March, 1962
ELECTROPHORETIC CONTRIBUTION TO ELECTROLYTE CONDUCTANCE
477
influenced appreciably, whereas the HCN yield rise to CNf: This CN+ ion can react in the gas was increased by about 30%. This suggests phase with methane to give CH,CN or, on neutralthat the ireaction between NIB carriers to give the ization, it gives the cyanogen radical which can final product does not involve the free radicals from radiation-damage but the reaction apparently occurs with the target gas or the materials on the surface of the reaction vessel. The increase in HCN yield in the presence of BrZprobably results from thermalization and charge-transfer processes. Although one cannot as a result of this study outline a complete mechanism by which the radioactive prloducts are formed, one can make some pertinent conclusions as to the nature of the mechanism. The fact that CH,, CHsOH, and C2H60H give HCN and CHsCN, and in addition, the ethanol gives C2H[&N, suggests that the cyanogen ion or radical is 1,heIYl3 carrier as far as the observed radiocyanides are concerned. The fact that neither ammonia nor amines are observed suggests that neither NH nor NHZ are the N13 carriers. The question then arises-how is the C-Pi bond formed? A possibility is that N13 ion can react to form ions such as CH4N+ when methane is the target gas, for example, and this ion on decomposition gives
abstract hydrogen mostly from materials deposited on the walls of the vessel to give HCN. It also can replace hydrogen on the wall-adsorbed molecules to form cyanides which remain on the walls of the vessel. Such activity has been found on the walls; however, the material has not been identified as yet. activity The fact that a significant amount of is present in one or more unidentified forms on the walls of the vessel is of some concern since it may have an important influence on the mechanisms of the reactions occurring. It also has prevented the realization of a satisfactory radioactivity balance and this point should be taken into consideration when comparisons are being made between the changes in yields of radioactive products as in Fig. 2. Such comparisons cannot be made in the present data; however, it is expected that this complication will be overcome in future work.
THE EFFECT OF TH:E EXPONENTIAL DISTRIBUTION FUNCTION ON THE E.LECTROPHORETIC CONTRIBUTION TO THE CONDUCTANCE OF 1-1 ELECTROLYTES' BY DAVIDJ. KARLAND JAMESL. DYE Kedzie Chemical Laboratory, Michigan State University, East Lansing, Michigan Received September 66, 1061
The use of the exponential ionic distribution function rather than the linear or quadratic expansion of it is demonstrated to have a large effect upon the electrophoretic contribution to conductance. Calculations by digital computer were made for a number of salts in dioxane-water and in ethanol-water mixtures, and it was found that much of the deviation from conductance theory usually attributed to ion-pair formation could arise from neglect of the higher terms in the distribution funct>ion.
Introduction The theoretical calculation of the conductance of electrolytes has attracted attention for many years and has been beset by many problems. Not the least of these has been the calculation of the degree of association of the ions to form ion-pairs. Before one can calculate the association constant, it is necessary i o know the proper conductance function for the ionic species. When association is marked, as for a weak acid or a salt in a medium of low dielectric constant, the Onsager limiting law can be used successfully. For cases of slight association, however, the constant is very sensitive to the theoretical conductance function used. A number of extensions of the limiting law have been propc~sed~-~ and Fuoss has used the Fuoss(1) This paper is based in part on a thesis presented by David J. Karl to the School for Advanced Graduate Studies of Michigan State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy, 1960. (2) E. Pitte, Proc. Roy. SOC.(London), ASl7, 43 (1953). (3) H. Falkeohagen, M. Leist, and G. Kelbg, Ann. Physik, (61 11, 61 (1953). (4) R. M. Fuoss and L. Onsager, Proc. Natl. Acad. Sei. U. S., 41, 274 1010 (1955): J . Phg8. Chem.. 61, 608 (1957); 62, 1339 (1958).
Onsager extension to calculate the association coiistant for ion-pair formation in mixed solvent systems.6s6 Since this treatment has been very successful in fitting conductance data and in correlating ion-size parameters, limiting mobilities, and association constants over a wide range of dielectric constants, the basic equations for the conductance of unassociated electrolytes appear to be sound, a t least for the case of large ions. This paper shows that terms which were dropped in the treatment of the electrophoretic effect are not small for those.cases requiring the introduction of an association constant. In fact, simply retaining these terms yields a surprisingly good fit of the data for reasonable and constant ion-size parameters in many cases, without requiring consideration of ionpair formation. It is suggested that the association constant calculated from conductance data is forced to include ionic interaction effects in addition to effects attributable to the formation of a distinct neutral species. The Distribution Function.-The terminology R. M. Fuoss, J . A m . Chem. Soc., 79, 3301 (1957). (6) R. M. Fuoss and C. A. Kraus, W d . , 79, 3304 (1957). (5)
DAVIDJ. KARLAXD JAMBS1). DYE
478
and symbols used in %hispaper are those of Harned and Owen.7 The Oiisager limiting law used t,he distribution function (for the unperturbed dist ribu tion)
ioJi = TLJL,
(1 - el ~ O J / k T )
(1)
obtained by expanding the exponential distribution function
so.. -
w
~
exp[j e , $0,/k7’1
(2)
in a power series and neglecting higher order terms. Fuoss and Onsager4 retained another term in the expansion of (2) as well as other terms from the equations of motion of order c to obtain an expression for the relaxation field. Their distribution function is
In this equation, the second term depends upon and the third term upon c. All terms of order Since the third ca/e and higher were discarded. term in eq. 3 cancels out of the electrophoretic integral, their results for this effect are the same as were obtained previously for a non-zero ion-size parameter. The functional form of $01 \vas obtained from the solution of the linearized Poisson-Boltzmann equation. Pitts2 used the Gronwall-La Mer-Sandved expression for the potential8 and the distribution function (3) in his treatment of both the relaxation and the electrophoretic eflects. Pitts notes that the approximations made in dropping higher terms limits the applicability of the equations in media of low dielectric constant; for example, to concentrations below about 5 X ill in ethanol (0 = 24.3). Falkenhagen, Leist, and Kelbg3 used a modification of (1) introduced by Eigen and Wickeg to correct the Boltzmann distribution for the fact that two atmosphere ions cannot simultaneously occupy the same region of space. It has been pointed out frequentlyl0-l2 that the exponential distribution function (2) is incompatible with the Poisson equation. On the other hand the linear function (1) is certainly incorrect at small values of r because it ignores the exponential dependence of the distribution function upon the interaction energy. Onsager13 has shown that the statistically correct expression for the time-average ionic distribution function is f,, = nln,exp
[%I
(4)
in which Wj, is the time-average interaction energy. At close distances of approach the most important contribution to Wj, is the pair-wise coulombic interaction of the two ions, and screening effects are (7) H. S . Harned and B. 13 Owen, “The Physical Chemistry of Electrolytic Solutions,” Third Edition, Reinhold Publishing Corpoiation, New York, PI;. Y., 1958. (8) T. H. Gronwall, V. K. La Mer, and K. Sandved, Piigszk. Z., 29, 358 (1928) (9) M. Eigen and E. Wicke, I~aturvJzssensehaflen,88, 453 (1951). (10) R. H. Fowler, “Statistical Mechanics,” Cambridge University Press, New York, N. Y., 1929, Chap. XIII. (11) R. H. Fowler and E A. Guggenheim, “Statistical Thermodynamics,” Cambridge University Press, New York, S. Y., 1952, Chap. IX. (12) R. A. Robinson and R. H. Stokes, ”Electrolqtic Solutions,” Butterworths Scientific Publications, London, 1955, pp. 76, 128, 142. (13) L. Onsager, J Chem. Phys., 2, 599 (1934).
Volt 66
of secondary importance. l 4 The distribution function of Bjerruni’j for r < g = [eiej]/2DkT includes only this pair-wise interaction At large values of r , e, rl.O,/lcT 4a, the linear function and the exponential function give nearly the same charge distribution, and that most of the atmosphere is outside of this distance. The diagram also shows that the form of the charge distribution curve obtained using the potential function of Groiiwall, La Mer, and Sandved and the exponential distribution function is similar to that obtained using the Debye-Hiickel potential function. It is to be noted that because of a cancellation of terms, the distribution function ( 3 ) gives the same charge density as the linear function (1). Recently, Fuoss and Onsager16 have proposed a new distribution function which includes the pairwise interactions at close distances and merges into an extension of the Debye function at the Bjerrum distance q. While this approach still involves the linear-superposition approximation, and the somewhat arbitrary choice of q as the matching distance, it has the advantage of being normalized to unit charge in the atmosphere. It would be of interest to use this new distribution function in the electrophoretic integral. We have chosen to examine the effect of the higher terms j n the distributioii function on the electrophoretic contribution to conductance by calculating the equivalent conductance for various salts over a range of dielectric constants and T’1SCOS’ ities using the exponential distribution function (2). It is not our intention to claim validity of the distribution function, but rather to show that the effects are appreciable and in such a direction that the true association constants are probably much smaller than generally has been thought. The assumptions inherent in the model and treatment used in this paper are: 1. The DebyeHiickel potential function is used; 2 . The treatment of the relaxation effect derived by Fuoss and Onsager4 is used. This involx-es the distribution function ( 3 ) and drops all terms in the relaxation field of order c8/a and larger; 3. The exponential distribution function (2) is used in deriving the expression for the electrophoretic effect17; 4. Stoke’s law for the movement of a sphere through a viscous medium is used in deriving the expression for the electrophoretic effect, although this is not expected to be valid at distances less than about iA.18 This assumption is expected to yield values (14) J. G. Kirkwood, %bzd.,2, 767 (1934). (15) N. Bjerrum, Kgl. Danske V d e n s k a b . Selslcab., Jilat.-jys. Medd , 7, No. 9 (1926). (16) R. M. Fuoss and L. Onsager, PTOC. Xatl. Acad. S a . U. 8. 47, 818 (1961). (17) J. L. Dye and F. H. Spedding, J . Am. Chem. SOC.,76, 888 ( 1984), (18) Reference 12 p. 118.
479
ELECTROPHORETIC CONTRIBUTION TO ELECTROLYTE CONDUCTANCE
March, 1962
of & which are too small; 5. The ions are considered to be hard, non-polarizable spheres, not subject to association effects. Breakdown of this assumption would tend to yield values of d which are too small, I n comparing the calculations with experiment, two parameters. B and Ao mere adjusted t o minimize the deviations. When fitting the data to include ion-pairing, Fuoss and c o - w o r k e r ~utilize ~~~~~~ three adjustable parameters, A*, &, and the association constant A. I n the case of large ions, both methods involve in addition a hydrodynamic radius parameter, R. Method of Calculation. -The calculations were programmed for the NIISTIC digital electronic computer in two parts. The first was the calculation of the electrophoretic Contribution to conductance, AA,, using A h , = AX+
+ AX.-
I
--.BJERRUM
1
-
I I
FUNCTION
EXPONENTIAL FUNCTION
I
LINEAR FUNCTION
I 1
---
I
I I I I
EXPONENTIAL FUNCTION WITH GRONWALL- LAMER-SANDVEO POTENTIAL
I
0125 5.298
I
i
x o =0100 0=40XlO‘~cu
(5)
in which
For symmetrical electrolytes, AA, = 2Ah+. The program also can be used for unsymmetrical electrolytes, in which case Ah- is calculated separately. I n equation 6, P = Ke*/DkT(lSx) and x = K U . Integration was performed numerically with the aid of the Newton-Cotes quadrature formula. 2o For small values of p, the integrand is very large and changes very rapidly with p. I n order to assure proper integration of the function, the integral was re-evaluated with successively smaller incre- /(Ah+),[ < E = 2 X ments until This gives a precision to the calculation of Ah, of d= 0.004 conductance units. Input for this program included D T , 7, &, and fi. The equivalent conductance then was calculated using the expression
I
2
4
3 H P
Fig. 1.-Charge distribution in the ionic atmosphere according to various distribution functions.
1.7
1.6
1.5
c“
-,p
1.4
4in which the relaxation field term, A X / X , the kinetic term, A.P/X, and the Einstein viscosity term, 5 / 2 Pc, are obtained from the equations of Fuoss and O n ~ a g e r ~as .’~
-
X
= aicl/z (1
- Al
+ A*) + pc’/*A3’/Ao
(8)
AP - ~ ~ a -~ 1) ( b _ X 126
P, 1.3
