THE THEORY OF LIMITING DIFFUSION-MIGRATION
CURRENTS IN PARTIALLY DISSOCIATED ELECTROLYTES
Yu.I.Kharkats, A.V.Sokirko
The A.N.Frumkin Institute of Electrochemistry,
Academy of Sciences of the USSR,
117071, Moscow V-71, Leninsky Prospekt, 31, USSR
The processes of diffusion-migration transport of ions in
solutions of completely dissociated electrolytes have been well
studied [1]. Of interest is to analyze the peculiarities of these
processes proceeding under the conditions, where the electrolyte
is only partially dissociated according to the reaction equation
k
z + |z |- 1
1 2
n A + n B "==' A B (1)
1 2 n n
k 1 2
2
z + |z |-
1 2
Here A and B are cation and anion, z and z are charge
1 2
numbers, A B is neutral molecule, n and n are stoichiometric
n n 1 2
1 2
coefficients, k and k are association and dissociation rate
1 2
constants. The stoichiometric coefficients n and n coinside with
1 2
|z | and z in the case when |z | and z are mutually simple
2 1 2 1
numbers ( do not have any common divisors ).
The diffusion transport of ions in systems with chemical
equilibria in the absence of electromigration was analyzed in [2-4].
The dependence of a limiting diffusion-migration current of
1
cation discharge on the equilibrium constant of partially disso-
ciated electrolyte was studied in [5] under an assumption, that
dissociation and recombination rate constants are rather high, so
that in the whole diffusion layer concentrations of cations C ,
1
anions C and non-dissociated neutral molecules C are related by
2 3
the equilibrium condition
n n
1 2
b C 3 C = C (3)
1 2 3
wrere b = k /k is equilibrium constant. One of the most
1 2
interesting results of [5] is the fact, that for sufficienly high
values of non-dissociated substance diffusion coefficient the
limiting current can be several times higher that the diffusion -
migration current in fully dissociated electolyte.
This paper presents the calculation of a limiting current of
cation discharge in partially dissociated electrolyte in a more
rigorous formulation without a supposition of equilibrium condi-
tion (2). Presented analysis is based on Nernst diffusion layer
model [1], which is widely used in electrochemical macrokinetics
and accounts in implicit form the convective transfer of ions. De-
veloped theory can also be applied to the systems with membrane
covered electrodes.
1. Statement of the problem and the general solution.
z +
1
Let us consider discharge of cations A reducing in a
steady state conditions to neutral species which does not
2
interact with any other substances in the solution. We shall also
z -
2
suppose that anions B are electrochemically inert and that
direct discharge of molecules A B is impossible in all the
n n
1 2
potential region.
The system of electrodiffusion equations describing distribu-
tion of component concentrations C , C , C and electric potenci-
1 2 3
al in the diffusion layer close to the electrode can be written as
dC dCd dJ i
1 3
----- ----- ---- -----
D + n D + z D C = , (3)
1 1 3 1 1 1
dx dx dx z F
1
dC dCd dJ
2 3
----- ----- ----
D + n D + z D C = 0, (4)
2 2 3 2 2 2
dx dx dx
2
d C
1 n n
----- 1 2
D = k ( C - b C 3 C ), (5)
3 2 2 3 1 2
dx
z C = |z | C . (6)
1 1 2 2
Here D , D , D are diffusion coefficients of corresponding
1 2 3
components, J=Ff/RT is the dimensionless potential, i is the ca-
tion discharge current density, x is coordinate, the remaining
designations are generally accepted.
The first and the third terms in equations (3) and (4)
describe diffusion and migration fluxes of cations and anions.
The second terms in these equations correspond to the transfer of
3
z + |z |-
1 2
substaces A and B due to diffusion of A B . Equation
n n
1 2
(5) descri- bes diffusion transport of A B molecules with
n n
1 2
account of reac- tion (1). Finally the equation (6) presents the
condition of local electroneutrality in the diffusiuon layer.
The system of equations (3)-(5) should be supplemented by
the boundary coudition
dC |
3 |
----- |
| = 0, (7)
|
dx | x=0
corresponding to the electrochemical inertness of molecules A B
n n
1 2
and the condition
C (0) = 0, (8)
1
corresponding to the limiting current of cation discharge. At the
diffusion layer boundary x = L concentrations C , C , C are equal
1 2 3
to their equilibrium values:
Ü
C (L) = C (b), i = 1,2,3. (9)
i i
Ü Ü Ü
The values of equilibrium concentration C , C and C can be
1 2 3
Ü
related with the total concenration C of substance A B in a
n n
1 2
solution and with the equilibrium constant b by equations
Ü n Ü n Ü
1 2
b (C ) 3 (C ) = C , (10)
1 2 3
Ü Ü
z C = |z | C , (11)
1 1 2 2
Ü Ü Ü
C + n C = n C . (12)
1 1 3 1
Combining equations (10)-(12), one obtains the equation that
4
Ü
determines the C (b) dependence
1
Ü Ü m n Ü
2
C + n b (C ) ( z /|z | ) = n C , (13)
1 1 1 1 2 1
where m = n + n is the formal order of the recombination
1 2
reaction. Substituting the solution of equation (13) into (11)
and (10), one determines the equilibrium concentrations appeared
in (9).
The calculations in [5], based on the solution of a system of
equations (2)-(4), (6) with boundary conditions (7)-(9), corres-
pond to the limiting case, when the dimensionless parameter
2
d = D /k L tends to zero, so that equation (5) should be
3 2
replaced by (2) for any 0 < x < L.
It follows from the local electroneutrality condition (6) and
from equations (3), (4), that
dC z1 * D n D n * dCd i
1 3 1 3 2 3
----- | ----| | ---- ---- | ----- -----
1 + + + = , (14)
| | | |
dx 7 |z |8 7 D D 8 dx z FD
2 1 2 1 1
After passing to dimensionless variables:
Ü
x = x / L, c = C / C , (15)
i i
equations (5), (14) and boundary conditions (7) - (9) are writen
in the form:
dc dc
1 3
----- -----
g + = j, (16)
dx dx
2
d c
3 m
------ _
d = c - b3c , (17)
2 3 1
dx
Ü Ü Ü Ü
c (1) = C / C = k, c (1) = C / C = l, (18)
1 1 3 3
5
dc |
3 |
----- |
c (0) = 0, | = 0, (19)
1 |
dx | x=0
wrehe the following designations for combinations of parameters
are introduced:
z1 * D n D n *
/ 3 1 3 2
| ---- | | ---- --- |
g = 1 + / + ,
| | | |
7 |z | 8/ 7 D D 8
2 1 2
D n D n *
iL / 3 1 3 2
-------- | ------ ------|
g = / + ,
0 | |
z FD C / 7 D D 8
1 1 1 2
m-1 n
_ Ü 2
b = b (C ) ( z /|z | ) .
1 2
Integrating (16), one obtains
g c + c = jx + b. (20)
1 3
Using conditions (19), one concludes that c (0) = b, and using
3
conditions (18), one has
Ü
j + b = j , (21)
where
Ü
j = gk + l.
Ü
Quantity j is the expression for a dimensionless current in the
case d = 0, i.e. under the conditions of equilibrium of the dis-
sociation-recombination reaction. Indeed, letting d = 0 in (17)
Ü
and (19), one obtains j = j . Quantity b can be treated in two
ways simultaneously: as a dimensionless concentration of non-
dissociated substance near the electrode and as a correction to
Ü
-
dimensionless current j for small d. As it was shown above, b ~0
6
-
at d ~ 0.
The system of equations (16) - (17) with boundary conditions
(18)-(19) is nonlinear ( c enters (17) in power of m > 2 ) and,
1
hence, it does not have any general analytical solution. Below
will be given analytical solutions for the limiting cases of re-
combination reaction d = 0, as well as for the cases of high and
low recombination-dissociation reaction rates ( d , 1 and d . 1).
The results of numerical solution of a system will be given for
the intermediate region of d values of the order of unity.
2. The case of the recombination-dissociation reaction equilibrium
Ü
As mentioned above, quantity j is the expression for dimen-
sionless limiting current in case of the dissociation-recombina-
tion reaction equilibrium d = 0, that was analyzed in paper [5].
The corresponding limiting current in dimensionless units can be
written as:
Ü
z FD z D n D n Cd
1 1 # 1 * Ü 3 1 3 2 * Ü 1 *$
------- ---- ---- ---- ----
i = ? |1 + |C + | + || C - |?.
1
L 3 7 |z | 8 7 D D 87 n 84 (22)
2 1 2 1
_ Ü _
In the expression of (22) current i depends on b via the C (b)
1
dependence only.
_
The i (b) dependence, determined by formulae (13), (22), is
_
shown in Fig.1. For a high dissociation degree, when b , 1, the
Ü
dimensionless concentration of cations is c ~ |z |, and the
1 2
current tends to the value
7
Ü
i = ( z + |z | ) z F D C / L,
1 2 1 1
that coincides with the value of i in a binary solution of fully
_
dissociated electrolyte. For a low dissociation degree, when b.1,
Ü
the dimensionless concentration of electroactive ions is c , 1,
1
and the limiting current tends to the value
Ü
*
z F D D C n n
--1---1 3--- |- 1 2 |
i = + |.
___ | ____ ____
L 7 D D 8
1 2
One should mention some important fact. Namely, though the
concentration of dischaging cations in a solution decreses with
lowering the dissociation degree, the limiting current tends to
the asymptotic constant value, that depends on the ratio of
diffusion coefficient D , D , D of components and parameters n
1 2 3 1
_ -
and n . In this case the value of i for b ~ 7 can be either hig-
2
_ -
her, or lower than its value for b ~ 0.
In the simpliest case, when all diffusion coefficients are
equal ( D = D = D ), dimensionless limiting current is equal
1 2 3
_
to 1 and does not depend on b.
_
A physical explanation for such a behavior of i(b) is the
fact, that electroactive cations are transferred in a diffusion
layer both via their diffusion and migration ( the distribution
_
of concentrations c (x) for some values of b is shown in Fig. 2 )
1
and due to diffusion transfer to the electrode with subsequent
dissociation of neutral molecules A B . ( The distribution of
n n
1 2
concentration c (x) is shown in Fig.3.) The rate of the latter
3
8
mechanism is proportional to the diffusion coefficient D of
3
neutral molecules, and the corresponding contribution into the
limiting current is given by the second term in formula (22).
In case of D = D = D the decrease of the contribution of
1 2 3
z +
1
diffusion-migration transfer of A cations, caused by decreas-
Ü _
ing c with growing b, is fully compensated by diffusion supply
1
of dissociating neutral A B molecules to the electrode, which
n n
1 2
_
just provides the independence of limiting current on b.
3. Analytical solution for the case of high reaction rates d , 1.
Since equation (17) contains a small parameter at a higher
order derivative and is nonlinear, we shall replace the dependent
and independent variables in this equation in such a manner, that
all terms of equation (17) be of the same order of magnitude
#_+ #_+
[6-8]. Let y = x/ed . As follows from (20), for x ed the sum
#_+
of concentrations gc + c is also of the order of ed . We shall
1 3
seek the solution of equation (17) in the form of
m/2 1/2
c = d W(y), c = d U(y), (23)
3 1
where W(y) and U(y) are functions of the order of unity. It fol-
m/2
lows from conditions (19) and (20), that b d . Neglecting in
m/2
relation (20) the terms of the order of d one obtains the
approximate expression for function U:
U(y) = j y / g. (24)
Substituting (23) and (24) into (17), one obtains the equation
9
for function W :
2 m
d W j y *
----- _ -----
= W - b | | (25)
2
dy 7 g 8
with boundary conditions
dW | -m/2
----
| =0, W (0) = b d . (26)
dy |y=0
The general solution W(y) of homogeneous equation (25) is
W = s exp (-y) + s exp (y). (27)
1 2
The particular solution W of a non-homogeneous equation can be
found by the method of constants variation. Summing up the gene-
ral solution of homogeneous equation with the particular soluti-
on of a non-homogeneous equation, one obtains the general soluti-
on of a non-homogeneous equation, that satisfies the W'(0) = 0
condition, in the form
7 y
# y -y m -y y m $
_ i i
W(y) = (b/2) (j/g) m ? e dt e t + e dt e t + m! ?, (28)
j j
3 4
y 0
where m! is the factorial function. Using (28) and the second co-
m
_ #_+
ndition in (26), one finds the value b = b (jed /g) m!. Substi-
tuting the latter one into (21), one obtains the equation for j:
Ü m/2 m
_
j = j - d b ( j/g ) m! (29)
Ü
One may neglect the small difference between j and j in the
right-hand side of (28) and write down the approximate exression
for dimensionless current in the form:
Ü m/2 Ü m
_
j = j - d b ( j /g ) m! (30)
10
#_+
Thus, for low values of parameter ed , i.e. for high dissoci-
ation rates the limiting diffusion-migration current decreases
(n +n )
1 2
proportionally to d .
4. The case of low reaction rates d . 1
In this case the solution can be sought in the form of expan-
-1
sion in powers of small parameter d :
-1
c = X + d Y, (31)
1
where X, Y are functions of the order of utity. Substituting this
expansion into (17), taking into account (16) and equating the
terms at d, one gets
2
d X
-----
= 0.
2 (32)
dx
After satisfying the boundary conditions (18), (19) one obtains
from the last equation the major part of solution for c :
1
X = k x. (33)
To find Y, we equate the terms not containing d and, substitu-
ting (33), we obtain
2
d Y m
---- _
g = b (kx) - ( jx + k +l - j - gkx ). (34)
dx
Relations (20) and (21) were also taken into account in deriving
(34). Function Y satisfies homogeneous boundary conditions:
Y (0) = 0, Y (1) = 0. (35)
Integrating (34) with using (35), one obtains:
11
m m+2 3 2 $
# _
1 b k (x - x) (gk-j)(x -x) (j-k-l)(x -x) ?
-- | ------------- - ------------ -------------
Y = ? + ?.
(36)
g ? ( m+2 )( m+1 ) 6 2 |
3 4
This expression and the j= dc /dx| condition give rise to the
1 |x=0
expression for a flux in case of low dissociation rates ( d . 1):
m
_
1 # 1 g * 1 b k $
--- --- - --- --- -------------
j = k + ? | |k - + ?. (37)
d g 3 7 2 6 8 2 (m+2) (m+1) 4
_
The j(b) dependence, given by (37), is mainly determined by
the first term and represents a monotoneously decreasing function.
5. The numerical solution.
The system of equations (16)-(19) has also been solved nume-
rically for some intermediate d values by using the Runge-Kutta
method and the optimization procedure of searching for the j
value satisfying the boundary conditions.
_
Fig. 4 shows the j (lg b) depedences, calculated by numerical
solution of the problem for some values of parameter d. As fol-
lows from numerical calculations and from the results of
approximate analytical solution of the problem, the limitig cur-
rent of reduction of cations decreaes as parameter d grows.
6. Conclusion.
The above investigation shows that limiting current in a
partially dissociated binary electrolyte depends, first, on the
electrolyte dissociation rate constant and, second, on the equi-
librium constant. Analytical expressions (30) and (37) for limi-
12
ting current, obtained for the cases of high and low ( d , 1,
d . 1 ) electrolyte dissociation rate constans, allow to determine
equilibrium constatn b from experimental values of i and k . In
2
case of intermediate d values constant b can be determined by
_
using the family of curves j (lg b) obtained by numerical
-
solution of the problem. In the d ~ 0 limit the calculated j (b)
dependence transforms into formula for j obtained in [5]. For low
values of dissociation rate constant ( d.1 ) the value of limi-
ting diffusion-migration current is mainly determined by the
value of equilibrium concentration of electroactive cations in
the solution.
Ü
Note in conclusion, that by changing the concentration C in
_
the solution, one can vary the value of parametr b, which is
Ü m-1
proportional to (C ) , whereas the value of parametr d does not
Ü
depend on C . This allows, in principle, to find the dissociation
rate constant k and the reverse recombinanation reaction rate
1
constant k from the comparison of the experimental dependence of
2
Ü _
limiting currrent on concentration C and calculated j (lnb)
curves for different values of d.
13
References.
1. Newman J. Electrochemical systems. New Jersey; Prentice-Hall,
1973.
"
2. Fetter K. Elektrochemische Kinetik. Berlin-Gottingen-Heidel-
berg: Springer-Verlag, 1961.
3. Galvell J.R. J.Electrochem. Soc., 1976, V.123, P.464.
4. Galvell J.R. Corros. Sci., 1981, V.21, P.551.
5. Kharkats Yu.I. Elektrokhimiya, 1988, V.24, P.539.
6. Nayfeh A.H. Intoduction to Perturbation Techniques.-
New York: Wiley Interscience, 1981.
7. Schlichting H. Grenzschicht-Theorie. Karlsruhe: Verlaq
G.Braun, 1965.
8. Vorotyntsev M.A. Elektrokhimiya, 1988, V.24, P.1239.
14
Figure Captions
_
Fig.1. The i(b) dependence for z = 2, |z | = 1 and for the
1 2
values of parameter D z /D + D |z |/D :
3 1 2 3 2 1
1 - 1; 2 - 2; 3 - 3; 4 - 4; 5 - 5.
Fig.2. The c (x) dependence for z = 2, |z | = 1 and D = D = D
1 1 2 1 2 3
_
and for values of b:
1 - 0.0343; 2 - 0.2187; 3 - 1; 4 - 6.481; 5 - 225.
Fig.3. The c (x) dependence for the values of parameters listed
3
in caption to Fig.2.
_
Fig.4. The dependence of cation flux to the electrode on lg (b)
for different values of d:
1 - d = 0.02; 2 - d = 0.1; 3 - d = 1; 4 - d = 10.
15
