Richard P. Wendt Loyola University N e w Orleans, Louisiona 70118
Simplified Transport Electrolyte Solutions
The nonequilihrium thermodynamic theory for transport of electrolytes has been thoroughly developed over the last 30 years (1-8). Among recent papers, probably that of Miller (9) best summarizes the present state of the art: i t is now possible to use data for binary systems to predict, within experimental error, diffusion coefficients, transference numbers, and conductivities for temary systems of electrolytes over a wide range of concentrations. Onsager's reciprocal relation (I, 10) for transport coefficients, sometimes called the "fourth law" of thermodynamics has been tested and confirmed numerous times for temary systems (II),' thereby giving additional credence to the premises of nonequilibrium thermodynamics. However, & little of the- theory is availaMe in physical chemistry textbooks for undergraduates. Perhaps this omission is understandable in light of the complexity of Onsager's derivation and the intricate formalism of the general theory. Such material might best be given as part of a graduate course in thermodynamics, using original papers as study guides. Undergraduates should, however, he made aware of the unified uiew that nonequilihrium thermodynamics takes of transport processes. Such a view is quite straightforward and relatively simple for dilute ideal solutions. In fact, for such systems nonequilibrium thermodynamics becomes identical with the old Nemst-Hartley (13) approach where the Onsaeer relation is never used. But even this simplified theory iH not completely developed in modem texts. in such a wav that the student clearlv sees the relation between diffusibn, transference, andconductance. It is the purpose of this article to develop these relations usine" the~te&inoloev "" of noneouilibrium thermodvnamics to show how the appropriate transport coefficients can be estimated from readily available values of limiting equivalent conductivities for ions. Flux Equations for Ideal Systems
For isothermal solutions with transport only in one dimension (the x-direction), the flux density J, of ion i is assumed to be proportional to the gradient of its electrochemical potential, ap,/ax Ji =-L&l Jx (1) where
Inserting eqns. (3) and (2) in eqn. ( I ) we find the "ideal" flux equation for ions
-
ui is assumed to he independent of Ci; thus in the limit of 0, the flux vanishingly small concentrations, i.e., as Ci JIalso approaches zero, a sensible result forced by the introduction of ui. The rigorous nonequilibrium thermodynamic treatment would have the solvent-fiied flux of ion 1 in a multi-ion system of ions given by
where ( L d o is a "main-term" coefficient similar to Li in eqn. ( I ) , and (Llzjo, (L13j0. etc., are "interaction" or crosscoefficients describing the effect of gradients of chemical potential of the other ions on the flux of ion 1. According to the Onsager reciprocal relation (I), the cross-coefficients (Li,)o = (Lji)o, when i Z j , in an array of coefficients for a set of fluxes (Ji)o. Each of the gradients $&/ax should include a term for the dependence of the activity coefficient activity, yi, of ion i on its own concentration and that of the other ions. Thus eqn. (4) includes two ohvious assumptions: transport ideality (LIZ = 0, Lla = 0, etc., or in general Lij = 0 for if j) and a kind of thennodynamic ideality, dlnyi/aC, = 0, for all i and j. The assumption that all cross-coefficients are identically zero makes the application of Onsager's relation hetween crosscoefficients a trivial exercise. The ideal eqn. (4) also ignores the reference frame problem inherent in a rigorous discussion of nonequilibrium thermodynamics. This last assumption produces little error for dilute solutions since then all reference frames become nearly identical (2). I t should he stated however, that the coordinate x in eqn. (4) and hence the measurement of J,, is fixed in a lahoratory or cell of reference; also, there is no hulk flow of solution. Thus the simple unified transport theory is nonequilihrium thermodvnamics s t r i p ~ e dto its hare hones: onlv the formalism and language now distinguish it from a ~ ~ m s t Hartley theory. Conductance Ohm's law is assumed to hold for electrolyte solutions, in the form
is specified as the numher of moles of i which pass through a unit area (em2) per unit time, the unit area being perpendicular to the x-direction; L, is called the phenomenological coefficient; C, is the concentration in mole/mP, and 2, the valence, including the sign, of ion i; R is the gas constant. taken here to he 8.314 x lo7 e r ~ / mole-degree; T is the absolute temperature; F is the F ~ G day constant, 96,493 coulombs/equiv; is the local electrical potential in volts; and the factor lo7 appears so that the product z,F x lo7 6 has dimensions of ergs/mole, the same asp,. The ionic mobility u, is defined by
Ji
+
646
/
Journal 01 Chemical Educa!ion
where k is the specific conductance of the solution and I is the electrical current density (coulombs/cm2 s) defined as net flux of positive charge. The observed conductance, R (reciprocal of observed resistance), is related to k by
For the most recent test, see Ref. (12).
Here A is the effective area of either electrode used in the conductance cell, and 1 is the effective distance hetween electrodes. The ratio A/1 is usually called the cell constant. T o relieve the strong dependence of k on Ci, the equivalent conductance, A, is introduced
The sum in eqn. (8) either runs over all cations or all anions, and represents total equivalents of either positive or negative charge (the two must of course he equal to ensure electrical neutrality). Absolute values of the valence, lz,(, are used to assure a positive sign for A. We relate A to the u , in eqn. (4) by noting that
Transference
Transference numbers ti, measured either by the Hittorf method or by using concentration cells with transference, have been shown by Miller (14) to be equal. In general, ti is the ratio of electrical current carried by ion i to the total current (20)
t, = I J I
Contribution of diffusion to the fluxes Ji is assumed negligible, soZi and I are given by eqn. (11); then
orusingeqns. (16) and (17) the sum running over all ions, and I;= zcJ,F
(10)
where z d i is the flux of equivalents of charge (remember zi = equivimole of ion i); thus ziJi x F = coulomhs/cmz s. Next we use the soecial condition for conductance exyerim~nts,namply, no concentration gradients exist in the conductance cell. This can be accompliihed experimentally by making conductance measurements whh an AC bridge operating at around 1OOO Hz. The term involving aCl/ax in eqn. (4) thus vanishes for all ions and eqn. (10) can be written as I;= z i 2 u i ~ i ~ 2 i 0 7 a $ / a ~
the index j again running over all cations or all anions. For a binary system, where eqn. (18) holds, eqn. (22) reduces to the familiar expression
Diffusion Potential
Concentration gradients must certainly exist during diffusion, hut to a very good approximation there is no net flow of electrical current. That is, during diffusion
111)
or using eqn. (10) and dividing both sides by F
Then
Turning again to the ideal flux equation we find that insertion of eqn. (4) into eqn. (25) leads to an expression for the local electrical potential gradient where i runs over all ions and j only runs over all cations or all anions. By knowing ui and Ci for all ions we could calculate A from eqn. (13). However, the transport parameter most commonly found in the literature is Xi, the equivalent conductance of ion i; it is related to the ionic specific conductance, ki by A,
-
kill\zi&,)
(14)
where hi = -z~carnla~) =
*i'u
