Abstract Generated abstract
This paper derives a general relation between an arbitrary applied electromotive force and the resulting current in an electrolytic cell with a spherical stationary microelectrode. Using diffusion kinetics, Tafel electrode behavior, double-layer capacitance, and solution resistance, it formulates the problem as a boundary-value system and reduces it to a nonlinear Volterra integral equation for several cases of reductant location: in solution, in the electrode, or in a thin surface layer. The work also develops an electrical analog based on distributed RC networks and analog-computing elements, allowing simulation of faradaic current, total current, electrode potential, and boundary concentration changes. A special implementation for semi-infinite linear diffusion and reversible processes is reported, illustrating the influence of solution resistance on polarographic wave shift and peak current.
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PHYSICAL CHEMISTRY
R. Sh. NIGMATULLIN
GENERAL EQUATION AND ELECTRICAL ANALOG OF AN ELECTROLYTIC CELL WITH A SPHERICAL STATIONARY MICROELECTRODE
(Presented by Academician A. N. Frumkin, 5 IV 1963)
The development of the theory of modern polarographic methods and electrochemical converters of electrical signals is associated with the study of the functional relation between the e.m.f. applied to an electrolytic cell and the current flowing \((^1)\).
Below, on the basis of the laws of diffusion kinetics, the dependence of the current \(i(t)\) on an e.m.f. \(\mathscr{E}(t)\) of arbitrary form is established for a cell with a spherical stationary microelectrode. It is assumed that: 1) the oxidation–reduction reaction at the electrode obeys the Tafel equation; 2) the capacitance of the double electric layer \(C_0\) is a prescribed function of the electrode potential; 3) ion migration is taken into account through the volume resistance of the solutions \(R_0\) \((^2)\). The paper also proposes an electrical model for investigating processes in the circuit of such a cell.
Fig. 1
Let the oxidant \(A_1\) and the reductant \(A_2\) on the surface of a microelectrode of radius \(a\) (Fig. 1) exchange \(n\) electrons: \(A_1 + n \ominus \rightleftarrows A_2\). Let \(A_1\) in this case always remain in the solution, while \(A_2\) either also remains in the solution (case I), or dissolves in the electrode material (II), or in its thin surface layer of thickness \(\bar a \ll a\) (III), i.e. we consider, respectively, a solid electrode, a mercury drop, or a solid electrode coated with a mercury film. In all these cases, finding the relation between \(i(t)\) and \(\mathscr{E}(t)\) reduces to solving a boundary-value problem for a system of two parabolic equations:
\[ \frac{\partial}{\partial t} C_\nu(r,t) = D_\nu\left[ \frac{\partial^2}{\partial r^2} C_\nu(r,t) + \frac{2}{r}\frac{\partial}{\partial r} C_\nu(r,t) \right] \quad (\nu=1,2), \tag{1} \]
where \(t \ge 0\), for \(C_1: r \ge a;\quad C_2:\ I — r \ge a,\ II — 0 \le r \le a,\ III — a-\bar a \le r \le a;\)
\[ C_\nu(r,-0)=\bar C_\nu=\mathrm{const} \quad (\nu=1,2); \tag{2} \]
\[ C_1(\infty,t)=\bar C_1;\quad I — C_2(\infty,t)=\bar C_2,\quad II — \frac{\partial}{\partial r}C_2(0,t)=0, \]
\[ III — \frac{\partial}{\partial r}C_2(a-\bar a,t)=0; \tag{3} \]
\[ \frac{\partial}{\partial r}C_1(a,t)=\frac{i_d(t)}{nFSD_1}; \quad I — \frac{\partial}{\partial r}C_2(a,t)=-\frac{i_d(t)}{nFSD_2}; \]
\[ II,\ III — \frac{\partial}{\partial r}C_2(a,t)=\frac{i_d(t)}{nFSD_2}; \tag{4} \]
\[ \frac{i_d(t)}{i_0} = \frac{C_1(a,t)}{\bar C_1}\exp \alpha \mathscr{E}_d(t) - \frac{C_2(a,t)}{\bar C_2}\exp \bar\beta \mathscr{E}_d(t); \tag{5} \]
\[ \text{a) }\quad i(t)=i_d(t)+\frac{d}{dt}\bigl[\mathscr{E}_d C_0(\mathscr{E}_d)\bigr], \qquad \text{b) }\quad \mathscr{E}(t)=R_0 i(t)+\mathscr{E}_d(t). \tag{6} \]
where \(C_1\) and \(C_2\) are the concentrations of \(A_1\) and \(A_2\); \(\overline C_1\) and \(\overline C_2\) are their initial, equilibrium values; \(i_d\) is the faradaic current; \(\mathscr E_d\) is the electrode potential (with the opposite sign), measured relative to a macroelectrode of the same material; \(\overline\alpha=\alpha nF/RT\), \(\overline\beta=-\beta nF/RT\), \(\beta=1-\alpha\), \(i_0=nFSk\overline C_1^{\beta}\overline C_2^{\alpha}\). The remaining notation is standard.
Fig. 2
The problem is solved as follows. From (1)—(4), by the operational method \({}^{(3)}\), we find the dependence of the boundary concentrations \(C_\nu(a,t)\) on the current \(i(t)\). Substituting \(C_\nu(a,t)\) into the Tafel equation (5), we establish the relation between \(\mathscr E_d\) and \(i_d\). Next, using Kirchhoff’s laws (6), we obtain the desired dependence:
\[ \frac{i_d}{i_0} = \frac{\exp \overline\alpha \mathscr E_d}{u_1} \left[ u_1 - \frac{d}{dt} \int_0^t F_1(t-\tau)i_d(\tau)\,d\tau \right] - \frac{\exp \overline\beta \mathscr E_d}{u_2} \left[ u_2 + \frac{d}{dt} \int_0^t F_2(t-\tau)i_d(\tau)\,d\tau \right], \tag{7} \]
where
\[ i_d=i-\frac{d}{dt}\left[\mathscr E_d C_0(\mathscr E_d)\right], \qquad \mathscr E_d=\mathscr E-R_0 i, \qquad u_\nu=nFSC_\nu\;(\nu=1,2), \]
\[ F_1(t)=\frac{a}{D_1} \left[ 1-\exp\frac{tD_1}{a^2}\operatorname{erfc}\sqrt{\frac{tD_1}{a^2}} \right], \]
and \(F_2(t)\) is different for the three cases:
\[ F_2^{\mathrm I} = \frac{a}{D_2} \left(1-\exp\delta\,\operatorname{erfc}\sqrt{\delta}\right), \qquad F_2^{\mathrm{II}} = \frac{a}{D_2} \left[ 3\delta+\frac{1}{5} - \sum_{m=1}^{\infty} \frac{2}{\mu_m^2}\exp(-\mu_m^2\delta) \right], \]
\[ F_2^{\mathrm{III}} = \frac{a}{D_2} \left[ \overline\delta+\frac{1}{3} - \sum_{m=1}^{\infty} \frac{2}{\mu_m^2}(-1)^m\cos\mu_m\exp(-\mu_m^2\overline\delta) \right]; \qquad \delta=\frac{tD_2}{a^2}; \qquad \overline\delta=\frac{tD_2}{a^2}; \]
\(\mu_m\) are the positive roots of the equation \(\operatorname{tg}\mu_m=\mu_m\).
The nonlinear integral equation of Volterra type that has been found establishes the dependence between the current \(i(t)\) and an arbitrary (within the class of functions of bounded variation) emf \(\mathscr E(t)\). In the general case, the solution of (7) cannot be expressed in closed form, i.e., in quadratures, through functions characterizing the original equation, without the use of infinite processes. Since the accuracy of electrochemical measurements rarely exceeds \(1\)—\(5\%\), i.e., is lower than the accuracy of electrical analog computers, the question of an electrical model of equation (7) naturally arises.
Proceeding from the analogy between the processes of diffusion and potential equalization in \(RC\) circuits with distributed parameters, one can show that the operational “diffusion resistance”
\[ f_\nu(p)=p\int_0^\infty F_\nu(t)\exp(-pt)\,dt =(-1)^\nu nFS\,\frac{[c_\nu(a,p)-C_\nu]}{i(p)} \qquad (\nu=1,2)^* \]
is equivalent to the electrical resistance of a two-terminal network \(Z_\nu(p)\), composed in the general case of a homogeneous \(RC\)-cable and an active resistance (see Table 1).
The derivative convolution of the functions \(F_\nu(t)\) and \(i_d(t)\) in (7) corresponds to the voltage at the output of a certain linear four-terminal network when the input voltage follows the law of variation \(i_d(t)\). Using (4), it is easy to show that, as such a four-terminal network, one may take an operational amplifier in which negative feedback is provided through a circuit with resistance \(Z_\nu(p)\). The results obtained make it possible to construct a structural model of equation (7) from \(RC\)-cables and standard blocks of analog computers: amplifiers, nonlinear converters, as well as differentiating, multiplying, and summing devices (Fig. 2).
Fig. 3. Oscillopolarograms of \(\mathrm{Cd}^{2+}\) in KCl, obtained on the model for three values of \(R_0\): 16.2; 273 and 1000 ohms (\(C_1=10^{-6}\ \text{g-ion}\cdot\text{cm}^{-3}\), \(E=0.5t_B,\ T=298^\circ\mathrm{K},\ S=3\cdot10^{-2}\ \text{cm}^2,\ D_1=0.72\cdot10^{-5}\ \text{cm}^2\cdot\text{sec}^{-1}\))
Equations for the voltages in the model circuit:
\[ \frac{U_{Mi}}{K_{M0}} = \frac{\exp\theta K_{M\alpha}U_{Me}}{r_{M1}} \left[ \frac{r_{M1}}{\theta} - \frac{d}{dt_M} \int_0^{t_M} F_{M1}(t_M-\tau_M)U_{Mi}(\tau_M)\,d\tau_M \right] - \]
\[ - \frac{\exp\theta K_{M\beta}U_{Me}}{r_{M2}} \left[ \frac{r_{M2}}{\theta} + \frac{d}{dt_M} \int_0^{t_M} F_{M2}(t_M-\tau_M)U_{Mi}(\tau_M)\,d\tau_M \right], \tag{8} \]
where
\[ U_{Mi}=U_M-\frac{d}{dt_M}\left[\vartheta U_{Me}\psi_M(\theta K_{M\gamma}U_{Me})\right], \qquad U_{Me}=\mathscr{E}_M-K_{MR}U_M, \]
\[ F_{M1}=R_{M1}\left(1-\exp\delta^I_{M1}\operatorname{erfc}\sqrt{\delta^I_{M1}}\right), \qquad F^I_{M2}=R_{M1}\left(1-\exp\delta^I_{M2}\operatorname{erfc}\sqrt{\delta^I_{M2}}\right), \]
\[ F^{II}_{M2}=R^{II}_{M2} \left[ 3\delta^{II}_{M2} + \frac{1}{5} - \sum_{m=1}^{\infty}\frac{2}{\mu_m^2}\exp(-\mu_m^2\delta^{II}_{M2}) \right], \]
\[ F^{III}_{M2}=R^{III}_{M2} \left[ \delta^{III}_{M2} + \frac{1}{3} - \sum_{m=1}^{\infty}\frac{2}{\mu_m^2}(-1)^m\cos\mu_m\exp(-\mu_m^2\delta^{III}_{M2}) \right], \]
\[ \delta^{(N)}_{M\nu} = t_M\beta^{-1}_{M\nu}(R^{(N)}_{M\nu})^{-2} \qquad (N=0,\ I,\ II,\ III;\ \nu=1,2), \qquad \theta=1b^{-1},\quad \vartheta=1\ \text{sec}. \]
Comparing the equations of the original system (7) and of the model (8), by known methods \((^5)\) it is easy to introduce similarity constants and to find similarity invariants.
\[ {}^*\ \text{The concentration transformed according to Laplace--Carson is} \]
\[ c_\nu(a,p)=p\int_0^\infty C_\nu(a,t)\exp(-pt)\,dt \qquad (\nu=1,2). \]
The proposed electrical analog of the cell makes it possible to obtain \(i(t)\), \(i_d(t)\), \(\mathscr{E}_d(t)\), as well as the time dependence of the change in the boundary concentrations of the oxidant and reductant for specified \(\mathscr{E}(t)\) and cell parameters. The corresponding voltages in the circuit are taken (relative to ground) from terminals 2–6 (Fig. 2).
Table 1
| Operational “diffusion resistance” \(f_\nu(p)\) | Electrical analog and operational resistance of the two-terminal network \(Z_\nu(p)\) |
|---|---|
| \(\displaystyle f_1(p)=\frac{1}{\sqrt{pD_1}+\dfrac{D_1}{a}}\) | A Semi-infinite \(RC\)-cable. \(R_{M1}>0\)—active resistance. \(\displaystyle Z_1=\frac{1}{\sqrt{p\beta_{M1}}+1/R_{M1}}\) \(\displaystyle \beta_{M1}=\bar c_{M1}/\bar r_{M1}\)* |
| \(\displaystyle f_2^{\mathrm{I}}(p)=\frac{1}{\sqrt{pD_2+\dfrac{D_2}{a}}}\) | B Semi-infinite \(RC\)-cable. \(R_{M2}^{\mathrm{I}}>0\)—active resistance. \(\displaystyle Z_2^{\mathrm{I}}=\frac{1}{\sqrt{p\beta_{M2}+1/R_{M2}}}\) \(\displaystyle \beta_{M2}=\bar c_{M2}/\bar r_{M2}\)* |
| \(\displaystyle f_2^{\mathrm{II}}(p)=\frac{1}{\sqrt{pD_2}\,\operatorname{cth}\!\left(\dfrac{a}{D_2}\sqrt{pD_2}\right)-\dfrac{D_2}{a}}\) | V Short-circuited segment of an \(RC\)-cable (length \(l\)). \(R_{M2}^{\mathrm{II}}<0\) active negative resistance. \(\displaystyle Z_2^{\mathrm{II}}=\frac{1}{\sqrt{p\beta_{M2}}\operatorname{cth}\!\left(\left|R_{M2}^{\mathrm{II}}\right|\sqrt{p\beta_{M2}}\right)+1/R_{M2}}\) \(\displaystyle \left|R_{M2}^{\mathrm{II}}\right|=r_{M2}l.\) |
| \(\displaystyle f_2^{\mathrm{III}}(p)=\frac{1}{\sqrt{pD_2}\,\operatorname{th}\!\left(\dfrac{a}{D_2}\sqrt{pD}\right)}\) | G Segment of an \(RC\)-cable (length \(l\)), open at the end. \(\displaystyle Z_2^{\mathrm{III}}=\frac{1}{\sqrt{p\beta_{M2}}\,\operatorname{th}\!\left(R_{M2}^{\mathrm{III}}\sqrt{p\beta_{M2}}\right)}\) |
* \(\bar r_{M\nu}\) and \(\bar c_{M\nu}\) are the distributed resistance and capacitance of the \(RC\)-cable \((\nu=1,2)\).
For the special case of semi-infinite linear diffusion \((D_\nu t \ll a^2)\) and reversible processes \((i_0\to\infty)\), the model was implemented.* The circuit used operational amplifiers from the IPT-5 integrator, a multiplication unit, and BN-3 universal functional converters. The external emf was specified by an NGPK-3 generator. As the semi-infinite \(RC\)-cable there was used an equivalent6, in the working frequency region, chain \(RC\)-line made up of a finite number of links. On the model, with an accuracy of 1.5–9%, the influence of \(R_0\) on the potential shift and the peak current of the polarographic wave (Fig. 3) under application of a linearly varying emf was studied.
In conclusion, I consider it a pleasant duty to express my gratitude to Academician A. N. Frumkin and Corresponding Member of the Academy of Sciences of the USSR V. G. Levich for their unfailing attention to the work and valuable discussion.
Kazan Aviation Institute
Received
27 III 1963
CITED LITERATURE
- Ya. P. Gokhshtein, DAN, 126, No. 3, 598 (1959); N. Matsuda, Y. Ayabe, Zs. Elektrochem., 59, 494 (1955); R. Sh. Nigmatullin, DAN, 150, No. 1, 138 (1963).
- A. N. Frumkin, V. S. Bogotskii, Z. A. Iofa, B. N. Kabanov, Kinetics of Electrode Processes, Moscow, 1952, p. 96; V. G. Levich, Physicochemical Hydrodynamics, Moscow, 1959, p. 262.
- V. A. Ditkin, P. I. Kuznetsov, Handbook of Operational Calculus, Moscow–Leningrad, 1951.
- A. A. Feldbaum, Computing Devices in Automatic Systems, Moscow, 1959, p. 209.
- I. M. Telbaum, Electrical Modeling, Moscow, 1959, p. 42.
- E. F. Bazlov, R. Sh. Nigmatullin, Transactions of the Kazan Aviation Institute, issue 79, 65 (1963).
* The subsequent part of the work was carried out jointly with E. F. Bazlov.