ROBERTN. GOLDBERG AND HENRY S. FRANK
1758 The Rotne-Prager improvements on the Oseen hydrodynamic interaction tensor were designed to deal with singularities in the intrinsic viscosity as noted by DeWames, et U Z . , ~ ~ and Zwanzig, et aL22 The singularities appear when the segmental friction constant is appreciably larger than that of a sphere obeying macroscOpic hydrodynamics* It is evident that these same singularities arise in eq 9 and 10 a t high values off. It
does not escape US that a further discussion of these singularities is in order in the light of our results, to which we hope to return in the future. (21) R. E. DeWames, W. F. Holland, and M. C. Shen, J . Chem. Phys., 46, 2782 (1967). (22) R. Zwanzig, J. Kiefer, and G. H. Weiss, Proc. N a t . Acad. Sci. u. s., 60,381 (1968).
Liquid Junction Potentials and Single-Ion Activities by Computer Simulation. I. The Concentration Cell with Transference by Robert N. Goldberg* and Henry S. Frank Department of Chemistry, Unizersitg of Pittsburgh, Pittsburgh, Pennsylvania 16216, and Mellon Institute of Carnegie-Mellon University, Pittsburgh, Pennsylvania 16816 (Received December 6, 1971) Publication costs assisted by the University of Pittsburgh and the National B u r e a u of Standards
The liquid junction potentials in KCl, NaC1, and HCl concentration cells with transference have been evaluated by computer simulation, making use of Onsager flux coefficients tabulated by Miller, and of a convenient formalism for single-ion activity coefficients to make possible the representation of osmotic driving forces. Varying assumptions regarding single-ion activity coefficients lead to varying values of a computed steadystate junction potential, but these differences are canceled by corresponding differences in calculated electrode potentials, so that the computed overall cell potential remains unchanged--i.e., measurement of the potential of a concentration cell with transference can give no information about single-ion activity coefficients. This expected result plus the agreement of computed cell potentials with experimental ones speaks both for the essential validity of the method and for the mutual self-consistency of the numerical values inserted for Onsager coefficients and for mean ionic activity coefficients. I n contrast to the steady-state cell potential, the time rise of the calculated potential is sensitive to the choice made for the single-ion activity coefficient parameter. There would thus seem to exist a possibility for determining single-ion activities in an experiment, if the time rise of the cell potential, after the formation of the junction, could be measured on a nanosecond time scale.
Introduction The liquid junction-individual ion activity problem remains, unsolved, as one of the classic problems of physical chemistry. Well-known thermodynamicists have at various times taken differing views on the problem. Guggenheiml has contended that individual ion activities and liquid junction potentials have “no physical reality,” on the ground that there is and can be no way of measuring them, but is also identified with a convention for dealing with them.2 Pitzer and Brewer3 write that “single-ion properties are potentially measureable.” Kirkwood and Oppenheim4 have made an equivalent statement in pointing out that “it is possible to measure absolute single-electrode potentials (by non-thermodynamic methods) .” Frank6 in advocating a return to the ascription of physical significance to single-ion activities, has deThe Journal of Physical Chemistry, Val. 76, N o . 12, 1972
fined (see below) a function useful for representing and manipulating individual ion activities. Unfortunately, there are to the best of our knowledge no data as yet available that lead rigorously to values of this function, although a number of ingenious proposals have been made. Most recently, Bates, Staples, and Robinson6
* Address correspondence to this author at the National Bureau of Standards, Washington, D. C. 20234. (1) E. A. Guggenheim, J . P h y s . Chem., 33, 842 (1929). (2) D. A. MacInnes, “Principles of Electrochemistry,” Dover Publications, New York, N. Y., 1961, p 242. (3) G. N. Lewis and M. Randall (revised by K . S. Pitaer and L. Brewer), “Thermodynamics,” McGraw-Hill, New York, N. Y., 1961, p 310. (4) J. G. Kirkwood and I. Oppenheim, “Chemical Thermodynamics,” McGraw-Hill, New York, N , Y., 1961, p 211. (5) H. S. Frank, J . Phys. Chem., 67, 1554 (1963). (6) R. G. Bates, B. R . Staples, and R . A . Robinson, A n a l . Chem., 42, 867 (1970).
CONCENTRATION CELLWITH TRANSFERENCE have assigned plausible values for single-ion activities in alkali halide solutions, derived on the basis of assumptions regarding single-ion hydration numbers, but these are not claimed to be either rigorous or unique. It has long been recognized that if liquid junction potentials could be unambiguously evaluated, it would be possible to use this information in combination with data from readily accessible electrochemical cells to obtain individual ion activities. The customary numerical formulas’ for liquid junction potentials depend, however, on prior assumptions regarding the individual ion activities, and so cannot be used to solve for them. Another approach to this problem is through the calculation of the charge distribution in the junction arising from the differential mobility of the ions. Early attempts of this sort were made by Taylor* and Longs~ v o r t h . ~Recently, HafemannlO has carried out an improved calculation of this kind, making use of three developments not available to earlier workers; namely, the Onsager formulation of nonequilibrium thermodynamics, accurate transport data, and the availability of high-speed computational methods. I n the present paper we have extended Hafemann’s calculation, for a junction of the type MX(c), MX(c’), t o include the cross coefficients in the Onsager equations, and have introduced a procedure t o take more flexible account of individual ion activity behavior. These calculations display an internal consistency that lends support to the formalism adopted and to the numerical values of the transport coefficients employed. As will be seen, however, the calculation here reported cannot be used to obtain unique values for either single-ion activities or liquid junction potentials.
1759 permittivity (mks units are used throughout this paper). If cl(x) and cz(x) are the concentrations, respectively, of cation and anion in particles per cubic meter and x1 and zz are their signed charges, then p(x) = xlcl(x) z2c2(z). If a length from x = 0 to x = 1 includes the whole of the region in which variations in cl, cz, p , or 4, are found, then the potential difference across the junction is given by the double integral
+
This integral becomes essentially constant with time when it has adjusted itself so that over its length the differential pull, which 4,1 - 4,o exerts on the positive and negative ions, suffices to compensate the forces of diffusion which result from the differences in concentration. The variation with time of the concentration of the ions is calculated by use of the Onsager phenomenological equations, using the formulation of Miller. l1 Assuming that bulk flow can be neglected, and choosing a solvent-fixed reference frame
where the J’s are fluxes given by J1
=
-zl [(g?)x
+ xlF(g),l
-
(4)
The Calculation We are concerned with the calculation of the liquid junction potential for the simplest possible case, when two different concentrations of the same binary electrolyte are in contact with each other through an initially sharp boundary. Such a junction is represented by iLIX(c), MX(c’) where we take M and X to be univalent cations and anions, respectively, and the end concentrations are c and c‘. The calculation considers the solutions to be infinite in extent in the y and x directions, so that all variations of composition and potential extend along the x direction. Representing the liquid junction potential as arising from a distribution of charged particles, the Poisson equation simplifies to
Here the laj’sare the Onsager transport coefficients, and his reciprocity relation states that 112 = 121 (the applicability of the reciprocal relation to concentration cells with transference is assumed here in general and has been demonstrated in the case of the silver nitrate concentration ce1112). bprc/bxis the gradient of the chemical potential of ion i. It, the I,, terms, and the gradient of the electrostatic potential, (b+/dx), = -j,” (p(x)/E)dx, are all to be evaluated at the appropriate position x. lL1illerll has given numerical values of Z i j for several aqueous systems as functions of concentration, derived from measured conductances, trans(7) See ref 2, Chapter 13. (8) P. B. Taylor, J . Phys. Chem,, 31, 1478 (1927).
where @ is the electrostatic potential in the interior of the solution, p(x) is the charge density, and e is the
(9) L. G. Longsworth, as quoted in ref 2, p 239. (10) D. Hafemann, J . Phys. Chem., 69, 4226 (1965). (11) D. G . ~ i 1 1 ibid., ~ ~ ,70, 2639 (1966). (12) M. J. Pika1 and D. G . Miller, ibid., 74, 1337 (1970).
The Journal of Physical Chemistry, Val. 76, No. 12, 1972
ROBERT N. GOLDBERG AND HENRYS. FRANK
1760 ference numbers, and diffusion coefficients. We employ his tabulated values in our calculation. The values of the bpl/bx and bpz/bx terms which appear in eq 4 are given by
tivity data from Wu and Hamer.14 These data were then fitted to polynomial expressions in the concentration, using a least-squares fitting program from Oak Ridge National Laboratory. l5 Hafemannlo has found that the value used for the permittivity affects only the time rise of the junction potential and not the final steady-state value. Our calculations confirmed this, and we have used a permittivity corresponding to a dielectric constant of 78.30, corresponding to that for pure water a t 25.Oo.l6 We have used a length increment of m and a time interval of 10-lO sec in the where E = (cl c2)/2. To obtain numerical values calculation. All computations were performed on the for the gradients, use is made of Frank’s5 function, CDC 1604 computer at Mellon Institute. Calculathe mean ionic activity deviation, defined on the molartions performed using the more exact double-precision ity scale by the equation In & = (l/v)[v+ In y+ routine on the computer yielded the same results as a v- In y-1, where each v is the appropriate ion number. single-precision calculation, for which computation For a 1-1 electrolyte, this becomes 6, = [ y ~ / y ~ ] ~ ’ ~ , time was of the order of 3 min, to simulate a junction which complements the definition of the mean ionic ( i e . , 100 “jumps”). It was found rise time of 10 nsec (molar) activity coefficient, y;t = (~lyz)~’*. that the junction potential had largely leveled off by Introducing this function, the b In y,/& terms in this time, and successive iterations yielded values eq 5 become linear in l/t3. This dependence was therefore used to extrapolate t o a virtual EJ at infinite time. This extrapolation correction was never more than a few per cent. and
+
b- -In_y2 _ _- _b -In yk
bF
bF
b In ba
(7)
To perform the calculation, it is necessary t o postulate a concentration dependence of 6,. For convenience, and on grounds of physical intuition, we have chosen a linear one for In 6,; namely In 6*
=
BF
(8)
Thus, if a value of the constant B is assumed and the experimental values of In y, are inserted, one can calculate the b In y , / b c values which are needed for the determination of the bpi/% terms. The integration of eq 2 is performed by what is essentially a computer simulation method where, beginning with the initial concentration profiles of the ions across the junction, @(x, to) the appropriate derivatives are computed and used to calculate the local fluxes and thence the changes over a sufficiently small period of time in the local concentrations of the ions. This gives a new distribution of the ions, to which there now corresponds a new al - @,. By repeating this process as long as is necessary, one is able to obtain the time rise of the junction potential. I n order t o perform the calculation on a digital computer, it is necessary to represent the analytical operations in terms of numerical approximations. I n this - integral (eq 2 ) is easily shown to reregard, the duce to Hafemann’slO numerical expression for the junction potential. All derivatives were calculated by the method of central differences.13 The lcj data were taken from the critical evaluation of Miller,11 and acThe Journal of Physical Chemistry, Vol. 76, N o . l$, 1972
Results and Discussion I n Table I are presented the results of calculations for several junctions involving aqueous NaC1, KCI, and HC1 as functions of the adjustable coefficients B. Also included are values of E J Acalculated, using the same B’s, combined with experimental data from the appropriate concentration cells with transference, e.g. Ag, AgC1, MCl(c), MCl(c’), AgCl, Ag
A The measured emf of cell A is given by
yciRT - In _I YCIF
c + RT - In 7 + EJA (9) F c
RT c RT & RT y;t E A =- l n - l + - l n ~ - - l n - , + E J ~ F y , F C F6,t
(10)
Turning eq 10 around and introducing the B coefficient, we obtain (13) M. G. Salvadori and M . L. Baron, “Numerical Methods in Engineering,” Prentice-Hall, Englewood Cliffs, N. J., 1961. (14) Y. C. Wu and W. J. Hamer, National Bureau of Standards Report No. 9908, U. 8 . Government Printing Office, Washington, D. C., Sept 1968. (15) L. H . Lietzke, ORNL-3132, Oak Ridge National Laboratory, Oak Ridge, Tenn., April 1961. (16) C. G. Malmberg and A. A. Maryott, J. Res. N u t . .Bur. Stand., 56, l(1956).
CONCENTRATION CELLWITH TRANSFERENCE
RT -In F
c c'
+ RT --B(c F
- c')
1761
(11)
We thus have two ostensibly independent methods, one the computer simulation and the other eq 11, which are capable of yielding values for the liquid junction potential, both involving the same unknown coefficient B , but employing experimental data from entirely un-
firmed that the values for EJA(computer) agreed with those found by means of eq 11 within a reasonable assignment of the accuracy of the I,, values. The two routes to the junction potential EJAare therefore somehow tautological and furnish no basis for solving for anything. That this sort of tautology should have been anticipated is to be understood in terms of the following argument. In the asymptotic limit in which the time rise of the junction potential has been completed, one has a condition of zero current flowing through the junction; that is
Table I : Results of Liquid Junction Potential Calculations
Junction
KC1 (0.10 M), KC1 (1.0 Ill) KCI (0.10 M), KC1 (0.50 M ) KC1 (0.10 M), KC1 (0.20 M ) KCl (0.10 M ) , KCl (0.08 M)
Assumed B
-4.0 0.0 2.0 -10.0
93.40 1.17 -44.95 103.35
0.0 10.0
0.80
-10.0 0.0 10.0 -10.0 0.0 10.0
KC1 (0.10 M ) , KCl (0.0501 M)
-10.0
KC1 (0.10 M ) , KCl (0.04 M )
-10.0
KC1 (0.10 M ) , KC1 (0.03 M )
EJA(C0mputer), mV
0.0 10.0 0.0
10.0 -10.0 0.0 10.0
KC1 (0.10 M ) , KC1 (0.01 M )
-10.0 0.0
KC1 (0.10 M ) , KC1 (0.005008 M )
-10.0
HCl (0.10 M), HC1 (0.003447 M )
-10.0 0.0
NaCl (0.09957 M ) , NaCl (0.049833 M )
-10.0
10.0 0.0 10.0
10.0 0.0
10.0
-101.76 26.01 0.34 -25.34 -5.24 -0.11
5.03 -13.13 -0.33 12.47 -15.81 -0.44 14.94 -18.47 -0.58
17.32 -23.44 -1.05 21.33 -23.90 -1.25 21.36 30.51 51.58 74.96 -16.49 3.73 9.03
EJA(eq 11)s mV
93. 51Q 1.02 -45.23 103.58'" 0.81 - 101.96 26.05" 0.36 -25.33 -5.22" -0.09 5.05 - 13.21. -0.39 12.43 -15.91" -0.49 14.92 - 18.63a -0.65 17.34 -24.23'" -1.11 22.01 - 25.83" - 1.42 22.98 29.12b 53.92 78.73 -16.560 -3.78 9.00
EA values from T. Shedlovsky and D. A. MacInnes, J . Amer. Chem. Soc., 59, 503 (1937). EAvalues from T. Shedlovsky and D. A. MacInnes, ibid., 58, 1970 (1936). EAvalues from A. S.Brown and D. A. MacInnes, ibid., 57, 1356 (1935).
related experiments. On the face of it, this would seem to afford a method of solving for B and thus obtaining values for the single-ion activities. As is obvious from an examination of Table I, however, the sets of values obtained by the two methods are in all cases quite close t o each other, and by introducing slightly "juggled" input data into the calculations, it was con-
I
= (ZJl
+
22J2)F =
0
(12)
Miller11 has shown that this condition leads to the classical expression for the liquid junction potential'
EJ
-El F
dhai
across junction
(13)
i Xi
where ti is the transference number of ion i. I n the case represented by cell A, the integrand in eq 13 reduces to tM+d1n a&(+- tci-d In aci-
=
d In acl-
+ 2 t ~ +Ind ah
(14)
where use has been made of the identities 2 In uk
=
In a b ( +
+ In acl-
(15)
and tx+
+ tc1- = 1
(16)
Therefore
The value of EJ, the junction potential calculated in the computer, although arrived a t by a seemingly independent route, is thus seen t o contain implicitly the term - ( R T / F ) In (ucI-/u'c~-), which cancels the corresponding term in eq 9. This produces the result that the remaining part of the computed value of the junction potential a t a virtual infinite time is fully represented by the last term in eq 17, which also represents the value of EA,the total cell emf of cell A and contains no residue depending on the value of the parameter B. It is thus seen that after the time rise of the junction potential has been completed, the emf of cell A contains, and can furnish, no information about individual ion activities. This corresponds to our empirical finding that changes in the assumed values of the B parameter produce changes in the junction potential and in the electrode potentials that exactly cancel. The exactness of this cancellation, as evidenced by the agreement of our results in Table I, supports the essential correctness both of the simulation procedure and of the Zi, values as tabulated by The Journal of Physical Chemistry, Vol. 7 6 , No. 12, 197%
1762
The insensitivity of a computed junction potential to the assumed value of the permittivity E presumably arises from a similar cancellation. While a rigorous analysis of this point would involve making E a function of the position x in the liquid junction and therefore seems a t present to be out of reach,I7the qualitative features of the compensation seem clear enough. A smaller E would result in stronger electrical forces, and thus a given concentration gradient would be accompanied by a smaller charge separation. I n exactly the same proportion, however, a given charge separation would produce a larger potential difference, and the result is understandable that a given concentration gradient would give the same potential regardless of the value of E. An extension of the simulation procedure to the three-ion junction, i e . , the Lewis and Sargent junction, was attempted by Hafemann.1° The rigorous performance of such a calculation, however, requires not only more than one B parameter, but also a means of treating the mixed electrolyte solutions in the junction. An argument analogous t o that given above leads to the expectation that the emf of the Lewis and Sargent junction will likewise prove incapable of yielding values for single-ion activities. A detailed analysis of this problem has been carried out by Chen and Frank.lg We have also attempted to treat cells with liquid junctions containing four different ions, but to date have been unable to devise any interpretation which is both rigorous and useful for the emf values for cells of this type, An additional result of the simulation calculations reported here is t o suggest a novel thought experiment that might lead to a solution of the single-ion problem. This may be illustrated by reference to Figure 1, which illustrates the fact that only after the time rise of the junction potential is complete is the cell emf independent of the assumed B parameter. If, therefore, the time rise of the junction potential could be measured, an experimental basis might exist for assigning a numerical value to B and thence to the single-ion activities. Since the time rise is of the order of nanoseconds, such an experiment involves difficulties which are at present insuperable. Another possibly serious difficulty arises from the fact that the computed time rise for a given B is sensitive to the values of E which underlies the computation, so that it might turn out that, if the variability of E across the junction matters, the measured time rise might not fit any curve calculated for a constant E . Insofar, however, as an extrapolation of EA back t o time t = 0 could be made, an estimate of B would seem feasible.
The Journal of Physical Chemistry, Vol. 76, N o . 12, 1972
ROBERT N. GOLDBERG AND HENRY S. FRANK I
1
4 -loor -1501
I
2
3
4
5
I
Time, nonoSeconds
Figure 1. Computed values of cell emf, EA, eq 9, as a function of time for the cell Ag, AgCI, KCl (0.10 M ) , KCI (0.50 M ) , AgC1, Ag. Computations are for three values of the parameter B, defined in eq 8.
Another thought experiment would involve the design of a cell in which the junction is far enough separated from the electrodes and well enough defined in extent to make possible a measurement of the dipole moment produced by its charge distribution. Here again, present experimental procedures fall far short of the sensitivity required for this measurement. Here also, medium or dielectric constant effects might present difficulties of interpretation. Mention should also be made of' Oppenheim'sZ0development of Kirkwood's suggestion for direct measurement of an individual electrode potential. I n that experiment, the electrode is brought into rapid oscillation and the intensity and angular distribution are measured of the quadrupole radiation produced. Oppenheim has derived equations that relate these quantities to the charge distribution a t the electrode and thence to the individual electrode potential.
Acknowledgments. Grateful acknowledgment is made of the support of this work by the Office of Saline Water, U. S. Department of the Interior. We are also grateful to Dr. Donald G. XIiller for helpful correspondence on a number of points in this paper. (17) Related t o the variability of e with x is the position sometimes taken that since solutions of concentration c and c' are, properly speaking, different media, the difference in electrostatic potential which we have called @Z- @O is not, in a strict sense, a well-defined quantity. As was pointed out by Onsager, however,lS there is no logical bar to the adoption of some convention for relating @ I to @o. From this point of view eq 2 may be regarded as a postulate which, among other things, defines the convention on which the present discussion is based. (18) L. Onsager, discussion at the H. S. Harned Memorial Symposium held by the Division of Physical Chemistry at the 160th National Meeting of the American Chemical Society, Chicago, Ill., Sept 4,1970. (19) C. H. Chen and H. S. Frank, manuscript in preparation. (20) I. Oppenheim, J . Phys. Chem., 68, 2959 (1964).
