On the Time of Metal Release from a Mercury Electrode during Its Polarization by Direct Current
Abstract Generated abstract
The paper examines the time required for metals accumulated in a mercury amalgam electrode to be released during anodic polarization by direct current, a condition relevant to repeated use of electrodes in anodic amalgam voltammetry. Starting from a previously derived diffusion expression for metal concentration in spherical and planar mercury electrodes, the authors derive formulas for the fraction of metal remaining in the amalgam at the transition time as a function of electrode size, diffusion coefficient, and dissolution time. The analysis indicates that this fraction is independent of the initial metal concentration and decreases much more rapidly for thin planar electrodes than for spherical electrodes. Experimental measurements with cadmium and zinc amalgams agree with the calculated curves within experimental error, showing that after about 90 seconds of direct current electrodissolution in the tested spherical electrodes, the residual metal fraction does not exceed 10%.
Full Text
ON THE TIME OF METAL EMERGENCE FROM A MERCURY ELECTRODE DURING ITS POLARIZATION BY DIRECT CURRENT
V. V. Pnev, M. S. Zakharov
(Presented by the scientific seminar of the MTP laboratory)
In the practice of anodic amalgam voltammetry with preliminary electrolytic accumulation of a metal in an amalgam, subsequent determinations are most conveniently carried out on one and the same electrode.
One of the conditions for the possibility of repeated use of an electrode is the practically complete emergence of the metal concentrated in it during the time of recording the anodic polarization \(\varphi—t\) curve.
Previously we [1] obtained the relation:
\[ C_R(X_1v_R)=C_R^0-\frac{1}{2}\lambda_R \left[ 2(2\gamma+2)v_R+X^2-\frac{2\gamma+2}{2\gamma+4} -4\sum_{n=1}^{\infty} \frac{(X)^{-\gamma} I_\gamma(\mu_n X)} {\mu_n^2 I_\gamma(\mu_n)} \exp(-\mu_n^2 v_R) \right], \tag{1} \]
where \(C_R^0\) is the initial concentration of metal atoms in the amalgam,
\[ \lambda_R=\frac{i r_0}{zFSD_R}, \]
\(i\) is the current, \(r_0\) is the radius of the spherical electrode or the thickness of the film, \(\gamma=1/2\) for a spherical electrode and \(\gamma=-1/2\) for a planar electrode,
\[ v_R=\frac{D_R t}{r_0^2}, \]
\(D_R\) is the diffusion coefficient of metal atoms in mercury,
\[ X=\frac{r}{r_0} \]
is the relative coordinate, \(I_\gamma(z)\) is the Bessel function of order \(\gamma\), and \(\mu_n\) are the roots of the equation:
\[ I_{\gamma+1}(z)=0. \tag{2} \]
Calculations show that for spherical and film electrodes, respectively at \(v_R \ge 0.12\) and \(v_R \ge 0.3\), the concentration at the surface of the amalgam is described by the equation:
\[ C_R^S=C_R^0-\lambda_R\left[(2\gamma+2)v_R-\frac{1}{2\gamma+4}\right]. \tag{3} \]
The average concentration in the amalgam \(\left[\overline{C}_R(r,t)\right]\) at any time \(t\) is equal to [2, p. 29]:
\[ \overline{C}_R(X_1v_R)=(2\gamma-2)\int_0^1 X^{2\gamma+1}C_R(X_1v_R)\,dX. \tag{4} \]
From equations (1) and (4), after integration, we obtain:
\[ \overline{C}_R(v_R)=C_R^0-(2\gamma+2)v_R. \tag{5} \]
From equations (3) and (5) it follows that the difference between the concentration at the surface and the average concentration is expressed by the equation:
\[ \overline{C}_R(v_R)-C_R^s=\frac{\lambda_R}{2\gamma+4}, \tag{6} \]
i.e., the difference between these concentrations at a given current does not depend on time. Thus, at \(i \to 0\) (infinitely slow dissolution), all of the metal initially concentrated in the amalgam will emerge from the amalgam.
At the moment the transition time is reached, we shall have:
\[ \overline{C}_R(\tau)=\frac{\lambda_R}{2\gamma+4}. \tag{7} \]
Since \(q_{\mathrm{coul.}}=zFV_{\mathrm{Hg}}C_R^0=it\), (where \(V_{\mathrm{Hg}}\) is the volume of mercury), it follows from equation (7) that
\[ \Theta_p\%=\frac{100r_0^2}{3D_R\tau}, \tag{8} \]
\[ \Theta_{\mathrm{sph}}\%=\frac{100r_0^2}{15D_R\tau}, \tag{9} \]
where \(\Theta=\dfrac{\overline{C}_R(\tau)}{C_R^0}\) is the fraction of the metal remaining at the moment \(\tau\) in the amalgam for planar (equation 8) and spherical (equation 9) electrodes. For a planar electrode (\(r_0=3\cdot10^{-4}\ \mathrm{cm}\)), already at \(\tau \ge 4\cdot10^{-2}\ \mathrm{sec}\) no more than 5% of the initial amount of metal remains in the amalgam. For a spherical electrode (\(r_0=3\cdot10^{-2}\ \mathrm{cm}\), \(D_R=1.5\cdot10^{-5}\ \mathrm{cm^2/sec}\)), the same amount of metal remains in the amalgam at \(\tau \ge 80\ \mathrm{sec}\). It follows from equations (8–9) that \(\Theta\) does not depend on \(C_R^0\).
Experimental verification of equation (9) was carried out using the example of Cd and Zn amalgams. \(\Theta\) was evaluated by the formula:
\[ \Theta=\left(1-\frac{q^1}{q}\right)\cdot100\%, \tag{10} \]
where \(q^1=it\), and \(q\) is the limiting value of the product \(it\). The results of the measurements are shown in Fig. 1. The same figure gives the curves calculated from equation (9). As can be seen from Fig. 1, the experimental and theoretical curves coincide within the experimental error. It is evident from Fig. 1 that after 90 seconds of electrodissolution of the amalgam by direct current, \(\Theta\) does not exceed 10%.
Fig. 1. Dependence of the fraction (\(\Theta\%\)) of metal remaining in the amalgam by the moment \(\tau\) on the value of \(\tau\), for Cd (1) and Zn (2) amalgams in a solution of \(0.5\ M\ K_2SO_4+5\cdot10^{-4}\ M\ MeSO_4\). \(1a\) and \(2a\)—theoretical curves according to (8).
Experimental conditions:
1) \(r_0=3.52\cdot10^{-2}\ \mathrm{cm}\), \(D_R=1.66\cdot10^{-5}\ \mathrm{cm^2/sec}\), \(\varphi_{\mathrm{e}}=-1.0\) (N.C.E.), \(t_{\mathrm{e}}=3\ \mathrm{min}\).
2) \(r_0=3.31\cdot10^{-2}\ \mathrm{cm}\), \(D_R=1.67\cdot10^{-5}\ \mathrm{cm^2/sec}\), \(\varphi_{\mathrm{e}}=-1.3\) (N.C.E.), \(t_{\mathrm{e}}=3\ \mathrm{min}\).
REFERENCES
- M. S. Zakharov, V. V. Pnev. Izv. TPI, 164, 1967, p. 79.
- A. V. Lykov. Theory of Heat Conduction. Moscow, Gostekhizdat, 1952.