PHYSICAL CHEMISTRY

E. A. STRELTSOVA

KINETIC EQUATIONS IN THE THEORY OF ELECTROLYTES

(Presented by Academician N. N. Bogolyubov, 7 V 1957)

As is known, in the derivation of Onsager’s kinetic equations ((^{1})) an essential assumption is that each ion is surrounded by a certain “ionic atmosphere” of ions of the opposite sign, which accompanies the ion in its motion. It is clear, however, that such a “model representation” cannot be regarded as fully satisfactory from the physical point of view and does not make it possible to obtain more exact equations.

In the present work the general problem is considered of finding kinetic equations for the distribution functions of the probabilities of position of particles situated in a liquid and interacting with one another by means of a given potential. The interaction of the particles with the liquid is represented by a certain stochastic process of the diffusion type.

The problem posed is solved by the statistical method due to N. N. Bogolyubov ((^{2})), which consists in introducing, instead of the Gibbs distribution function of all (N) particles of the system, a set of distribution functions for groups of (s) particles, with an arbitrary distribution of the remaining set of (N-s) particles.

The general scheme developed is applied to the case of a strong electrolyte. As the equations of the first approximation, the known Onsager equations are obtained.

Let us have a system consisting of particles of (m \geqslant 2) different classes. We denote the number of particles of class (a) by (N_a), where

[
\sum_{1\leqslant a\leqslant m} N_a = N .
]

The particles move in a liquid of volume (V) in the presence of an external field with potential (U_a(t,q_a)), (q{q^\alpha}), and interact with one another pairwise by means of the potential (\Phi_{ab}(|q_a-q_b|)). In deriving the equations we shall proceed from the idea that, owing to the interaction of the particles with the chaotically moving molecules of the liquid, the motion of the particles themselves proceeds according to the scheme of a Markov stochastic process and can be described by the diffusion equation

[
\frac{\partial D_N}{\partial t}
=
\sum_{\substack{1\leqslant \alpha \leqslant 3\ 1\leqslant i\leqslant N_a\ 1\leqslant a\leqslant m}}
\frac{\theta}{\lambda_a}
\frac{\partial}{\partial q_i^\alpha}
\left{
\frac{\partial D_N}{\partial q_i^\alpha}
+
\frac{1}{\theta}
\frac{\partial U_N}{\partial q_i^\alpha}
D_N
\right}.
\tag{1}
]

Here (D_N = D_N(t,q_1\ldots q_N)) denotes the probability distribution function of the positions of all (N) particles of the system at the time (t), (\lambda_a) is the resistance coefficient in the expression for the Stokes braking force,

[
\theta = kT
]

is the thermal function, and (U_N) is the potential energy of the system

[
U_N
=
\sum{}' \Phi_{ab}(|q_i-q_j|)
+
\sum_{\substack{1\leqslant i\leqslant N_a\ 1\leqslant a\leqslant m}}
U_a(t,q_i).
\tag{2}
]

The symbol (\sum') denotes summation over all distinct pairs of particles.

Introducing the distribution functions (F_{a_1\ldots a_s}(t,q_1\ldots q_s)) and defining them in the same way as was done in [2], we derive equations for these functions from equation (1), which, after the natural limiting transition (N\to\infty,\ V\to\infty,\ N/V=n), take the form:

[
\frac{\partial F_{a_1\ldots a_s}(t,q_1\ldots q_s)}{\partial t}
=
\sum_{\substack{1\leq \alpha \leq 3\ 1\leq j\leq s}}
\frac{\theta}{\lambda_{a_j}}
\frac{\partial}{\partial q_j^\alpha}
\left{
\frac{\partial F_{a_1\ldots a_s}}{\partial q_j^\alpha}
+
\frac{1}{\theta}
\frac{\partial U_{a_1\ldots a_s}}{\partial q_j^\alpha}
F_{a_1\ldots a_s}
\right}
+
]

[
+
\sum_{\substack{1\leq \alpha \leq 3\ 1\leq j\leq s\ 1\leq b'\leq m}}
\frac{n_{b'}}{\lambda_{a_j}}
\frac{\partial}{\partial q_j^\alpha}
\int
\frac{\partial \Phi_{a_j b'}\bigl(|q_j-q'|\bigr)}{\partial q_j^\alpha}
\cdot
F_{a_1\ldots a_s b'}(t,q_1\ldots q_s q')\,dq',
\tag{3}
]

[
s=1,2,\ldots,
]

where (U_{a_1\ldots a_s}) is the potential energy of a system of (s) isolated particles.

To the differential equations obtained it is necessary to add the conditions of decay of correlations. These conditions mean that if a given aggregate of (N) particles is divided into two groups of (s) and (\sigma) particles, respectively, then the distribution function for the entire aggregate is represented as the product of the distribution functions of each group separately when these groups are moved infinitely far apart from one another,

[
F_{s+\sigma}=F_s F_\sigma .
]

We shall now regard the particles under consideration as electrolyte ions that are in an electric field with strength depending only on time, (E(t)), so that

[
e_a E^\alpha(t)=-\frac{\partial U_a(t,q)}{\partial q^\alpha},
\tag{4}
]

where (e_a) is the charge of an ion of the (a)-th class; the total charge of the ions is equal to zero and they interact pairwise by means of central forces characterized by the Coulomb potential

[
\Phi_{ab}\bigl(|q_a-q_b|\bigr)=\frac{e_a e_b}{K|q_a-q_b|},
\tag{5}
]

where (K) is the dielectric constant of the solution. Since in what follows we shall confine ourselves to obtaining equations of the first approximation, we shall not explicitly take into account the potential of short-range forces.

We consider translationally invariant solutions

[
F_{a_1\ldots a_s}(t,q_1+x\ldots q_s+x)=F_{a_1\ldots a_s}(t,q_1\ldots q_s);
\qquad
F_a(t,q)=1.
]

In this case equation (3) for (s=1) is satisfied identically, since

[
\int
\frac{\partial \Phi_{ab}\bigl(|q_1-q_2|\bigr)}{\partial q_1^\alpha}
\cdot F_{ab}(t,q_1 q_2)\,dq_2
]

does not depend on (q_1).

If now to equation (3) one applies the same device as in [4]—introducing dimensionless coordinates (\xi), (q^\alpha=r_d \xi^\alpha), where (r_d) is the Debye radius—then it is not difficult to show that the expression (\partial \Phi/\partial q^\alpha), entering as a term in the derivative of the potential energy in the second term of the right-hand side, is proportional to the dimensionless volume (\varepsilon=1/nr_d^3), which may be regarded as a small parameter if one restricts the study to solutions of sufficiently low concentrations. The remaining terms of equation (3) are of order unity relative to (\varepsilon).

In order to obtain equations of the first approximation, in the exact equations we shall discard those terms that are proportional to

small parameter (\varepsilon). Then we obtain the approximate equations:

[
\frac{\partial F_{a_1\ldots a_s}(t,q_1\ldots q_s)}{\partial t}
=
\sum_{\substack{1\leq \alpha \leq 3\ 1\leq j\leq s}}
\frac{\theta}{\lambda_{a_j}}\,
\frac{\partial}{\partial q_j^\alpha}
\left{
\frac{\partial F_{a_1\ldots a_s}}{\partial q_j^\alpha}
-\frac{1}{\theta} e_{a_j}E^\alpha F_{a_1\ldots a_s}
\right}
+
]

[
+
\sum_{\substack{1\leq \alpha \leq 3\ 1\leq j\leq s\ 1\leq b'\leq m}}
\frac{n_{b'}}{\lambda_{a_j}}\,
\frac{\partial}{\partial q_j^\alpha}
\int
\frac{\partial \Phi_{a_j b'}\bigl(|q_j-q'|\bigr)}{\partial q_j^\alpha}
F_{a_1\ldots a_s b'}(t,q_1\ldots q_s q')\,dq',
\tag{6}
]

[
s=1,2,\ldots
]

It is easy to show that all these equations can be satisfied by setting

[
F_{a_1\ldots a_s}(t,q_1\ldots q_s)
=
1+
\sum_{1\leq i\ne j\leq s}
g_{a_i a_j}(t,q_i q_j).
\tag{7}
]

Here (g_{ab}(t,q_a q_b)) are binary functions satisfying the symmetry conditions

[
g_{ab}(t,q_a q_b)=g_{ba}(t q_b q_a)
\tag{8}
]

and the correlations (g_{ab}(t,q_a-q_b)\to 0) as (|q_a-q_b|\to\infty). In view of translational invariance, the solutions of the functions (g_{ab}(t,q_a q_b)), in essence, depend not on the coordinates themselves but only on their differences.

Substituting (7) and (5) into (6), we have:

[
\frac{\partial g_{a_1a_2}(t,q_1 q_2)}{\partial t}
=
\sum_{1\leq \alpha\leq 3}
\frac{\theta}{\lambda_{a_1}}\,
\frac{\partial}{\partial q_1^\alpha}
\left{
\frac{\partial g_{a_1a_2}}{\partial q_1^\alpha}
-\frac{1}{\theta}e_{a_1}E^\alpha g_{a_1a_2}
\right}
+
]

[
+
\sum_{1\leq \alpha\leq 3}
\frac{\theta}{\lambda_{a_2}}\,
\frac{\partial}{\partial q_2^\alpha}
\left{
\frac{\partial g_{a_1a_2}}{\partial q_2^\alpha}
-\frac{1}{\theta}e_{a_2}E^\alpha g_{a_1a_2}
\right}
+
]

[
+
\sum_{\substack{1\leq \alpha\leq 3\ 1\leq b'\leq m}}
\frac{n_{b'}}{\lambda_{a_1}}\,
\frac{\partial}{\partial q_1^\alpha}
\int
\frac{\partial}{\partial q_1^\alpha}
\left(
\frac{e_{a_1}e_{b'}}{K|q_1-q'|}
\right)
g_{a_2b'}(t,q_2 q')\,dq'
+
]

[
+
\sum_{\substack{1\leq \alpha\leq 3\ 1\leq b'\leq m}}
\frac{n_{b'}}{\lambda_{a_2}}\,
\frac{\partial}{\partial q_2^\alpha}
\int
\frac{\partial}{\partial q_2^\alpha}
\left(
\frac{e_{a_2}e_{b'}}{K|q_2-q'|}
\right)
g_{a_1b'}(t,q_1 q')\,dq',
\tag{9}
]

whence:

[
\frac{\partial g_{a_1a_2}}{\partial t}
=
\frac{\theta}{\lambda_{a_1}}\Delta_{q_1} g_{a_1a_2}
+
\frac{\theta}{\lambda_{a_2}}\Delta_{q_2} g_{a_1a_2}
-
\frac{e_{a_1}}{\lambda_{a_1}}E\,\operatorname{grad}{q_1} g
-
]

[

\frac{e_{a_2}}{\lambda_{a_2}}E\,\operatorname{grad}{q_2} g

\frac{4\pi}{K}
\sum_{1\leq b'\leq m}
n_{b'}e_{b'}
\left(
\frac{e_{a_1}}{\lambda_{a_1}}g_{a_2b'}
+
\frac{e_{a_2}}{\lambda_{a_2}}g_{a_1b'}
\right).
\tag{10}
]

The equation obtained coincides with the well-known Onsager equation. To verify this, it suffices, instead of the functions (g_{ab}), to introduce the functions (f_{ab}), as was done in work ((^3)). They are related to our (g_{ab}) by the simple relation

[
f_{ab}=n_a n_b g_{ab}.
]

The method considered can also be generalized to the derivation of equations of higher approximations, similarly to how this was done for the case of statistical equilibrium in work ((^4)).

Kiev Technological Institute
of Light Industry

Received
27 IV 1957

CITED LITERATURE

(^1) L. Onsager, Phys. Zs., 28, 277 (1927).
(^2) N. N. Bogolyubov, Problems of Dynamical Theory in Statistical Physics, Moscow, 1946.
(^3) H. Falkenhagen, Elektrolyte, Leipzig, 1953.
(^4) E. A. Streltsova, ZhETF, 26, no. 2, 173 (1954).