Abstract
Full Text
Physical Chemistry
Corresponding Member of the Academy of Sciences of the USSR V. G. Levich, B. I. Khaikin, and V. A. Kir’yanov
FARADAIC IMPEDANCE FOR REVERSIBLE ELECTRODE PROCESSES PROCEEDING ACCORDING TO THE SCHEME OF CATALYTIC HYDROGEN EVOLUTION
The present work considers a depolarization process described by the following scheme:
[
\mathrm{B}\ \mathop{\rightleftarrows}^{\rho}_{\sigma\rho}\ \mathrm{A};
\tag{I}
]
[
\mathrm{A}+\mathrm{e}^{-}\ \mathop{\rightleftarrows}^{\mathrm{el}}\ \mathrm{C};
\tag{II}
]
[
2\mathrm{C}\xrightarrow{k}2\mathrm{B}+\mathrm{H}_2.
\tag{III}
]
According to this scheme ((^{1,2})), the catalyst exists in two forms, A and B, which are in protolytic equilibrium. Form A is discharged reversibly at the electrode, and the product of the electrode reaction, C, regenerates B by a bimolecular mechanism with simultaneous evolution of hydrogen. The quantities (\rho) and (\sigma\rho) are the effective rate constants of the reversible monomolecular reaction (I), and (k) is the rate constant of the bimolecular process (III).
The purpose of the present work is to derive general relations for the Faradaic impedance, making it possible to estimate from experimental data the rate constant of the bimolecular process and the normal potential of the electrochemical reaction.
For a dropping electrode the kinetic scheme (I)—(III) (in the presence in solution of a large excess of indifferent electrolyte) is described by the following system of differential equations:
[
\begin{aligned}
\frac{\partial a}{\partial t}
&=D\frac{\partial^2 a}{\partial x^2}
+\frac{2}{3}\frac{x}{t}\frac{\partial a}{\partial x}
+\rho(b-\sigma a),\
\frac{\partial b}{\partial t}
&=D\frac{\partial^2 b}{\partial x^2}
+\frac{2}{3}\frac{x}{t}\frac{\partial b}{\partial x}
-\rho(b-\sigma a)+kc^2,\
\frac{\partial c}{\partial t}
&=D\frac{\partial^2 c}{\partial x^2}
+\frac{2}{3}\frac{x}{t}\frac{\partial c}{\partial x}
-kc^2,
\end{aligned}
\tag{1}
]
where (a,b,c) are the concentrations of the substances A, B, C.
Assuming that the electrode reaction is the rapid stage ((^{1})), one may write the initial and boundary conditions for the system of equations (1) in the form:
[
t=0\ \text{or}\ x\to\infty:
]
[
a+b+c=\alpha,\qquad \sigma a=b,\qquad c=0;
]
[
t>0,\ x=0:
]
[
\frac{\partial a}{\partial x}+\frac{\partial c}{\partial x}=0,\qquad
\frac{\partial b}{\partial x}=0,\qquad
a=\lambda c,
\tag{2}
]
where (\alpha) is the total concentration of the substances A and B before the start of electrolysis;
[
\lambda=\lambda_0\exp\left(\frac{F\varphi_1}{RT}\sin\omega t\right),\qquad
\lambda_0=\exp\left[\frac{F}{RT}\left(\varphi_0-\varphi^{(0)}\right)\right];
\tag{3}
]
(\varphi_0) is the aperiodic component of the cathode potential; (\varphi_1) is the amplitude of the small sinusoidal potential; (\varphi^{(0)}) is the normal potential of the electrochemical reaction.
If the volume reactions are sufficiently fast, then the thickness of the kinetic layer is considerably smaller than the thickness of the diffusion layer. Therefore the influence of convection (motion of the drop into the depth of the solution is equivalent to motion of the solution) will be small ((^3)), and in the right-hand sides of equations (1) the second terms may be neglected. In what follows we shall assume that, in the course of electrolysis, a stationary state is established, determined by the rate of the volume reactions and by the diffusion process.
We seek the solution of the system of equations (1) in the form of a series in powers of (\varphi_1):
[
a=a_0+\bar a e^{i\omega t},\qquad
b=b_0+\bar b e^{i\omega t},\qquad
c=c_0+\bar c e^{i\omega t},
\tag{4}
]
where (a_0, b_0, c_0) are the aperiodic parts of the concentrations; (\bar a, \bar b, \bar c) are complex amplitudes proportional to (\varphi_1). Taking into account the assumptions made above, one can obtain equations for the quantities (\bar a) and (\bar c):
[
i\omega \bar a
=
D\frac{d^2\bar a}{dx^2}
-
l\left(\bar a+\frac{\bar c}{1+\sigma}\right),
\qquad
i\omega \bar c
=
D\frac{d^2\bar c}{dx^2}
-
\frac{\eta}{(1+\nu x)^2}\bar c,
\tag{5}
]
where
[
l=\rho(1+\sigma),\qquad
\nu=\left[\frac{k(c_0){x=0}}{6D}\right]^{1/2},
\qquad
\eta=2k(c_0).
]
The boundary conditions for equations (5) and (6) are found from conditions (2):
[
(\bar a){x\to\infty}=(\bar c)=0;
\tag{6}
]
[
\left(\frac{d\bar a}{dx}\right){x=0}
+
\left(\frac{d\bar c}{dx}\right)
=0;
\tag{7}
]
[
(\bar a){x=0}
=
(\bar c)\lambda_0
+
\frac{nF\varphi_1}{2kRT}.
\tag{8}
]
To determine the rate constant of the bimolecular process, one should consider the low-frequency case ((\omega\ll \eta,l)), which is done here.
Solving the system of differential equations (5), one can find the following expressions for the quantities ((d\bar c/dx){x=0}) and ((d\bar a/dx)):
[
\left(\frac{d\bar c}{dx}\right){x=0}
=
-
\left(
0.86\sqrt{\frac{\eta}{D}}
+
0.69\frac{i\omega}{\sqrt{D\eta}}
\right)
(\bar c);
\tag{9}
]
[
\begin{aligned}
\left(\frac{d\bar a}{dx}\right){x=0}
={}&
-
\left(
\sqrt{\frac{l}{D}}
+
0.5\frac{i\omega}{\sqrt{Dl}}
\right)
(\bar a)
\
&-
\frac{1}{1+\sigma}\sqrt{\frac{l}{D}}\,
\beta
\left{
\left[
\frac{1}{2}
-\frac{\beta}{2}
-\frac{\beta^2}{2}e^\beta \operatorname{Ei}(-\beta)
\right]
\right.
\
&\left.
\qquad
+
1.73\frac{i\omega}{\sqrt{\eta l}}
\left[
-\frac{1}{2}
-\frac{2}{5}\beta
-\frac{\beta^2}{10}
-\beta e^\beta \operatorname{Ei}(-\beta)
\left(
\frac{\beta^2}{10}
+
\frac{\beta}{2}
+
\frac{4}{5}
\right)
\right]
\right}
(\bar c)_{x=0};
\tag{10}
\end{aligned}
]
where
[
\beta=\sqrt{\frac{12l}{\eta}},
\qquad
\operatorname{Ei}(-\beta)
=
-\int_{\beta}^{\infty}\frac{e^{-t}}{t}\,dt.
]
The parameter (\beta) entering (10) is, in order of magnitude, equal to the ratio of the thickness of the kinetic layer of the bimolecular reaction (III) to the thickness of the kinetic layer of the monomolecular reaction (I), and therefore characterizes the relative rate of the volume reactions.
From the system of algebraic equations (7), (8), (9), and (10) one can determine ((dc/dx)_{x=0}), which is related to the complex amplitude of the current by the formula
[
\bar I = FD \left( \frac{d\bar c}{dx} \right)_{x=0}.
\tag{11}
]
The complex amplitude of the current can also be expressed in terms of the values of the pseudocapacitance (C_s) and the polarization resistance (R_s) ((^4)). If the electrochemical cell polarized by an alternating current is represented in the form of an equivalent circuit in which the pseudocapacitance and the polarization resistance are connected in series, then the complex amplitude of the faradaic current is given by the well-known expression:
[
\bar I =
\frac{\varphi_1}{R_s - i \dfrac{1}{\omega C_s}} .
\tag{12}
]
The polarization resistance and the pseudocapacitance are found by comparing expressions (11) and (12). In particular, if the faradaic impedance is determined mainly by the rate of the bimolecular process ((\beta \gg 1)), then
[
R_s =
\frac{k^{1/3}RT}{F^{4/3}D^{1/6}\lambda_0^{2/3} j_0}
\left[
\frac{0.86}{l^{1/2}}
+
0.67 \frac{F^{2/3}D^{1/3}}{k^{2/3}j_0}
\left(
\lambda_0 + \frac{1}{1+\sigma}
\right)
\right] ;
\tag{13}
]
[
C_s =
\frac{F^{1/3}\lambda_0 j_0^{5/3} k^{2/3} l^{3/2}}
{D^{1/3}RT \omega^2
\left[
0.19 \dfrac{k j_0}{FD^{1/2}}
+
0.93 l^{3/2}
\left(
\dfrac{1}{1+\sigma} + \lambda_0
\right)
\right]} ,
\tag{14}
]
where (j_0) is the density of the constant component of the current. If, however, the slow stage of the process is a monomolecular reaction ((\beta \ll 1)), then
[
R_s =
\frac{k^{2/3}RT}{F^{4/3}D^{1/6}\lambda_0 j_0^{2/3}}
\left[
\frac{0.87}{l^{1/2}}
+
0.67 \frac{F^{2/3}D^{1/3}\lambda_0}{k^{2/3}j_0}
\right] ;
\tag{15}
]
[
C_s =
\frac{F^{1/3}\lambda_0 j_0^{5/3} k^{2/3} l^{3/2}}
{D^{1/3}RT \omega^2
\left(
0.19 \dfrac{k j_0}{FD^{1/2}}
+
0.93 \lambda_0 l^{3/2}
\right)} .
\tag{16}
]
Formulas (13), (14) or (15), (16) can be used to determine the unknown constants that characterize the kinetics of the process. For example, if the rates of the forward and reverse reaction (I) are known (i.e., the quantity (l)), then from experimental data on the constant components of the potential and current density ((\varphi_0) and (j_0)), as well as on the pseudocapacitance and polarization resistance, one can find the rate constant of the bimolecular process (III) and the normal potential (\varphi^{(0)}).
The formulas given above also determine the dependence of (R_s) and (C_s) on frequency: the pseudocapacitance is inversely proportional to the square of the frequency, while the polarization resistance does not depend on frequency. It should be noted that such a dependence is valid only for low frequencies ((\omega \ll \eta, l)).
From formulas (13), (15) and (15), (16), the constant component of the current density can be eliminated with the aid of the previously obtained ((^2)) current–voltage characteristics. In the general case, however, cumbersome relations are obtained. Therefore, we shall restrict ourselves to the particular case, especially important experimentally, when the slow stage is the bimolecular process ((\beta \gg 1)), and the concentration of the inactive form of the catalyst B considerably exceeds the concentration of the active form A ((\sigma \gg 1)). The formulas for the polarization resistance and the pseudocapacitance then take the following form:
[
R_s = 0.82 \frac{RT \sigma^{3/2}\lambda_0^{3/2}}
{F^2D^{1/2}k^{1/2}\alpha^{3/2}} ;
\tag{17}
]
[
C_s = 3.04 \frac{F^2D^{1/2}k^{3/2}\alpha^{5/2}}
{RT\omega^2\sigma^{5/2}\lambda_0^{5/2}} .
\tag{18}
]
The last formulas are valid for potentials satisfying the inequality (\beta \lambda_0 \gg 1), i.e., at least for potentials not too much more negative than the normal potential (\varphi^{(0)}).
Expressions (17) and (18) contain, in explicit form, the parameters characterizing the system under consideration, which makes it possible to directly estimate the influence of these parameters on the magnitude of the Faradaic impedance.
This work was carried out in connection with the experimental investigations of S. G. Mairanovskii. The authors express their gratitude to S. G. Mairanovskii for discussion of the results of the calculations.
A comparison of the results with experimental data will be given in the latter’s publication.
Institute of Electrochemistry
Academy of Sciences of the USSR
Received
10 V 1961
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