A Study of the Adsorption-Electrochemical Theory of the Ammonium Amalgam Effect in the APN Method
Abstract Generated abstract
This paper develops a kinetic adsorption and electrochemical model for the ammonium amalgam effect in the APN method at constant solution temperature. It describes the growth of a mercury drop containing ammonia as the balance between formation of gas foam within the drop and diffusion of gas bubbles through the electrode surface, relating these rates to current, effective surface area, bubble surface coverage, and drop radius. Using an empirical relation between surface coverage and electrode dimensions, the authors derive equations for the time dependence of drop growth during electrolysis and for shrinkage after the current is switched off. The resulting expressions satisfy limiting kinetic cases and are reported to agree with previously published experimental radius measurements.
Full Text
Studies on the Adsorption–Electrochemical Theory of the Ammonium Amalgam Effect in the APN Method
Part 1. Constant Temperature of the Analyzed Solution
Yu. A. Karbainov, G. N. Sutyagina
(Presented by the scientific seminar of the Department of Analytical Chemistry)
If \(V_1\) is the rate of the process of formation of gas foam inside the mercury drop and \(V_2\) is the rate of the process of diffusion of gas bubbles through the electrode surface into the solution, then for the rate of increase in the size of the ammonium amalgam drop one may write the expression
\[ \frac{dm}{dt}=V_1 - V_2, \tag{1} \]
where \(m\) is the amount of ammonia inside the mercury dropping electrode, in moles. We assume, as previously [1], that the process determining the rate \(V_1\) is the delivery of ammonium ions to the electrode surface. Consequently, the rate \(V_1\) is proportional to the electrolysis current, which, in turn, is proportional to the effective surface of the a.a. drop (not occupied by gas bubbles), that is,
\[ i_{\mathrm{э}}=K_{\mathrm{э}}\cdot S(1-\Theta)\cdot C_{\mathrm{ш}}=F\cdot V_1, \tag{2} \]
where \(\Theta\) is the fraction of the surface of the a.a. drop occupied by gas bubbles. The remaining notation is generally accepted.
Taking into account that at a given moment \(M\) ammonium ions participate in the electrode reaction, and that only a fraction of the total number of discharged radicals, equal to \(m=aM\) [1], forms ammonium amalgam, we obtain
\[ V_1=\frac{a i_{\mathrm{э}}}{F}=K_1\cdot S\cdot(1-\Theta), \tag{3} \]
where
\[ K_1=a\,\frac{K_{\mathrm{э}}\cdot C_{\mathrm{ш}}}{F} \tag{4} \]
We assume that the rate \(V_2\) is limited by the diffusion of gas bubbles through the surface of the a.a. drop and is proportional to the fraction of the surface occupied by gas bubbles:
\[ V_2=K_2^{*}\cdot S\cdot\Theta \tag{5} \]
where
\[ K_2^{*}=K_2\cdot\frac{m}{V_z} \tag{6} \]
Taking (3) and (5) into account, expression (1) has the form
\[ \frac{dm}{dt}=K_1\cdot S\cdot(1-\Theta)-K_2^{*}\cdot S\cdot\Theta \tag{7} \]
Let us express in (7) the proportionality coefficient \(K_2^*\) through \(K_1\). As \(t \to \infty\), \(\dfrac{dm}{dt}=0\). Under these conditions, according to (7),
\[ K_1 \cdot S_\infty (1-\Theta_\infty)=K_2^* \cdot S_\infty \cdot \Theta_\infty \]
and
\[ K_2^* = K_1 \frac{1-\Theta_\infty}{\Theta_\infty}. \tag{8} \]
In the case where the ideal-gas laws are applicable to the mixture in the bubbles of an a.a. drop, for \(m\) one may write
\[ m=\frac{P}{1.5RT}\cdot \frac{4\pi}{3}\left(r^3-r_0^3\right) \tag{9} \]
and
\[ m_\infty=\frac{P}{1.5RT}\cdot \frac{4\pi}{3}\left(r_\infty^3-r_0^3\right), \tag{10} \]
where \(r_0\), \(r_\infty\), and \(r\) are the radii of the a.a. drop, respectively, at the times \(t_0\), \(t_\infty\), and \(t\). The remaining notation is generally accepted.
Taking (8) into account, expression (7) may be written differently:
\[ \frac{dm}{dt} =K_1\cdot S\cdot(1-\Theta) -K_1\frac{1-\Theta_\infty}{\Theta_\infty}\cdot S\cdot \Theta \tag{11} \]
or
\[ \frac{dm}{dt}=K_1\cdot S\left(1-\frac{\Theta}{\Theta_\infty}\right). \]
Taking (9) and (10) into account, expression (11) may be simplified:
\[ \frac{dr}{1-\dfrac{\Theta}{\Theta_\infty}} = K\,dt, \tag{12} \]
where
\[ K=\frac{1.5RT}{P}\cdot K_1 . \tag{13} \]
In expression (12), the degree of filling \(\Theta\) is a function of the electrolysis time and, correspondingly, of the electrode dimensions.
In order to relate the quantity \(\Theta\) to the electrode dimensions, one may proceed as follows. In [1], two series of photographs of an a.a. drop were obtained at different intervals of electrolysis time. From these data, \(\Theta\) can be estimated independently and this quantity can be related to the radius of the a.a. drop. We made such an estimate. On the basis of these data the following relation was obtained, connecting the degree of filling \(\Theta\) with the electrode dimensions:
\[ \Theta=K_s\left(1-\frac{S_0}{S}\right), \tag{14} \]
where \(K_s\) is a proportionality coefficient, a quantity independent of the electrode dimensions.
The value of \(K_s\), found on the basis of (14), is equal to 0.181.
Formula (14) satisfies the limiting cases that follow from analysis of the kinetics of the process in the presence of the ammonium amalgam effect. Indeed, when \(t\to 0\) and \(S\to S_0\), it follows from formula (14) that \(\Theta\to 0\). Conversely, when \(t\to \infty\) and \(S\to S^\infty\), according to (14) \(\Theta\to \Theta^\infty\).
Taking (14) into account, expression (12), after the corresponding transformations, can be reduced to the form
\[ \frac{r^2dr}{r_\infty^2-r^2}=K^*dt, \tag{15} \]
where
\[ K^*=\frac{r_0^2}{r_\infty^2-r_0^2}\cdot K, \tag{16} \]
or, after integration, we have
\[ r=r_{\infty}\, \frac{e^{\frac{2}{r_{\infty}}(K^{*}t-r_{0})}-\xi e^{-\frac{2}{r_{\infty}}r}} {e^{\frac{2}{r_{\infty}}(K^{*}t-r_{0})}+\xi e^{-\frac{2}{r_{\infty}}r}} \]
where
\[ \xi=\frac{r_{\infty}-r_{0}}{r_{\infty}+r_{0}} . \tag{18} \]
Introducing the dimensionless radius \(\rho=\dfrac{r}{r_{\infty}}\) and \(\rho_{0}=\dfrac{r_{0}}{r_{\infty}}\), expression (17) can be written differently:
\[ \frac{\xi}{e^{-2\rho_{0}}}\cdot \frac{1+\rho}{1-\rho}\,e^{-2\rho}=e^{\tau} \tag{19} \]
where
\[ \tau=\frac{2}{r_{\infty}}\,K^{*}\cdot t . \tag{20} \]
Expressions (17) and (19) satisfy the limiting cases that follow from analysis of the kinetics of the process under EAA conditions.
The theoretical curve calculated from equation (19) is presented in Fig. 1. To compare the theory with experiment, by analogy with [1] one can proceed as follows. Let us estimate the angular coefficient \(\dfrac{dr}{dt}\), using expression (15), at \(t=\Theta\) and at any other value of the electrolysis time:
Fig. 1. Dependence of the radius of an ammonia amalgam droplet on electrolysis time.
\[ \left(\frac{dr}{dt}\right)_{t=0} =K^{*}\left(r_{\infty}^{2}\cdot r_{0}^{-2}-1\right), \tag{21} \]
\[ \left(\frac{dr}{dt}\right)_{t} =K^{*}\left(r_{\infty}^{2}\cdot r^{-2}-1\right). \tag{22} \]
The ratio of the angular coefficients will be equal to
\[ A=\frac{r_{\infty}^{2}\cdot r_{0}^{-2}-1} {r_{\infty}^{2}\cdot r^{-2}-1} = \frac{\rho_{0}^{-2}-1}{\rho^{-2}-1}. \tag{23} \]
From (26), after minor transformations one can obtain
\[ \lg r=\lg r_{\infty}+\tfrac{1}{2}\lg y, \tag{24} \]
where
\[ y=\frac{A}{A+\rho_0^{-2}-1}. \tag{25} \]
According to (24), the plot in the coordinates \(\lg r,\ \lg y\) should give a straight line with a slope equal to \(1/2\) and an intercept on the ordinate axis equal to \(\lg r_\infty\).
Using the experimental data on the dependence of the radius of a drop of a.a. on the electrolysis time, obtained in [1], we constructed a plot in the coordinates \(\lg r,\ \lg y\). This plot is shown in Fig. 2. The data of
Fig. 2. Dependence of \(\lg r\) on \(\lg y\).
Fig. 2 are in complete agreement with the conclusions following from the analysis of expression (24).
Let us obtain an expression for the decrease in the radius of a drop of a.a. after the electrolysis current is switched off. In this case, since \(i_3=0\), from (11) we have
\[ -\frac{dm}{dt}=K_1\frac{1-\Theta_\infty}{\Theta_\infty}\,S\cdot\Theta . \tag{26} \]
Substituting into (26), instead of \(\Theta\), its value from (14), and taking (9) into account, one can obtain
\[ \frac{r^2\,dr}{r_0^2-r^2}=\alpha\cdot dt, \tag{27} \]
where
\[ \alpha=K^0\cdot r_\infty^2, \tag{28} \]
\[ K^0=K\frac{1-\Theta_\infty}{r_\infty^2-r_0^2} \tag{29} \]
After integrating (27) in the limits from \(r_\infty\) to \(r\), we have
\[ -r+\frac{r_0}{2}\ln \xi\frac{r+r_0}{r-r_0}=\alpha t_*-r_\infty, \tag{30} \]
where
\[ t_*=t-t_{\mathrm{n}} \tag{31} \]
is the time elapsed since the electrolysis current was switched off. In exponential form, expression (30) has the form
\[ r=r_0\, \frac{ e^{\frac{2}{r_0}(t_*\alpha-r_\infty)}+\xi e^{-\frac{2}{r_0}r} }{ e^{\frac{2}{r_0}(t_*\alpha-r_\infty)}-\xi e^{-\frac{2}{r_0}r} }. \tag{32} \]
Introducing the dimensionless radius \(\rho^*=r/r_0\), from (32) we finally obtain
\[ \frac{\xi}{e^{-\frac{2r_\infty}{r_0}}}\, \frac{1+\rho^*}{1-\rho^*}\,e^{-2\rho^*} = e^{\tau^0}, \tag{33} \]
where
\[ \tau^0=\frac{2}{r_0}\alpha\cdot t_* . \tag{34} \]
Equation (33) also satisfies the limiting cases following from the analysis of the kinetics of the process in the presence of the ammonia amalgam effect.
For \(t=0\) or \(t=t_{\mathrm{n}}\), it follows from (33) that \(r=r_{\infty}\). For \(t_{*}\to\infty\) or \(t\gg t_{\mathrm{n}}\), \(r\to r_{0}\).
Figure 3 presents the theoretical curve of the dependence of \(\rho^{*}\) on \(\tau\),
Fig. 3. Dependence of the radius of an ammonia amalgam droplet on time after switching off the electrolysis current.
calculated from equation (33). According to these data, after the electrolysis current is switched off the radius of an a.a. droplet will exceed \(r_{0}\), for example, by a factor of 2 at \(\tau^{0}=6.18\) and by a factor of 3 at \(\tau^{0}=3.76\).
REFERENCES
- A. V. Konkova. Dissertation. Tomsk, 1969.