On the Theory of the Resolving Power of Amalgam Polarography with Accumulation on a Mercury Film Electrode
Abstract Generated abstract
This paper develops a theoretical estimate of the resolving power of amalgam polarography with accumulation on a mercury film electrode, a method noted for greater sensitivity and separation capability than the stationary mercury drop. Considering the limiting case of a thin mercury film, slow linear potential sweep, and an irreversible electrode process controlled by electrochemical reaction rate, the authors derive analytical expressions for the anodic dissolution current, peak current, and resolving power coefficient. The resulting formula relates resolving power to the difference in peak potentials, charge-transfer coefficient, and number of electrons, while indicating that under these assumptions it is independent of diffusion coefficient, sweep rate, and mercury film thickness.
Full Text
ON THE THEORY OF THE RESOLVING POWER OF THE METHOD OF AMALGAM POLAROGRAPHY WITH ACCUMULATION ON A MERCURY FILM ELECTRODE
A. G. Stromberg, V. F. Yankauskas
(Presented by the scientific seminar of the Department of Physical Chemistry)
A mercury film electrode [1, 2] is finding ever wider application in the method of amalgam polarography with accumulation (APA), owing to its higher sensitivity and resolving power in comparison with the stationary mercury drop.
By the resolving power \((j)\) of the APA method, for a given difference in the potentials \((\Delta \varphi_{12})\) of the two elements, one should understand the ratio of the maximum current of anodic dissolution (the peak current, \(I_1\)) of the interfering element to the peak current \((I_2)\) of the element being determined, at a specified error \((g)\) arising from the superposition of the peak current \((I_1)\) of the element being determined on the current \((i_1)\) of the descending branch of the anodic-dissolution current of the interfering element, if the interfering element is more electropositive, or on the current of the ascending branch of the anodic-dissolution current of the interfering element, if the interfering element is more electronegative (in comparison with the element being determined). Hence:
\[ j = \frac{I_1}{I_2} = \frac{I_1}{I_2}\cdot \frac{i_1}{i_1} = \frac{I_1}{i_1}\cdot \frac{i_1}{I_2} = P \cdot g. \tag{1} \]
\[ P = \frac{I_1}{i_1}; \qquad g = \frac{i_1}{I_2}. \tag{2} \]
where \(P\) is the coefficient of resolving power.
Thus, the problem of the theoretical calculation of the resolving power \(j\) reduces to deriving a mathematical relation for the dependence of the coefficient of resolving power \(P\) on the difference in peak potentials \(\Delta \varphi_{12}\) of the two elements and on other factors. For this it is necessary to know the theoretical expressions for the peak current and the current of the descending branch on a mercury film electrode.
At present, a rigorous expression for the current of anodic dissolution of a metal from an amalgam film has been obtained only in integral form [1, 3, 4], and numerical calculations have been made only for the peak current [3, 4]. Therefore, in the present article the limiting case of an irreversible electrode process will be considered, for which it is possible to obtain a comparatively simple analytical expression when the dissolution of the metal from the amalgam film is limited only by the rate of the electrochemical reaction. In this case the concentration of metal atoms at the surface of the amalgam may be taken as approximately equal to the concentration of the metal inside the amalgam film. And the distribution of ions
metal in the solution in a completely irreversible process will not affect the kinetics of the electrode process. For the case under consideration one can, obviously, write
\[ i=z\cdot F\cdot S\cdot K_b\cdot C_a^s,\qquad K_b=K_s e^x;\qquad C_a^s=C_a=\frac{q}{z\cdot F\cdot v}; \tag{3} \]
\[ x=\frac{\beta}{b^1}(\varphi-\varphi^\circ)=\frac{\beta}{b^1}Wt;\qquad b^1=\frac{RT}{zF}, \tag{4} \]
where \(K_b\), \(K_s\) are the rate constants of the electrochemical reaction, respectively, at potentials \(\varphi=0\) and \(\varphi=\varphi^\circ\); \(\varphi^\circ\) is the standard potential; \(C_a^s\) is the concentration of metal atoms at the surface of the amalgam, \(g\text{-atom}/\text{cm}^3\), equal in the case considered to the mean concentration \(C_a\) of the metal in the amalgam; \(q\) is the quantity of electricity corresponding to the amount of metal that still remains in the amalgam at time \(t\); \(t\) is the time of anodic dissolution of the metal under a linear change of potential, reckoned from the standard potential; \(v\), \(S\) are the volume and surface area of mercury in the film electrode; \(\beta\) is the charge-transfer coefficient in the anodic process; \(W\) is the rate of change of potential, volts/sec; \(z\) is the number of electrons participating in the electrode process per one metal atom; \(F\), \(R\), \(T\) have their usual meaning. Taking into account that \(i=-dq/dt\) and \(v/S=l\), from (3) and (4) we obtain
\[ \frac{dq}{q}=-Ne^x dx;\qquad N=\frac{RT}{F}\frac{1}{l\cdot W}\cdot \frac{K_s}{z\cdot \beta} \tag{5} \]
(\(l\) is the thickness of the amalgam film). Integration of equation (5) gives
\[ q=q_0 e^{-N(e^x-1)}. \tag{6} \]
Substituting (6) into (3), we obtain*
\[ \left. \begin{aligned} i&=M\exp\!\left[x-N(e^x-1)\right];\qquad M=\frac{K_s}{l}q_0,\\[4pt] \ln i&=\ln M+x-N(e^x-1). \end{aligned} \right\} \tag{7} \]
Equation (7) gives the expression for the descending branch of the current of anodic dissolution of the metal from the amalgam film at \(\varphi>\varphi_n\), and the expression for the ascending branch at \(\varphi<\varphi_n\) (\(\varphi_n\) is the potential of the anodic peak). In order to find the value \(\varphi=\varphi_n\), we set the derivative \(\partial \ln i/\partial x=0\) equal to zero. We then obtain
\[ e^{x_n}=\frac{1}{N};\qquad x_n=-\ln N. \tag{8} \]
The expression for the peak current is obtained by substituting formula (8) into (7) and expressing \(M\) and \(N\) from relations (5) and (7):
\[ \ln I=\ln \frac{M}{eN}+N;\qquad I=\frac{M}{eN}e^N \tag{9} \]
\[ I=\frac{F}{RT}\frac{\beta\cdot z\cdot W}{e}\, q_0 \exp\!\left[\frac{RT}{F}\frac{1}{l\cdot W}\cdot \frac{K_s}{z\cdot \beta}\right]. \tag{10} \]
The area \(q_0\) under the \(i,\varphi\) curve during anodic dissolution of the metal, which enters into formula (10), can be expressed, in the absence of depletion, by means of the obvious relations
\[ q_0=z\cdot F\cdot C_a\cdot t=z\cdot F\cdot C_a\cdot V\cdot B=i_c t_c=K_c\cdot S\cdot C_u\cdot t_c \tag{11} \]
* A similar equation in another form and by another method was obtained by Brainina [5] for thin solid films under the assumption that the activity of the metal in the film is proportional to the amount of metal in the thin film, and by Itoginskii [6] for a mercury film.
or in the presence of depletion of the solution during the preliminary electrolysis [7]:
\[ \left. \begin{gathered} q_0=z\cdot F\cdot C_n\cdot v=z\cdot F\cdot C_n\cdot V\cdot \gamma=\int_0^{t_e} i_e\,dt \\ \gamma=1-e^{-B};\qquad B=\frac{K_e}{z\cdot F}\cdot \frac{S t_e}{V} \end{gathered} \right\}, \tag{12} \]
where \(\gamma\) is the degree of depletion of the solution; \(B\) is a dimensionless parameter characterizing the degree of depletion of the solution; \(i_e\) is the electrolysis current from a stirred solution at a constant potential corresponding to the limiting current of cathodic deposition of the metal; \(K_e\) is the electrolysis-current coefficient; \(V\) is the volume of the solution; \(C_n\) is the concentration of ions in the solution, g-ion/cm\(^3\); \(t_e\) is the duration of electrolysis, sec.
Taking relation (8) into account, we transform formula (9):
\[ \ln I=\ln M+x_n-Ne^{-x_n}. \tag{13} \]
From formulas (7) and (14), taking (4) into account, we obtain an expression for the resolving-power coefficient
\[ \lg P=\lg \frac{I}{i}=0.434\ \exp\left[\frac{F}{RT}y\right]-\frac{F}{2.3RT}\cdot y-0.434, \tag{14} \]
where
\[ y=\beta\cdot z\cdot \Delta\varphi_{12}=\beta z(\varphi_{n3}-\varphi_{n1})>1. \]
It follows from formula (15) that the resolving-power coefficient at a given temperature depends only on one parameter, \(y\), i.e., on the difference of the peak potentials, the ionization coefficient \(\beta\), and the number of electrons \(z\) (on the charge of the metal ions). It is interesting to note that, in the limiting case under consideration, unlike the mercury drop, the coefficient \(P\) does not depend on the parameter \(\varepsilon=(D/z\cdot W\cdot l^2)^{1/2}\), i.e., it does not depend on the diffusion coefficient of the metal in mercury \(D\), the rate of change of potential \(W\), or the thickness of the mercury film \(l\).
Calculating from formula (14) the resolving-power coefficient \(P\) for given \(\beta\), \(z\), and \(\Delta\varphi_{12}\) (i.e., for a given \(y\)), one can, by formula (1), estimate the resolving power of the method of amalgam polarography with accumulation on a mercury film electrode.
Conclusions
For the limiting case of a thin mercury film and a slow linear change of potential in an irreversible electrode process, a theoretical expression has been obtained for the resolving power of the method of amalgam polarography with accumulation.
References
- V. A. Igolinskii, Dissertation, TPI, Tomsk, 1963.
- V. A. Igolinskii, A. G. Stromberg. In the collection Methods of Analysis of Chemical Reagents. All-Union Scientific Research Institute of Chemical Reagents and Especially Pure Substances, issue 5–6, Moscow, 1963.
- W. De-Vries, E. van Dalen. J. Electroanal. Chem. 8, 366, 1964.
- W. De-Vries. J. Electroanal. Chem. 9, 448, 1965.
- Kh. Z. Brainina, E. M. Roizenblat, V. B. Belyavskaya. Zavod. labor., 28, 1047, 1962.
- V. A. Igolinskii, A. G. Stromberg. Izv. TPI (in press).
- A. G. Stromberg. Izv. SO AN, No. 5, 76, 1963.