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Physical Chemistry
A. Ya. Gokhshtein
On the Theory of \(I-t\) Curves in the Reduction of Anions at a Dropping Electrode in the Presence of an Adsorbing Catalyst
(Presented by Academician A. N. Frumkin on 1 XII 1960)
A number of interesting results in the theory of polarographic currents with simultaneous adsorption of a surface-active substance, the rate of which is determined by diffusion, were obtained in the works of Weber, Koutecký, Koryta, and Kuta. These authors studied in detail the case of a linear dependence of the reaction rate on coverage \((^1)\), and for an exponential dependence, without taking the influence of concentration polarization into account, gave a calculation of the initial segment of the \(I-t\) curve \((^2)\).
The direction of the present work was chosen in connection with the investigations of A. N. Frumkin, O. A. Petrii, and N. V. Nikolaeva-Fedorovich \((^3)\), who obtained, in the reduction of multiply charged anions in the presence of adsorbing cations, \(I-t\) curves with a well-pronounced maximum. The aim of the present work is to take concentration polarization into account and to derive relations between the characteristic points of the \(I-t\) curve over its entire course and the quantities that determine the kinetics of ion discharge.
The current \(I\), whose dependence on time \(t\) is to be determined, is caused by an irreversible reaction proceeding at a constant potential \(\varphi\) on the surface of an expanding spherical electrode. The volume of the sphere increases at a constant rate \(m/\gamma\); \(m\) (in g/sec) is the rate of flow of mercury, \(\gamma\) (in g/cm\(^3\)) is the specific gravity of mercury. The initial concentrations of the anion and catalyst are \(C\) and \(C_0\), and the diffusion coefficients are \(D\) and \(D_0\). With the addition of a supporting electrolyte and for a diffusion-layer thickness small in comparison with the electrode radius, the equation of convective diffusion has the form \((^4)\)
\[ \partial c/\partial t-(2x/3t)\partial C/\partial x=D\,\partial^2 C/\partial x^2; \tag{1} \]
\(C(x,t)\) is the concentration of the anion at a distance \(x\) from the electrode surface. The boundary condition follows from the equation of A. N. Frumkin \((^5)\)
\[ \frac{I}{nFS}=D\,\frac{\partial C(0,t)}{\partial x} = C(0,t)\,k\exp\frac{(n+\alpha)F\psi_1}{RT}\exp\left[-\frac{\alpha F\varphi}{RT}\right], \tag{2} \]
where \(\alpha\) is the barrier coefficient, \(n\) is the number of electrons, and \(k\) is the reaction-rate constant. The surface density of the adsorbed substance can be determined from the Ilkovič equation \((^1)\).
\[ Q(t)=2(3/7\pi)^{1/2}D_0^{1/2}C_0t^{1/2}\quad (\mathrm{mol/cm^2}). \tag{3} \]
Let us denote the \(\psi_1\)-potential in the absence of catalyst by \(\psi_{10}\); then one may approximately take
\[ \psi_1(t)=\psi_{10}+\sigma Q(t), \tag{4} \]
where the constant coefficient \(\sigma\) is determined by the kinetics of the process.
Combining (2), (3), and (4), we obtain
\[ D\,\frac{\partial C}{\partial x}(0,t) = C(0,t)k\exp\left\{\frac{F}{RT}\bigl[(n+\alpha)\psi_{10}-\alpha\varphi\bigr]\right\} \exp\left[ 2\left(\frac{3}{7\pi}\right)^{1/2}\frac{F}{RT}\sigma(n+\alpha)D_0^{1/2}C_0t^{1/2} \right]. \tag{5} \]
In the new variables
\[ y=t^{2/3}x,\qquad h=st^{7/3},\qquad s=\left[ 2\left(\frac{3}{7\pi}\right)^{1/2} \frac{F}{RT}\sigma(n+\alpha)D_0^{1/2}C_0 \right]^{14/3}. \tag{6} \]
(1) and (5) take the form
\[ \frac{\partial C}{\partial h}=\frac{3D}{7s}\frac{\partial^2 C}{\partial y^2},\qquad \frac{\partial C}{\partial y} =s^{2/7}\frac{k}{D}\exp\left\{\frac{F}{RT}\bigl[(n+\alpha)\psi_{10}-\alpha\varphi\bigr]\right\}h^{-2/7}e^{h^{3/14}}C(0,h). \tag{7} \]
The first of these equations, for \(C(y,0)=C\), is equivalent to the following:
\[ C(0,h)=C-\left(\frac{3D}{7\pi s}\right)^{1/2} \int\limits_0^h \frac{[\partial C(0,\xi)/\partial y]\,d\xi}{\sqrt{h-\xi}} . \tag{8} \]
Eliminating \(C(0,h)\) from (7) and (8), introducing the new unknown function
\[ f(h)=\frac{1}{C}\left(\frac{3D}{7\pi s}\right)^{1/2}\frac{\partial C(0,h)}{\partial y} \tag{9} \]
and, taking (6) into account, reducing the constants, we obtain for \(f(h)\) the integral equation
\[ f(h)=\frac{1}{p}h^{-2/7}e^{h^{3/14}} \left[1-\int\limits_0^h \frac{f(\xi)\,d\xi}{\sqrt{h-\xi}}\right], \tag{10} \]
which contains the parameter
\[ p=\frac{2F}{RT}\, \frac{\sigma(n+\alpha)D_0^{1/2}D^{1/2}C_0} {k\exp\frac{F}{RT}\bigl[(n+\alpha)\psi_{10}-\alpha\varphi\bigr]} ; \tag{11} \]
\(f\), \(h\), and \(p\) are dimensionless. For what follows it is convenient to single out, from (11), groups of coefficients:
\[ p=\frac{\nu C_0}{\omega},\qquad \nu=\sigma(n+\alpha)D_0^{1/2},\qquad \omega=\frac{RT}{2F}D^{-1/2}k\exp\frac{F}{RT}\bigl[(n+\alpha)\psi_{10}-\alpha\varphi\bigr]. \tag{12} \]
In a series of experiments in which \(\nu\) and \(\omega\) are constant, the parameter \(p\) varies with the catalyst concentration \(C_0\). Estimation of the integral standing in the square brackets in (10) shows that for \(h>10^3\) and \(p>10^4\) (these limits do not constrain practical calculations) the singularity of the function \(f\) at \(h=0\) may be neglected without risk of making an error in the fourth digit.
Equation (10) was solved for various values of \(p\) by the numerical method described earlier \((^6)\). Having determined the function \(f(h)\) from (10), returning to the old variables and collecting the coefficients, we obtain for the current flowing from the entire surface \(S(t)\) of the electrode at time \(t\) the expression
\[ I=nFDS(t)\,\partial C(0,t)/\partial x =6.458\,F(F/RT)^{2/3}(m/\gamma)^{2/3}nCD^{1/2}(\nu C_0)^{7/3}ft^{4/3}. \tag{13} \]
Owing to the relation (6) between \(t\) and \(h\), the function \(f\) may be assigned to each of these variables, \(f(h)=f(st^{7/3})\). The current is proportional to \(ft^{4/3}\), and, consequently, to \(fh^{4/7}\). It follows from this that \(I\) and \(f\) attain maxima at different values of \(h\). We shall denote the values of \(f\) and \(h\) at which the current is maximal (\(I_{\max}\)) by \(f_m\) and \(h_m\). To determine them it is sufficient to multiply the calculated function \(f(h)\) by \(h^{4/7}\). The position of its maximum will give \(h_m\), after which \(f_m=f(h_m)\). Using the relations obtained earlier for the descending branches of polarograms \((^7)\), the following representation of \(f(h)\) at some distance from its maximum may be given:
\[ f(h)=0.318\,(h-h_a)^{-1/2}. \tag{14} \]
Here only \(h_a\) depends on \(p\); thus, for \(p=10^6\), \(h_a=0.972\cdot 10^5\). From (14) it is seen that after the current maximum on the \(I-t\) curve there should follow a minimum (\(I_{\min}\)). The corresponding \(h_{\min}\) is determined from the condition for the minimum of the function \(fh^{4/7}\). According to (14),
\[ h_{\min}=8h_a,\qquad f_{\min}=0.318(7h_a)^{-1/2}. \tag{15} \]
(14) makes it possible to continue \(f(h)\) after the value of this function has been calculated at \(h\approx 3h_a\). Following \(f(h)\), \(f(h)h^{4/7}\) and \(f(st^{7/3})t^{4/3}\) are computed (Fig. 1); the latter, when substituted into (13), gives the current as a function of time. Figure 1 also shows the change in the near-electrode concentration of the vo
time. Multiplication of the right-hand side of (13) by \(h^{1/2}s^{1/2}t^{1/6}=1\) (see (6)) gives another expression for the \(I-t\) curve:
\[ I=13.093\,F\,(m/\gamma)^{2/3}nCD^{1/2}h^{1/2}ft^{1/6} =2.30\,Fm^{2/3}nCD^{1/2}h^{1/2}ft^{1/6}. \tag{16} \]
It follows from (14) that, as \(h\) increases, \(\left(h^{1/2}f\right)\to 0.318\). Substituting this limit into (16), we obtain, as a special case, the well-known Ilkovič equation. Thus, at large \(h\),
\[ I/I_{\mathrm{lim}}=\sqrt{h/(h-h_a)}; \]
for \(h\gg h_a\) the \(I-t\) curve merges with the limiting-current curve. Nomograms for constructing \(I-t\) curves at any value of \(p\), \(10^3 \leq p \leq 10^9\) cm, are shown in Fig. 2.
Fig. 1. 1 — theoretically calculated \(I-t\) curve; 2 — limiting current \(I_{\max}\); 3 — \(C(0,t)/C\). \(p=10^6\)
We now proceed to derive relations for the maximum on the \(I-t\) curve. The corresponding values \(f_m\), \(h_m\), \(C(0,t_m)/C\), calculated from (10) for different values of \(p\), are shown in Fig. 3. The interval of values chosen there, \(10^3 \leq p \leq 10^9\), over which the corresponding functions are defined, includes the values of \(p\) encountered in practice. It is noteworthy that \(h_m^{1/2}f_m\) depends almost linearly on \(\lg p\),
\[ h_m^{1/2}f_m=0.290+0.0341\lg p,\qquad 10^3<p<10^9. \tag{17} \]
Fig. 2. \(I-t\) curves for values \(p\): \(10^3\); \(10^4\); \(10^5\); \(10^6\); \(10^7\); \(10^8\); \(10^9\), and the limiting current. At right—interpolation nomogram
Substitution of (17) into (16) leads to the basic equation for the current at the maximum of the \(I-t\) curve:
\[ I_{\max}=(0.666+0.0782\lg p)\,Fm^{2/3}nCD^{1/2}t_m^{1/6}. \tag{18} \]
Dividing (18) by the value of the limiting current at the same instant gives a relation convenient for applications: \(I_{\max}/I_{\text{lim}} = 0.91 + 0.107 \lg p\). (19) From (19) one can determine \(p\) without knowing many experimental constants. The limiting current can be obtained experimentally either (more accurately) reproduced from a portion of the \(I-t\) curve sufficiently far from the peak. At \(p = 10^6\), \(I_{\max}/I_{\text{lim}} = 1.55\) (Fig. 1). For the time of appearance of the maximum, from (6) we obtain
\[ t_m=\frac{7\pi}{12}\left(\frac{RT}{F}\right)^2 \frac{h_m^{3/7}}{v^2 C_0^2}, \]
\[ t_{m,(20^\circ)}=1.169\cdot 10^{-3} \frac{h_m^{3/7}}{v^2 C_0^2}, \]
\[ t_{m,(25^\circ)}=1.209\cdot 10^{-3} \frac{h_m^{3/7}}{v^2 C_0^2}. \tag{20} \]
Fig. 3. \(f_m\) (1), \(h_m\) (2), \(h_m^{1/2} f_m\) (3), \(C(0,t'_m)/C\) (4) as functions of the parameter \(p\)
The exact values of \(h_m\) are given in Fig. 3; for \(10^4 < p < 10^9\) the representation \(h_m^{8/7} = 4(\lg p)^{2.04} \simeq 4\lg^2 p\) is admissible. For two \(I-t\) curves with different catalyst concentrations \(C_{01}\) and \(C_{02}\), from (20)
\[ t_{m1}/t_{m2}=(C_{02}/C_{01})^2(h_{m1}/h_{m2})^{3/7}. \tag{21} \]
When \((C_{02}/C_{01}) < 2\), \((h_{m1}/h_{m2})^{3/7}\) is close to 1 and \((t_{m1}/t_{m2}) \simeq (C_{02}/C_{01})^2\), i.e., the time of the maximum \(t_m\) is very sensitive to a change in \(C_0\), which can be used for analytical determinations. Conversely, relation (19) changes slowly, as \(\lg C_0\). To determine from experimental data the quantities \(v\) and \(\omega\), it is sufficient to find \(p\) from (19) and then use equations (20) and (12) successively. Since errors are possible in determining the limiting current (10%), the value of \(p\) found from (19) must be refined using the nomogram in Fig. 2, using points of the ascending branch. For the shift of the \(\psi_1\)-potential at the point of maximum, \(\Delta\psi_{1m}=[RT/(n+\alpha)F]h_m^{3/14}\simeq 2[RT/(n+\alpha)F]\lg p\). Substitution of \(C(0,t_m)/C\) (Fig. 3) into (2) gives a simple relation between the maximum current and the rate constant.
Fig. 4. Experimental points and the curve calculated at \(p=10^6\), on a relative scale
Figure 4 shows the results obtained by (3) in an experiment with a solution of \(5\cdot 10^{-4}\ M\) \(K_2S_2O_8 + 2.5\cdot 10^{-3} M Na_2SO_4 + 3\cdot 10^{-5} M (C_4H_9)_4NJ\) at \(20^\circ\), compared with the curve calculated for \(p=10^6\). From formulas (20) and (12) for this case \((C_0 = 3\cdot 10^{-8}\ \text{mol}/\text{cm}^3)\) \(v = 0.75\cdot 10^7\ \text{cm mol}^{-1}\text{sec}^{-1/2}\); \(\omega = 2.25\cdot 10^{-7}\ \text{sec}^{-1/2}\); \(\sigma = 1.44\cdot 10^9\ \text{cm}^2\text{mol}^{-1}\); \(\Delta\psi_{1m}=0.14\ \text{V}\).
The author expresses gratitude to Academician A. N. Frumkin for guidance of the work.
Institute of Electrochemistry
Academy of Sciences of the USSR
Received
29 I 1960
CITED LITERATURE
- J. Weber, J. Koutecky, J. Koryta, Zs. Elektrochem., 63, 583 (1959).
- I. Kuta, J. Weber, J. Koutecky, Coll. Czechoslov. Chem. Comm., 25, 2376 (1960).
- A. N. Frumkin, O. A. Petrii, N. V. Nikolaeva-Fedorovich, DAN, 135, No. 5 (1960).
- V. G. Levich, Physicochemical Hydrodynamics, Moscow, 1959, p. 538.
- A. N. Frumkin, V. S. Bagotskii, Z. A. Iofa, B. N. Kabanov, Kinetics of Electrode Processes, Moscow, 1952, p. 209.
- Ya. P. Gokhshtein, A. Ya. Gokhshtein, DAN, 128, 985 (1959).
- Ya. P. Gokhshtein, A. Ya. Gokhshtein, ZhFKh, 34, 1654 (1960).