EFFECT OF TEMPERATURE ON THE REVERSIBILITY OF ANODIC AND CATHODIC PEAKS IN THE APN METHOD

Yu. A. Karbainov, L. A. Chernova, S. N. Karbainova

(Presented by the scientific seminar of the Department of Analytical Chemistry)

The effect of temperature on the characteristics of anodic peaks in the method of amalgam polarography with accumulation (the APN method) has been studied in detail in [1]. However, questions connected with the effect of temperature on the reversibility of anodic and cathodic peaks still remain poorly studied. The present work is devoted to investigations in this area.

According to [2], the dependence of the coefficient of an irreversible anodic peak on temperature may be described by an equation of the form

\[ K_a^* = K_{aT}^* \cdot \exp\left(-\frac{E_a}{RT}\right), \tag{1} \]

where \(E_a\) is the activation energy of the electrode process at the potential of the anodic peak.¹

If the pre-electrolysis stage in the APN method is carried out under identical conditions, thereby ensuring constancy of the concentration of metal atoms in the amalgam, while the anodic dissolution is carried out at different temperatures of the solution being analyzed, then all changes in the height and shape of the anodic peaks obtained in this way will be connected with the influence of temperature on the rate-limiting stage in the anodic cycle through the value \(K_a^*\). In this case, if the change in \(K_a^*\) with temperature is related to the corresponding change in the theoretical value of the coefficient of a reversible anodic peak, the degree of reversibility of anodic peaks at different temperatures can be estimated.

Under the working conditions of the APN method, the dependence of the coefficient of a reversible anodic peak on temperature is given by the relations [3]:

\[ K_a = K_0^* \frac{T^{-1/2}}{A + l - \dfrac{E_{DR}}{2RT}} \quad \text{(stationary mercury drop electrode);} \tag{2} \]

where

\[ K_0^* = 0.446 \frac{z^{3/2} F^{3/2}}{R^{1/2}} \cdot \omega^{1/2} \cdot D_{RT}^{1/2}, \tag{3} \]

\[ A = 66.9 \frac{R^{1/2}}{z^{1/2} F^{1/2}} \cdot \frac{1}{r \omega^{1/2}} \cdot D_{RT}^{1/2}. \tag{4} \]

constants independent of the change in temperature.

\[ K_a = H_T \cdot \frac{1}{T} \quad \text{(Hg film electrode),} \tag{5} \]

where

\[ H_T = 0.333 \frac{z^2 F^2}{R} \cdot l \cdot \omega . \tag{6} \]

In (3)—(6), \(r\) is the radius of the stationary dropping electrode, cm; \(l\) is the thickness of the Hg film electrode, µm; \(\omega\) is the rate of change of potential, V/sec; the remaining notation is standard.

Using (1), (2), and (5), one can obtain

\[ \frac{K_a - K_a^*}{K_a} = \Theta \frac{ \exp\left(\dfrac{E_a}{RT}\right) }{ A + \exp\left(-\dfrac{E_{DR}}{2RT}\right) } \cdot T^{-1/2} - 1 \quad \text{(stationary Hg dropping electrode),} \tag{7} \]

where

\[ \Theta = \frac{K_0^*}{K_a^* \cdot T}, \tag{8} \]

\[ \frac{K_a - K_a^*}{K_a^*} = \frac{ B \exp\left(\dfrac{E_a}{RT}\right) }{T} - 1 \quad \text{(Hg film electrode),} \tag{9} \]

where

\[ B = \frac{H_T}{K_a^* \cdot T}. \tag{10} \]

In logarithmic form, expressions (7) and (9), respectively, have the form:

\[ \lg \Phi = \lg \Phi_{1/2} + \lg \frac{K_a - K_a^*}{K_a^*}, \tag{11} \]

\[ \lg P = \lg P_{1/2} + \lg \frac{K_a - K_a^*}{K_a^*}, \tag{12} \]

where

\[ \Phi = \Theta \frac{ \exp\left(\dfrac{E_a}{RT}\right) }{ A + \exp\left(-\dfrac{E_{DR}}{2RT}\right) }, \tag{13} \]

\[ \lg \Phi_{1/2} = \frac{1}{2} \lg T_{1/2}, \tag{14} \]

\[ P = B \exp\left(\frac{E_a}{RT}\right) - T, \tag{15} \]

\[ \lg P_{1/2} = \lg T_{1/2}. \tag{16} \]

In expressions (14) and (16), \(T_{1/2}\) denotes the temperature at which the quantity \(K_a\) reaches a value equal to one half of the coefficient of the reversible anodic peak, that is,

\[ K_a^* = \frac{1}{2} K_a. \]

Let us analyze the limiting cases of (11) and (12). When \(\dfrac{K_a^*}{K_a} \ll 0.05\), then, with an error not exceeding 5%, it may be assumed that the anodic peak is practically irreversible. In this case

\[ \lg \frac{K_a-K_a^*}{K_a^*}=1.28. \]

Conversely, when, with a change in temperature, the ratio reaches a value equal to \(\dfrac{K_a^*}{K_a} \ll 0.95\), the anodic peak becomes practically reversible. In this case

\[ \lg \frac{K_a-K_a^*}{K_a^*}=-1.28. \]

Thus, on the basis of the temperature dependence of the anodic-peak coefficient, the following method can be proposed for investigating the reversibility of anodic peaks in APN. From an anodic peak recorded under certain conditions (electrode dimensions, rate of potential change, temperature), the value of the anodic-peak coefficient is estimated. Under these same conditions, using (2) and (5), the theoretical value of the anodic-peak coefficient is estimated. The quantity

\[ \lg \frac{K_a-K_a^*}{K_a} \]

is evaluated. If

\[ \lg \frac{K_a-K_a^*}{K_a^*} \gg 1.28, \]

the anodic peak is practically irreversible; if

\[ -1.28 \leq \lg \frac{K_a-K_a^*}{K_a^*} \leq 1.28, \]

it is quasi-reversible; if

\[ \lg \frac{K_a-K_a^*}{K_a^*} \ll -1.28, \]

it is practically reversible (with an error not exceeding 5%).

Using (7) and (9), we shall make some numerical estimates. Let us estimate at what value of the quantity \(E_a\) one can, solely by changing the temperature from 20 to \(80^\circ\)C, pass from an irreversible to a practically reversible anodic peak.

According to the condition, at \(20^\circ\)C the anodic peak is practically irreversible, that is,

\[ \frac{K_a-K_a^*}{K_a^*}=19. \]

In this case, at \(E_{DR}=10^3\ \text{cal/mol}\), for a stationary mercury drop electrode we have

\[ \frac{\exp\left(\dfrac{E_a}{586}\right)}{A+0.43}\cdot \Theta =342, \]

and for a mercury film electrode

\[ B\cdot \exp\left(\frac{E_a}{586}\right)=5863. \]

At \(80^\circ\)C the anodic peak, according to the condition, is practically reversible. In this case, for a stationary mercury drop electrode we obtain

\[ \frac{\exp\left(\dfrac{E_a}{706}\right)}{A+0.49}=19.73, \]

and for a mercury film electrode

\[ B\cdot \exp\left(\frac{E_a}{706}\right)=370.7. \]

Solving the obtained relations with respect to \(E_a\), we finally obtain for a stationary mercury drop electrode \(E_a=9800\ \text{cal/mol}\); for a mercury film electrode—\(E_a=9500\ \text{cal/mol}\).

We calculated the dependence of $\dfrac{K_a^*}{K_a}$ at 80°C on the value of $E_a$, found from expressions (7) and (9). In this case, at 20°C the ratio $\dfrac{K_a^*}{K_a}$ was taken to be

\[ \frac{K_a^*}{K_a}=0.05. \]

From analysis of the data obtained it follows:

1. The change in the ratio $\dfrac{K_a^*}{K_a}$ with temperature depends mainly on the value of $E_a$ and is practically independent of the conditions under which the experiment is carried out. This is also confirmed by data on the dependence of $\dfrac{K_a^*}{K_a}$ at 80°C on the value of $E_a$ under conditions of linear semi-infinite diffusion.

2. At $E_a = 10 \cdot 10^3$ cal/mol, anodic peaks that are practically irreversible at 20°C become practically reversible at 80°C. In this case the height of the anodic peak, and consequently the sensitivity of the APN method, increases by a factor of 19.

The influence of temperature on the reversibility of anodic peaks can also be considered through the change in the effective width of half-peaks with temperature. According to [2], the dependence of the effective width of reversible anodic peaks on temperature under the working conditions of APN has the form

\[ b_{1/2}^* = 3 \frac{R}{zF} T \quad \text{(for a mercury film electrode);} \tag{17} \]

\[ b_{1/2}^* = a \frac{T^{1/2}}{\exp\left(\dfrac{E_{DR}}{2RT}\right)} + \rho T^{1/2} \quad \text{(for a stationary mercury drop electrode).} \tag{18} \]

Here

\[ a = 0.747 \frac{R^{1/2}}{z^{1/2} F^{1/2}}\, r \cdot \omega^{1/2} \cdot \frac{1}{D_{R,T}^{1/2}} \tag{19} \]

and

\[ \rho = 50 \frac{R}{zF}. \tag{20} \]

The dependence of the effective width of irreversible anodic half-peaks on temperature can be obtained using expression (1). For a mercury film electrode

\[ b_{1/2}^{(*)} = zFl\omega \frac{1}{K_a^*} = m \cdot \exp \frac{E_a}{RT}, \tag{21} \]

where

\[ m = \frac{zFl \cdot \omega}{K_{a,T}^*}. \tag{22} \]

For a stationary mercury drop electrode

\[ b_{1/2}^{(*)} = zF\omega \cdot \frac{2}{3} \cdot \frac{1}{K_a^*} = n \cdot \exp\left(\frac{E_a}{RT}\right), \tag{23} \]

where

\[ n = \frac{zF}{3K_{a,T}^*}\, r \cdot \omega. \tag{24} \]

From (17), (18), (21), and (23) it follows that

\[ \frac{b_{1/2}^{(*)}-b_{1/2}^{*}}{b_{1/2}^{*}} = NT^{-1}\exp\left(\frac{E_a}{RT}\right)-1 \quad \text{(for a mercury film electrode)} \tag{25} \]

and

\[ \frac{b_{1/2}^{(*)}-b_{1/2}^{*}}{b_{1/2}^{*}} = \frac{ n\cdot \exp\left(\frac{E_a}{RT}\right) }{ \alpha\cdot \frac{T^{1/2}}{\exp\left(\frac{E_{DR}}{2RT}\right)+\rho T^{1/2}} } -1 \quad \text{(for a stationary mercury dropping electrode).} \tag{26} \]

In (25)

\[ N=\frac{z^2F^2}{3R}\cdot l\cdot \omega^2\cdot \frac{1}{K_{a,T}^{*}}. \tag{27} \]

The conclusions following from an analysis of expressions (25) and (26) are analogous to those given above from (11) and (12).

Let us consider the effect of temperature on the reversibility of cathodic peaks in the APN method. For the coefficient of reversible cathodic peaks one may write:

\[ K_{\mathrm{e}}=K_{\mathrm{e}}^{0}\cdot D_0^{1/2}. \tag{28} \]

Or, as a function of temperature,

\[ K_{\mathrm{e}}=K_{\mathrm{e},T}\cdot \exp\left(-\frac{ED_0}{2RT}\right). \tag{29} \]

We assume that the dependence of the coefficient of irreversible cathodic peaks on temperature is described by the equation:

\[ K_{\mathrm{e}}^{*}=K_{\mathrm{e},T}^{*}\cdot \exp\left(-\frac{E_{\mathrm{e}}}{RT}\right), \tag{30} \]

where \(E_{\mathrm{e}}\) is the activation energy of the electrode process at the potential of the cathodic peak, cal/mol. Solving (29) and (30) together, we obtain

\[ \frac{K_{\mathrm{e}}-K_{\mathrm{e}}^{*}}{K_{\mathrm{e}}^{*}} = \sigma\cdot \exp\left(\frac{2E_{\mathrm{e}}-ED_0}{2RT}\right)-1, \tag{31} \]

where

\[ \sigma=\frac{K_{\mathrm{e},T}}{K_{\mathrm{e},T}^{*}}, \tag{32} \]

\[ \lg \frac{K_{\mathrm{e}}}{K_{\mathrm{e}}^{*}} = \lg \sigma+ \frac{2E_{\mathrm{e}}-ED_0}{4.6R}\cdot \frac{1}{T}. \tag{33} \]

Let us estimate the value of \(E_{\mathrm{e}}\) that is necessary in order to pass from irreversible cathodic peaks at \(20^\circ\text{C}\) to practically reversible cathodic peaks at \(t=80^\circ\text{C}\). From (31) it follows that

\[ \frac{K_{\mathrm{e}}}{K_{\mathrm{e}}^{*}} = \sigma \exp\left(\frac{2E_{\mathrm{e}}-ED_0}{2RT}\right). \tag{34} \]

We shall regard the cathodic peak as practically irreversible if \(K_{\mathrm{e}}^{*}/K_{\mathrm{e}}\ll 0.05\). Conversely, it is practically reversible if \(K_{\mathrm{e}}^{*}/K_{\mathrm{e}}\leq 0.95\). Proceeding from these considerations and using relation (34), we find that the value of \(E_{\mathrm{e}}\) in this case must be equal to \(E_{\mathrm{e}}=3.5\cdot 10^3\) cal/mol (if \(ED_0=5\cdot 10^3\) cal/mol) and \(E_{\mathrm{e}}=6\cdot 10^3\) cal/mol (if \(ED_0=10\cdot 10^3\) cal/mol).

We calculated the dependence of \(K_{\mathrm{e}}^*/K_{\mathrm{e}}\) at \(80^\circ\mathrm{C}\) on the activation energy \(E_{\mathrm{e}}\), found from (34). In this case, as before, at \(20^\circ\mathrm{C}\), \(K_{\mathrm{e}}^*/K_{\mathrm{e}}=0.05\).

It follows from an analysis of these data, in particular, that the higher the value of \(E_{D_0}\), the weaker the effect of temperature on the change in the reversibility of cathodic peaks (for a given value of \(E_{\mathrm{e}}\)).

REFERENCES

1. S. N. Karbainova. Abstract of Candidate’s dissertation, Tomsk, 1971.
2. J. Heyrovský, J. Kůta. Principles of Polarography. Moscow, “Mir,” 1965, p. 203.
3. Yu. A. Karbainov, S. N. Karbainova, A. G. Stromberg. “Electrokhimiya,” 10, 1409 (1971).

1. Here and below, we shall neglect the change in the value \(E_a\) caused by the change in the peak potential with temperature. ↩