Zi,/N, The cation coefficients L / N are shown vs. S1la in Figure 1, where the ionic strength S is S=

+

(~1x1~ 2

- N(ZI - 22)

~ 2 . 2 2 ~ ) ~

2

I..

L

i:

Y

c

(46)

Each declines slowly with increasing concentration, much more slowly than the ionic conductance t d , and there is a slight inflection point. This decline is consistent with an increasing negative i-i coulombic contribution, which is about the same as the corresponding increase in Zlz/N. The relative values for each salt are in the expected order with the more solvated (bulkier) Li+ and higher valence types having lower “intrinsic” mobilities. The drop for H+ is somewhat more dramatic, undoubtedly due to its different conduction mechanism. Figure 2 shows the anion coefficients &2/N us. S‘”. They all come to a common point at c = 0, as would be expected since the anion is the same. They appear similar to the Zll/N, declining slowly with increasing concentration and having inflection points. These Zz2/N have some curious crossovers at higher concentrations, with HC1 the most anomalous. One would not expect the C1- mobility to be so influenced by the H+ since this interaction is primarily accounted for by 112/N and since coupling through interaction with the solvent should not be that large. The decline in is again slower than the ion conductance tzA, but the percentage spread of tzh for these salts is less. Thus the “intrinsic” mobilities show the C1- ion specificities better than the ion conductances. One possible explanation for the fact that the “intrinsic” mobility of C1- is smaller in LiCl solutions than in KC1 solutions at a given S (as is tzh) is that the more heavily solvated Li+ ion orients a larger portion of the solvent more rigidly than does K+. Therefore LiCl solutions will have fewer small, easily exchangeable solvent units, making solvent-C1- exchanges more difficult and t,hereby reducing the C1- “intrinsic” mobility. More interesting are the interaction coefficients 112/N, shown vs. S‘/’ in Figure 3. Their properties are: (1) they are specific even close to infinite dilution; (2) they are zero at c = 0 and rise rapidly with concentration to as high as 60% of a main coefficient l i i / N ; (3) at higher concentrations llz/N goes through a maximum for NaCI, LiCI, and BaC12 and probably would for the others if data extended to higher concentrations; (4)the slopes at c = 0 seem slightly higher than Lorenz’s calculated values;lO and ( 5 ) Z12/Na/’ is fairly constant, especially below 0.01N. Identical limiting behavior near infinite dilution

-‘i

5.0

4.5.f t

l

0

)I

l

l

l

l

l

,

.2 .3 .4 .5 .6 .? .8

l

:?

I~

;

10 . 1.1

I

;

1

;

[

; t

12 . 1.3 1.4 1.5 1.6 1.7 1.8

might have been expected for Zl2/N because the ions are so far apart that they appear as point charges to one another. However, each ion type has a specific interaction with the solvent which persists even at great dilution, namely, s ~ l v a t i o n . ~ ~ * ~ ~ ~

~~

(44) Solvation is used in the more general sense which includes the

possibility of breaking up the solvent structure as well as orienting it. Also included in this notion are the polarization of the solvent by the ion and its alteration of the dielectric constant in its immediate surroundings.


2650

The l12/Nhave certain curious aspects aside from the strange crossovers for HC1 compared to the other 1-1 salts. I t is puzzling at first sight that the interaction at a given concentration is largest for KC1 and decreases for KaCl and LiCl in that order. Moreover this order persists to c = 0 and is predicted by the theoretical limiting law.l0 The greater interaction of K + and C1- compared to Lif and C1- could occur if the Lif ion were properly “screened” by its larger solvation sphere. However for Li+ the solvated solvent is (1) more rigidly oriented and (2) subject to partial dielectric saturation. Both effects reduce the effective dielectric constant around the ion (as well as the over-all bulk value). However, a lower dielectric constant means a larger average coulombic force, which would in turn suggest a larger interaction for Li+ and C1- rather than the smaller one observed. On the other hand, the larger solvate sheath of the L i t prevents the C1- from getting as close to the Lif as it can to the less solvated K f . This larger distance of closest approach reduces the average coulombic force between Li+ and C1- compared to K f and The dielectric constant and distance effects oppose each other, but it is plausible that the distance effect is substantially larger, thus explaining the order of l12/N.47 Also curious is the decrease in b 2 / N at higher concentrations for NaC1, LiCl, and BaC12. It would be expected for coulombic reasons to increase continuously with concentration ( i e . , as the ions get closer on the average) for all salts. However, a bulky solvate structure for the above cations would prevent the C1from getting too close compared to K + even a t higher concentration, thereby reducing their interaction per unit concentration increase. The competition for solvate water a t high concentrations must also influence the interaction because at 3 N , for example, the ions are separated by only two water molecules.

VII. Importance of the Ztj The E,, are the fundamental transport coefficients because they arise from a fundamental thermodynamic theory. Any isothermal transport process in our binary system is completely characterized by eq 11 together with a knowledge of the l,, as functions of c, T , and P . Moreover, the same lij apply when the phenomenon is two or three dimensional, e.g., where an electric field is perpendicular to the diffusion direction. Consequently even though the concentration dependence of the lij i s determined from the onedimennumbers t’ ” a‘nd D l the can be applied to any process no matter how complex. The Journal of Physical Chemistry

DONALD G. MILLER

We note that the It, mobilities are independent of the phenomenon (even in the presence of gravitational fieldss4). In contrast, the mobilities usually considered are different in diffusion than in conductance, essentially because they are different combinations of the l , j . Particularly important is the cross term 112. For example, Kohlrausch’s law of independent ion mobilities is valid a t c = 0 only because Zl2/N vanishes there. Because it appears negatively in the expressions for A and tih owing to z2 and because it rises rapidly with concentration, it is the principal cause of the rapid decline of A with increasing concentration. A second exampleg is the Nernst-Hartley equation. Its derivation implicitly or explicitly assumes Zl2 = 0,48and it is valid only at infinite dilution. Attempts to apply it at higher concentrations by the empirical use of A, ti, and the activity term appropriate to the actual concentration yield49p50

where in tohepast the correct reference frame for D has not always been specified. This formula is in serious error at concentrations as low as 0.01 N . If the c = 0 values of ti and A are used but the correct activity factor is retained,4gs50better agreement with experiment is obtained but the result is theoretically un(45) The reference frame is not a contributor a t c = 0 because all of the usual frames become identical there. (46) This line of thought was suggested privately by Professor R. H. Stokes from the standpoint that h / N might be such a sensitive measure of ion interaction that it reflects the “incipient” ion association or ion pairing of a strong electrolyte. Therefore “incipient” ion association would be larger for KC1 than for LiCl because the C1- ion can get closer t o K +. Actual ion pairing, however, is fairly unlikely because the pair is bulkier than the ions individually, resulting in a lower D ,and because the pair would not be acted on by a field gradient. Both of these notions indicate lower 111 for KC1 than for LiC1, contrary to fact. (47) Another tentative possibility suggested privately by Professor H. S. Frank is that h / N should contain a factor qn, with n depending on the nature of the interaction which 112 measures. If the effect is mostly electrophoretic, n = 2 ; if mostly relaxational, n = -2. Comparison shows that an n even more negative than - 2 is necessary to alter the order, which, if this view is correct, implies 1n/N is essentially relaxational. Similar considerations indicate that n = - 1 is proper for Z,JN. For Zll/iV, this slows the decline. For h / N , it brings the different salt values closer together, slows the decline, and causes some of them t o go through a shallow minimum. Closer coincidence could be achieved only with a power of q less than 1, wh+h is different for each salt and moreover is different for Zn/N and l a / N for the same salt. These ideas are suggestive, but no simple, general quantitative correlation appears t o exist between Z, and v, in agreement with the analysis in section VIII. (48) H . J. V. Tyrell, “Diffusion and Heat Flow in Liquids,” Butterworth and Co. Ltd., London, 1961, pp 55-57. (49) G. S. Hartley, Phil. Mag., (8) 12, 473 (1931). (50) R. A. Robinson and R. H. Stokes, “Electrolyte Solutions,” 2nd ed, Academic Press Inc., New York, N. Y., 1959, pp 286-290.


justifiable. Examination of eq 41 with i = 1, j = 2 shows that there is a term missing from eq 47, namely

c

+

:+)Il2

1 0 3 ~ r r ~1z ~ (

-

N

rlrz

(48)

The failure of the Nernst-Hartley equation is thus due solely to the neglect of L / N . Because l12/N rises rapidly with c, so does the error. Finally because lI2 can become so large with respect to a main coefficient, any simplified electrolyte theory which sets 112 = 0 can be only a crude approximation at finite concentrations.

VIII. Viscosity and Vector Transport Properties As noted earlier, viscosity contributions to T u are small. If included, then T u in terms of SF flows becomes 0

n

T U = C J i X , - II div v -

IT:

%El

(grad v ) ~- n a - ( r o t v - 20)

(49)

where the new viscosity quantities (including an asymmetric pressure tensor) mentioned in this section are defined in ref 3, pp 307, 308. However, Curie's theorem51 states that in an isotropic system, vector flows cannot result from tensor forces and vice versa; more precisely an analysis for these linear systems similar to that in ref 3, p 308, leads to

2651

N-H equation) among t, A, and D because they all arise from vector equations. However such a connection cannot exist between q and D or A because q comes from an equation of one tensor order and D and A from equations of another, the two classes being completely decoupled macroscopically. Hence, it is not surprising to find the quantitative failure of Walden's rule and other viscosity correction factors. There must of course be some connections between D, t, A, etc. and q, because all are ultimately calculable from the same microscopic force laws between the particles of the solution. Consequently qualitative correlations may be expected. However the point is that specific force laws for an individual substance cannot be completely eliminated in any rigorous expression for q in terms of D, A, etc., whereas these force laws, no matter how complex, must always cancel out of the Nernst-Hartley expression at c = 0. Statistical mechanical derivations leading to relations between D and q, etc., are based on very specialized models. Objections that Stokes' law and Walden's rule can be derived from fundamental principles are met as follows. These derivations are based on the motion of a large macroscopic object through a continuous medium. These conditions do not apply when an ion moves through a very lumpy medium consisting of other ions and solvent molecules of not greatly differing sizes. Therefore correlations based on such rules, though suggestive, are necessarily qualitative.

IX. The R I j Representation Some workers42154-62 consider an inverse description

II

= -7" div

v

(51)

0

n*= -2 7 (grad VY no= -T,(rot v - 2 0 )

(52)

(53) i.e., there are no viscosity terms with the vector equations and no vector terms in the viscosity equations.52 This implies, for example, that viscous flow can never result in setting up temperature or diffusion gradients in a system, and c o n v e r ~ e l y . ~ ~ There are further implication^.^ If q were connected to D, t, or A by the macroscopic equations, one might expect rules such as Walden's to be valid, perhaps as a limiting case, in the same way the Nernst-Hartley equation is valid in a limiting case. However, because the scalar, vector, and tensor equations are separate, the vector properties t, A, and D are completely determined by the l t j of eq 50, and q is macroscopically independent of them. Therefore, there may be a universal direct macroscopic connection (the

(51) See ref 3,pp 31-34, 57-64, 308. (52) B. Baranowski and T. Romotowski, P h y s . Fluids, 7, 763 (1964),and B u l l . A c a d . Polon., Ser. S c i . Chim., 12, 71, 127 (1964), have shown by a more detailed analysis that a third-order part of the internal angular momentum tensor (arising from an assumed asymmetric pressure tensor) and Ji should have diffusion and angular rotation terms in common. This represents an indirect coupling between vector flows and the third-order part of the angular momentum "flow," contrary to this statement. However, the a p propriate transport coefficients are not related to q", q , or qr; hence, the remarks of this section are still valid. (53) (a) If, however, the pressure tensor is asymmetric, then there can be such gradients parallel to hydrodynamic flow;6* (b) a second instance of Curie's theorem is that scalar processes do not interact with vector ones. Thus a homogeneous chemical reaction (a scalar process) can alter the temperature uniformly throughout the system but can never induce temperature gradients ( L e . , vector quantities) in it. (54) 0. Lamm, A c t a Chem. Scund., 11, 362 (1957). (55) A. Klemm, Z.Naturforsch., Sa, 397 (1953). (56) R. W.Laity, J . P h y s . Chem., 63, 80 (1959). (57) R. W.Laity, ibid., 67, 671 (1963). (58) S. IJjunggren, Trans. R o y . I n s t . TechnoZ., Stockholm, 172, 1 (1961). (59) (a). R. J. Bearman, J . P h y s . Chem., 66, 2072 (1962); (b) J. Chem. P h y s . , 28, 136 (1958); (c) ibid., 28, 662 (1958); (d) ibid., 31, 751 (1959).

V o l u m e 70, N u m b e r 8 A u g u s t 1966

DONALD G. MILLER

2652

of transport process using "friction coefficients" Rt5 to be more desirable for solutions than the Zf, formalism used systematically above.e2 The author does not share this view for reasons given in section D below. However certain aspects of the R formalism will be discussed and the final equations and numerical results given. A . Generalities. The inverse linear laws of irreversible thermodynamics for an isothermal n 1 component system are

+

n

i

X f = xRt5(J,)*

= 0,

. . ., n

(54)

j-0

where the X i are those of eq 2 and the R1,are the transport coefficients inverse to the Zfl. The ( J J * can be written more explicitly as

(JJ* =

- v'*)

Ci.(v"t

(55)

where Bt and v"* are the velocity of i and the velocity of the arbitrary reference frame, respectively, with respect to a cell-fixed reference frame, and have the units l./cm2 sec. Since neither the X,nor the J f are independent, the Rt5 are not unique.5s However, if one arbitrarily assumes

fc5R,,

j -0

=

0

i

=

0, . . ., n

(56)

then the R1, become uniquely specified. Moreover with this assumption, it is easily shown that the Rt, are reference-frame independent. Thus, multiply eq 56 by bi - v'* and subtract from eq 54. One obtain^^^*^^ n

X i = CRtjcj(fij - 5,)

i

= 0,

j-0

. . ., n

(57)

Since 6* cancels, the Rt, do not depend on the reference frame.6s We now show that the ORR for the Zt5 are equivalent to the ORR for the Rti. Since the X, of eq 54 are the same as the X,used for the SF flows in eq 6 and because the Rt5 are the same for the SF frame as any other, let us consider eq 54 written for SF J's. In this case JO = 0 so that the R0 terms vanish, and X Ois dependent because of eq 3. Therefore an independent set of equations can be wri t t e P n

X r = j-1 2Rt,J5

i

= 1,

..., n

(58)

H ~ eq 6 is~the exact ~ inverse ~ of eq ~ 58. since ~ J and X are independent, 111 # 0 and IRI # 0, where

11

is the determinant of the appropriate coefficient matrix. Hence The J O U Tof~Physical ~ Chemislry

R = 1-1

1 = R-1 (59) The Rt, in terms of Zi, or conversely are easily found by matrix inversion or by Cramer's rule. So far, we have not used the ORR, which are equivalent to saying 1 is symmetric. If this is so, then 1-1 and hence R are also symmetric. Therefore the validity of the ORR for eq 6 (i = 1, . . . , n) implies the ORR

Rt,

=

R5f

i, j

= 1,

. .., n

(60)

for eq 58 and conversely. This part of the argument is due to K l e m ~ nand ~ ~L j ~ n g g r e n . ~ ~ There remain the coefficientsRfo and Rot of the dependent eq 54. These may be shown to satisfy the ORR by basing a new SF reference frame on constituent 1 and repeating the above argument and then doing it again using constituent 2 (any other 2 constituents will-do just as well). This technique is valid because eq 5-7 are valid for any choice of solvent constituent and because the Rti are independent of how the reference frame is c h ~ s e n . ~ ~ ~ ~ ~ Conversely if the ORR are valid for the singular R matrix, then they will be valid for the TZ 1 nonsingu1 different choices for lar 1 matrices based on the the solvent. Alternatively, if we assume both

+

+

i, j

CcjRir = 0; C C ~ R ,=, 0

=

0,1, . . .,

(61)

then it is easily shown that eq 60 and 61 imply

Rto

=

i

Ror

=

1, . . ., n

(62)

The above results are valid for both electrolytes and nonelectrolytes. B. Binary System. Let us now specialize to a binary electrolyte system. The complete dependent set is Xo = RoJo RoiJi R d z Xi = RiJo RiiJi RizJz (63) X2 = R2Jo R2iJ1 Rid2

+ + +

+ + +

(60) P. J. Dunlop, J . Phys. Chem., 68, 26 (1964). (61) J. G. Albright and R. Mills, ibid., 69, 3120 (1965). (62) Most of these authors have been concerned with pure diffusion. Laity'ZvL' and Bearman,hg however, have applied this formulation to electrolytes. (63) Cf.ref 15, footnote 37. (64) Alternatively the JoXo term disappears from Tu and the remaining terms are independent. Hence, eq 58 can be written immediately as a legitimate set of linear relations. (65) However (1ij)o # (Zij)i # (2;j)i because the 1's do depend on the reference frame. (66) In other words, any three choices of solvent constituents in the , 1 representation correspond to the inverses of three nonsingular submatrices of rank n of the original singular R matrix of rank n 1. The validity of the ORR for any three such submatrices covers all possible ORR of the original R matrix. Therefore, in ref 58 it was not neceeaary to define Roi Rio arbitrarily.

+


Given values of the Ru, i, j = 1, 2, then Rot = R,o are given by eq 61 as

RZO=

- (rlR12 + r2Rz2)co C

ROO= -(rlRlo

+ r&d-

(64)

C

co

where ct = rtc, and co is the solvent concentration in moles per liter given by Cf,

=

1000d-cM, -mMo Mo lOoOc

2653

We have used the sets (R11, R12, Rzz) and (RIo,Ri2, Rzo). The former comes directly from an analysis of the experiments based on eq 58. The latter set is based on eq 57, where the Rt, are omitted because they multiply zero terms. Equations similar to (69), (70), and (71) have been given by Laity42in terms of r10, r12, rzol where the rfb are Klemm's mole fraction coefficients. The solution of eq 69, 70, and 71 for the Rij lead to the results

R 1 1 =z12 - + - =t-2 2- (Y r12L

~ ~ (X g )

1 0 ~ 5 2 ~ ~ 2

Nh

rlr2 zZ2

(65)

where d is the density and M , and M o are the molecular weights of salt and solvent, respectively. C . Expressions and Values for R f j . Experimental values of the R{,, i, j = 1, 2, can be gotten in two ways. The first is to invert eq 11 for the Z's to get the independent R set

+ Ri2J2 = R21Ji + R2zJ2

X i = RiiJi X2

The results are

RII

=

&; 111

- 112 R - -; - 121 111 RIZ= -; Ill 21 - 111 R22 -- 1-11 (67)

For the seven salts previously considered, this route and eq 64 have been used to obtain the experimental values of Rtj. These values have been multiplied by an appropriate concentration factor to bring them to relative constancy, the results being given in Tables I-VI1 and shown in Figures 4-9. The second way is to derive expressions for Rtr, i, j = 1, 2, in terms of experimental quantities in the same way the Z t j were obtained. The results, using the ORR, are

L =

1

rI2R1l

+ 2rlrzR12 + rz2R22 Laity6' has given related but not quite so explicit results for Klemm's coefficients no,~ Z O ,and r12together with some numerical values for NaC1. D. Comments on the Behavior of the Rtj. Examination of Figures 4-9 shows that the R1o are in the order expected from the bulk of the solvated ion and are relatively constant. For R20 there is a peculiar crossover at S G 0.5. It is not clear whether the hump in N'/'R12 is real or not because RIZdepends on the subtraction of two large smoothed quantities to yield a Volume YO, Number 8 August 1966

DONALD G. MILLER

2654

small one. (A similar hump is exhibited by E12/N’/’.) to infinity a t infinite dilution, and that the solvent fricNRll and NR22 have inflection points, but the latter tion Roobehaves peculiarly. does not have unusual crossovers. Roo,” is relatively X. Remarks constant to S E 0.5. Irreversible thermodynamics gives the study of elecThe R,, are usually interpreted as “friction coeff i c i e n t ~ , ” ~where ~ , ~ ~R,, - ~ is ~ the “friction” of i against j . However, examination of Tables I-VI1 shows that while the R,, are positive, the R,,, i # j , are negative. The significance of a negative friction is somewhat difficult to understand. To avoid negative Rill i # j , LaityS7 and DunlopGObase their definition of friction coefficients on

x,= CC,Ri3(5,- a,)

(78)

(or alternatively using mole fractions), where R,, = -R,,. However, this opposite sign no longer comes from a direct linearization by irreversible thermodynamics, which is undoubtedly why Dunlop@’ discussed the empirical aspects of this formulation at length. If the phenomenological equations are used in the form of eq 57 or 78, then the Rtr, i # j , are of more interest than the Rii, since the latter are coefficients of terms which vanish. However the R,, appear in a nontrivial way in the equally valid formulation of eq 54 and should equally well be interpretable as the friction of i against i. However, if R,, are positive, then R,, art: negative, and the problem of negative frictions has not disappeared. One of the advantages of using R,,, i # j , is that Rlo and Rzo are quite constant with on cent ration,^^ more so than the l,,/N. I t is intuitively reasonable that the frictions of ions 1 and 2 with the solvent, RIOand RZo1 respectively, would be relatively concentration independent. It is also reasonable that Ru, R12, and R22, the frictions of the ions with themselves and with each other, would depend on concentration (although their , a t infinite dilution is difficult to interpret). value, However, up to a few tenths normal, the solvent is the predominant species, so that its friction with itself should not be altered by the presence of a few ions. Therefore, Rooshould not be concentration dependent; yet it depends very strongly on e. Furthermore, a true solvent-solvent friction should have the same value at infinite dilution in any salt solution, yet the limiting Roovaries by a factor of 3 for the salts considered. The question of which formalism is preferable, E or R, is of course a matter of taste. I n the opinion of this author, the notions of conductances or mobilities and of interactions which vanish at c = 0 are intuitively simpler than frictions. The advantages that R’s are reference-frame free and that Rlo and Rzo are constant are counterbalanced by the disadvantages that some R’s are positive and others negative, that some go --a3

The Journal of Physical Chemistry

trolyte solutions a general macroscopic framework which can be applied to any process no matter how compIex and in which no necessary quantities are neglected. The E,, give new insight into ionic properties and are clearly more fundamental than more common quantities such as t, A, and D which are composites of the E,,. It is urged that microscopic theories be applied to electrolyte transport properties in the spirit of Onsager and FUOSS’S classic paper,12 i.e., the calculation of all of the Z, or R,, to the same order of approximation. If any transport quantity is to be given special theoretical attention, it should be Z12. I n this paper we have limited our considerations to binary isothermal systems. The next paper will consider ternary isothermal systems of strong electrolytes.

Acknowledgments. The author wishes to thank Joseph Brady, Robert Carpenter, and Guy 3lcMillan for their aid in computer calculations; Fred AIcMurphy for his aid in critically reviewing the experimental data; and Professors H. S. Frank and R. H. Stokes for their comments on the behavior of the l z j . Appendix 1 Proof That I is Independent of ReferenceFrame. Con-

sider a pair of arbitrary reference frames R and s. The relation between the flows for the ith constituent is’s

+

( W where E , is in moles/cm3 and USR is the velocity of frame S with respect to frame R in cm/sec. Now ( I ) Rand ( I ) sare given by (Jf)R

=

(Ji)S

CtUSR

where z, is zero for nonelectrolytes. eq 1A in the ( I ) Rexpression yields

+

(OR = Czi(Ji)~U S R x Z i c i

Subst,itution of = (1)s

(3A) term vanishes owing to electroneu-

because the USR trality. Because S and R were arbitrary, I is independent of reference frame. Consequently so are a, A, and A. This however is not. the case for ti and D.

(67) That NRu, NRzz, Roo/N, and N ’ / ~ R I(shown Z in Figures 6-9) are also fairly constant can be derived from the fact that ZU/N, h / N . and 11z/Na/2are fairly constant.


lo

2655

I'"'''''''-"'-I

a CL

&

\

'0

LaCI3

t

3 t KC!

2 -4I _-

HCP

0 1 1 0 I

I

I

:

I

:

I

2

3 4

5

6

7

;

I

/

I

e

1

i

I I I I j I ! 8 9 I O I1 12 13 14 15 16 17 I 8 $2

Figure 4.

10-9(-Rlo) us. S'/2. HCP

: I ; ; ; ; I ; i ; ; ; I ; ; ; I 0 .I .2 3 9 .5 .6 .7 .8 .9, 1.0 I,/ 1.2 1.3 1.4 1.5 1.6 1.7 1.8 SG Figure 6 .

10-10N'/z(--R12)

us. S'I2.

3.2-3.0--

a n" 2.8--

*

in our present notation. obtain

'O_ 2.G-

2.2-

1

2.0--

"t

t

0 .I .2 .3 .4 .5 .6 .7 .8 .9 1.0 1.1 1.2 13 14 1.5 1.6 1.7 18 .

&

Figure 5.

- R ~ o us. ) S'/*.

Appendix 2 Derivation of Eq 35. Equations 49 and 28 of ref 38 specialized to a single electrolyte are d In y VU d In y dm dc CPO dm dc

+--

dm dc

c

(4-4)

Using these equations, we

+ c-d In y = (1 + %)(I dC

+ m-

In

dm

')

(6A)

Dividing eq 34 by eq 6A yields eq 35, the desired result.

Appendix 3 Sources of Experimental Data. The Zij/N have been calculated from the smoothed data described below, but are themselves unsmoothed. More places than are significant have been retained for consistency and for reduction in accumulation of errors in subsequent calculations. The uncertainties in the primary data are given below. At infinite dilution, D has been calculated using the Nernst-Hartley equation (eq 47) and ti' and hofrom the tables. We have used k = 8.3144 joules/mole deg, 5 = 96,493 coulombs/equiv, and 25°C = 298.15"K. HCl. c/m = 0.99707 - 0.0003117~"~ - 0.017733~ - 0.0008983~"~,from a least-squares fit of ICTB8 data ford as a function of weight per cent. Volume 70,Number 8 Auguat 1966

DONALD G. MILLER

2656

20

I f

:I

+

& 13 v

4--

/

16 15

'Q

12

3 3--

2

KCP

I

HCL'

0 , ;I : ; I i I ; I I ; I I I I : ! , 0 .I 2 3 4 .5 .6 7 .8 .9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 5

9

8

4

Figure 7. 10-llNR1l us. S'l2.

23

2,2

0 .I .2 3 .4 .5 .6 .7 .8 .9,1.0 1.1 1.2 1.3 1.4 1.5 1.6 17 1.8

4

2.1

Figure 9.

10-7Rpo/N us.

tl is from the best line through the moving boundary (MB) data of Longsworth70and the emf data of Harned and Dreby71 and of Giber, et aLT2 The uncertainty is a few figures in the fourth place to c = 0.2 M and probably one in the third place above. (The notation M in connection with concentration refers to molarity.) A from the best line through the deviation function w = A lOOc'/' is based on the data of Stoked3 and Owen and Sweeton.14 The uncertainty is about 0.05 conductance unit. M = (I&/(l md In y/m) is from the best line through M calculated from 1 md In y/dm above and (D)v from conductometric and Gouy measurements

+

+

0

.I

.2 .3 4 .5 .6 .7 .8 .9 10 . 1.1

1.2 13 j.4 1.5 1.6 1.7 1.8

Sh

Figure 8. 10-1lNR12 us. SI/%.

+

1 md In r/dm from the derivative of eq 12-5-3 of Harned and Owenagis based on emf data using the density equation above. The uncertainty is a few figures in the third place at higher concentrations. The JOUTTZU~ of Physical Chemistry

+

(68) "International Critical Tables," Vol. 111, McGraw-Hill Book Co., Inc., New York, N. Y., 1929,p 54. (69) See ref 39, pp 509, 510. (70) (a) L. G. Longsworth, J. Am. Chem. SOC.,54, 2741 (1932); (b) ibid., 57, 1185 (1935). (71) H. S. Harned and E. C. Dreby, ibid., 61, 3113 (1939). (72) J. Giber, S. Lengyel, J. Tamas, and P. Tahi, Mugy. Kem. Folyozrut, 66, 170 (1960); Acta Chim. Hung.,32, 429 (1962). (73) R. H. Stokes, J . Phya. Chem., 65, 1242 (1961). (74) B. B. Owen and F. H. Sweeton, J. Am. Chem. SOC., 63, 2811 (1941).

ISOTHERMAL VECTORTRANSPORT PROCESSES IN BINARYELECTROLYTE SYSTEMS

of Harpst, et U Z . , ~ ~and diaphragm cell measurements of Stokes.76 is from the smoothed values of M using the 1 md In y/dm above. The uncertainty is between 0.01 and 0.02. LiCZ. c / m = 0.99707 - 0.01746~ - 0.000698c2, from the density equation of Jones and B r a d ~ h a w . ~ ~ 1 md In y/dm is from eq 12-5-3 of Harned and Owed9 using the above density equation up to 0.9 M . Above 0.9 M , numerical derivatives (third differences) were used, based on a sixth-order polynomial in ml/' fit to $ from Robinson and Stokes.78 The uncertainty may be as much as 0.01 above 1.0 M . tl is from the best line through the MB data of Longsat low concentrations and the Hittorf data of Jones and Bradshaw77 above 0.5 M . These data run smoothly into each other but the Hittorf values are increasingly high below 0.5 M . The uncertainty is a few figures in the fourth place below 0.5 M and one or two in the third place above 0.5 M . A is from the best line through AO' from the data of Jervis, et U Z . , ~ to ~ 0.005 M and from the data of Shedlovskyso (corrected to Jones and Bradshaw's standard8') from 0.005 to 0.1 M . These data are in good agreement, but those of Krieger and Kilpatrick82 are not. Above 0.1 M , the best line is through (AICT 0.3)83 ~ ~1.0 M , and Jacopetti's datas4 above to 0.7 M , A I C T at 1.0 M . The uncertainty is within 0.02 to 0.1 M , 0.1 above. M = (D)v/(l md In r/dm) from 1 md In y/dm above and (D)vbelow. (0)" is from the best line through the conductometric data of Harned and Hildreths5 and the Gouy data of Vitagliano.86 The uncertainty is about 0.002. The diaphragm cell data of Stokes76are less accurate. NaCl. c/m = 0.9970 - 0.0183m, from ref 39, p 725. 1 md In r/dm is from eq 12-5-36gwith the above density equation to 1.3 M . Above 2.0 M the best line through the numerical derivatives of +ms7 was used, where 4 is the osmotic coefficient and

+

+

+

+

+

+

d(@m)/dm = 1

+ md In y/dm

(7-4) I n between, the best line consistent with the emf and isopiestic data was used. The uncertainty is about 0.005. tl is from the best line through the MB data of Allgood and Gordonss and L o n g s w ~ r t h , and ~ ~ ~the , ~ ad~ justed indicator data of Currie and Gordon.go The latter were Calculated using the smoothed KC1 results of Table IV and the volume corrections of Spir0.~1 The uncertainty is a few figures in the fourth place up to 0.5 M , and 0.001 above 0.5 M . A is from data of Shedlovsky, et U Z . , ~ to ~ 0.2 M , which

2657

are in good agreement with Benson and Gordon.93 Above 0.2 M the data of Chambers, et a l l g 4were used. The uncertainty is 0.03-0.05. M = (D)"/(l md In r/dm) is from the best line md In r/dm above through M calculated from 1 and (D)v from conductometric data of Harned and Hildreths5 and Gouy data of Vitagliano and Lyons.95 Stokes' diaphragm cell data76are less accurate. ( D ) v is from smoothed values of M and (1 md In -y/dm) above. The uncertainty is about 0.0030.005. KCZ. c/m = 0.9970 - 0.0284m 0.0003m2,from ref 39, p 725. 1 md In r/dm is from the derivative of Gosting's equationg6to 1.0 M . Above 1.0 M the best line through the numerical derivatives of 4mg7 was used. The uncertainty is two or three figures in the third place. tl is from the equation

+

+

+

+

+

tl

=

0.4905 - 0.00254C

(8A)

based on the hiIB data of Shedlovsky and M a c I n n e ~ , ~ ~ Longsworth and M a ~ I n n e s ,and ~ ~ Allgood, et (75) J. A. Harpst, E. Holt, and P. A . Lyons, J . Phys. Chem., 69, 2333 (1965). (76) R. H. Stokes, J . Am. Chem. SOC., 72, 2243 (1950). (77) G. Jones and B. C. Bradshaw, ibid., 54, 138 (1932). (78) See ref 50, p 483. (79) R. E. Jervis, D. R. Muir, J. P. Butler, and A. R. Gordon, J . Am. Chem. SOC., 75, 2855 (1953). (80) T. Shedlovsky, ibid., 54, 1411 (1932). (81) G. Jones and B. C. Bradshaw, ibid., 55, 1780 (1933). (82) K. A. Krieger and M. Kilpatrick, ibid., 59, 1878 (1937). (83) See ref 68, Vol. VI, p 233. (84) SI. Jacopetti, Gam. Chim. Ital., 72, 241 (1942). (85) H. S. Harned and C. L. Hildreth, J . Am. Chem. SOC.,73, 650 (1951). (86) V. Vitagliano, Gam. Chim. Ital., 90, 876 (1960). (87) See ref 50, p 476. (88) R. W. Allgood and A. R. Gordon, J . Chem. Phys., 10, 124 (1942). (89) L. G. Longsworth, J . Am. Chem. SOC., 65, 1755 (1943) (90) D. J. Currie and A. R. Gordon, J . Phys. Chem., 64, 1751 (1960). (91) SI. Spiro, J . Chem. Phys., 42, 4060 (1965). (92) T. Shedlovsky, A. S. Brown, and D. A. MacInnes, Trans. Electrochem. SOC.,66, 165 (1934). (93) G. C. Benson and A. R. Gordon, J . Chem. Phys., 13, 473 (1945). (94) J. F. Chambers, J. hI. Stokes, and R. H. Stokes, J . Phys. Chem., 60, 985 (1956). (95) V. Vitagliano and P. A. Lyons, J . -4m.Chem. SOC., 78, 1549 (1956). (96) L. J . Gosting, ibid., 71, 4418 (1950). (97) See ref 50, p 476. (98) T. Shedlovsky and D. A . MacInnes, J . Am. Chem. SOC.,59, 503 (1937). (99) D. A. MacInnes and L. G. Longsworth, Chem. Rev., 11, 171 (1932).


DONALD G. MILLER

2658

+

and on the Hittorf data of MacInnes and Dole.lol The 1 md In y/dm is from the derivative of Shedlovsky’s equation for log y114 and the density equation above. uncertainty is 0.0002 to 1.0 M and 0.0015 above 1.0 The uncertainty is a few figures in the third place. M. Our smoothed values differ slightly from those of S h e d l o ~ s k y ’ sdata ~ ~ ~differ from those of McLeod and Currie and Gordon.go A is from the data of Shedlovsky, et to 0.1 M, Gordon115because, although both are based on concentration cells, the transference numbers used to obChambers, et thereafter. These are in good agreetain y or y are different. ment with Gunning and Gordonlo2 and Benson and Gordon.93 The uncertainties are about 0.03-0.05. tl is from best line through the data of Long~worth.’~b M = (D),J(l md In y/dm) is from the best line longs worth'^'^^ data are preferred to those of Keenan, et aZ.,l16both because they extend to higher concentrathrough M calculated from 1 md In y/dm above and is from the conductometric data of Harned and tion and because they are consistent with the selection Nuttalllo* and Harned and Blander,lo4 and optical of y. There seems to be no criterion for choosing bedata of Gostingg6and Longsworth.lo5 tween them on experimental grounds. The sets of (D)y is from smoothed values of M and 1 md In data differ by as much as O.OOl5, which is taken as the y/dm above. The uncertainty is about 0.003-0.005. probable uncertainty. BaCZ2. c/m = 0.99707 - 0.02349~- 0.00068~”~~ A is from the best line through the deviation function from the density equation of Jones and Ray.loB x = A 150c’/’, based primarily on the data of Benson 1 md In y/dm is from the density equation above and Gordon117to 0.01 M and on the data of Shedlovsky and the equation and Brownlll above 0.01 M. Where they overlap, both sets of data are in excellent agreement; the uncertainty is about 0.03. M = (D)v/(l md In y/dm) is from the best line 0.11124c(l In c) - c X through M calculated from 1 md In y/dm above and (D)yfrom conductometric data of Harned and Parker”* 0.03057 - 0.00102~”’ and Gouy data of Lyons and Riley.11g Earlier con[0.99707 0.03057~- O.O0068c’/’ ductometric and diaphragm cell data are inaccurate. (9A)

+

+

+

+

+

+

+

+

1

+

~

This equation is based on a least-squares fit of y from m = 0.1 to l.21°7with weights 1 and the y’s from diffusion at the four lowest concentrationslo8with weights 2, using the density equation above. This fit is good to 0.002 in y , and the derivatives are within 0.5% of numerical derivatives of mrp from ref 107. tl is from the best line through the Hittorf data of Jones and Dolelog (no MB data exist). The uncertainty is about 0.001. tI0 is obtained from Ao and Aocl = 76.35. A is from the best line through the deviation func175~’”to 0.05 M and x = A 46c”’ tions y = A from 0.05 to 1.0 M. These lines were primarily based on the values of Jones and Dolello adjusted to the Jones-Bradshaw standards1 and of Shedlovsky and Brown (rounded) The uncertainty is about 0.05. The data of Calvert, et a1.,l12are quite scattered. M = (.ZI),J(l md In y/dm) is from the best line through M calculated from 1 md In y/dm above and ( D ) y from conductometric data of Harned and Polestral/3arid Gouy data of Vitagliano and Lyons.e5 (D)v is from smoothed values of M and 1 md In y/dm above. The uncertainty is a few figures in the t,hird place. CaC12. clm = 0.99707 0.0924~ - O.O042c’/’, from the density equation of Shedlovsky and Brown. 111

+

+

+

+

+

+

The Journal of Physical Chemktry

(100) R. W.Allgood, D. J. Leroy, and A. R. Gordon, J . Chem. Phys., 8, 418 (1940). (101) D. A. MacInnes and hi. Dole, J . Am. Chem. SOC.,53, 1357 (1931). (102) H. E. Gunning and A. R. Gordon, J . Chem. Phys., 10, 126 (1942). (103) H. S. Harned and R. L. Nuttall, J . Am. Chem. SOC., 71, 1460 (1949). (104) H. S. Harned and M. Blander, J . Phys. Chem., 63, 2078 (1959). (105) L. G. Longsworth in “Structure of Electrolyte Solutions,” W. J. Hamer, John Wiley and Sons, Inc., New York, N. Y., 1959, p 194. (106) G. Jones and W. A. Ray, J . Am. Chem. SOC.,63, 288 (1941). (107) See ref 50,p 498. (108) See ref 39,p 252. (109) G. Jones and M. Dole, J . Am. Chem. SOC.,51, 1073 (1929). (110) G. Jones and hl. Dole, ibid.,52, 2245 (1930). (111) T.Shedlovsky and A. S. Brown, ibid., 56, 1066 (1934). (112) R. Calvert, J. A. Cornelius, V. S. Griffiths, and D. I. Stock, J . Phys. Chem., 62, 47 (1958). (113) H. S. Harned and F. M. Polestra, J . Am. Chem. SOC., 76,2064 (1954). (114) T.Shedlovsky, ibid., 72, 3680 (1950). (115) H.G. McLeod and A. R. Gordon, ibid., 68, 58 (1946). (116) A. G. Keenan, H. G. McLeod, and A. R. Gordon, J . Chem. Phys., 13, 466 (1945). (117) G. C. Benson and A. R. Gordon, ibid., 13, 470 (1945). (118) H. S. Harned and H. W. Parker, J . Am. C h m . Soc., 77, 265 (1955). (119) P.A. Lyons and J. F. Riley, ibid., 76, 5216 (1954).


+

(0)"is from smoothed values of M and 1 md In y/dnz above. The uncertainty is about 0.003-0.004. L a C h c/m = 0.99707 - 0.01587~- 0.01191ca'/', from the density equation of Jones and Ray.loB 1 nzd In y/dm is from the derivative of Shedlovsky's equation for log y114 and the density equation above. These are in reasonable agreement (0.002) with those of Spedding, et a1.120 21 is from the best line primarily through the data of Longsworth and i\!tacInnes121at low concentrations and Spedding, et a1.,122at higher concentrations. The uncertainty is about 0.0005. The Hittorf data of Jones and P r e n d e r g a ~ t lare ~ ~in poor agreement a t lower concentrations (as much as 0.008). t10 is obtained from Aa = 145.95 of James and R!I0nk1~~ and A%, = 76.35. A is from the best lines through the deviation functions x = A $. 300c'/' and, above 0.003 M , y = A 125c'/', averaged where they overlap. These lines are based primarily on the data of Longsworth and iUacInnes,121Jones and Bickford,IZ5and James and all of which are in excellent agreement. The uncertainty is about 0.02-0.04. The data of Saeger and Spedding126and Spedding, et a1.,127 are in less good agreement. ill = (D)o/jl cd In y/dc) is from the best line cd In y/dc of Shedthrough M calculated from 1 lovsky's equation114and ( D ) ousing the density equation above and ( D ) y from the conductometric data of Harned and Blake.128 The difference between ( D ) o and (0)" is everywhere less than 0.0005 a t these low concentrations. (I)),,is from smooth values of M and the 1 md In y / dm above. The uncertainty is about 0.002.

+

+

+

+

+

Appendix 4 Explicit Equations for l i j / N . For convenient refer-

2659

ence, eq 41 is written out below, omitting the distinction between th and t'. 111 --I t12A -

We note also that eq 72-74 and eq 76 and 77 can be written in condensed form, respectively, as

1 - tt

(i

=

1, 2)

(14A)

(120) F. H. Spedding, P. E. Porter, and J. M. Wright, J . Am. Chem. SOC., 74, 2781 (1952). (121) L. G.Longsworth and D. A. MacInnes, ibid., 60, 3070 (1938). (122) F.H. Spedding, P. E. Porter, and J. hl. Wright, ibid., 74, 2778 (1952). (123) G.Jones and L. T. Prendergast, ibid., 58, 1476 (1936). (124) J. C. James and C . B. Monk, Trans. Faraday Soc., 46, 1041 (1950). (125) G. Jones and C. F. Bickford, J . Am. Chem. Soc., 56, 602 (1934). (126) V. W. Saeger and F. H. Spedding, 18338, Nov 1960. (127) F.H. Spedding, P. E. Porter, and J. M. Wright, J . Am. Chem. Soc., 74, 2055 (1952). (128) H . S. Harned and C. A. Blake, ibid., 73, 4255 (1951).


Application of Irreversible Thermodynamics to Electrolyte Solutions. I

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