On the Nature of the Isotope Effect in the Electrosynthesis of Ozone
Abstract Generated abstract
This paper examines the isotope effect observed during ozone formation from oxygen in an electric discharge, where ozone becomes enriched in oxygen-18 and the separation coefficient increases with temperature. The authors argue that the effect cannot be explained by equilibrium isotope exchange between oxygen and ozone, and instead calculate kinetic isotope effects for the elementary stages of ozone formation and decomposition using the accepted discharge mechanism and approximate transition-complex assumptions. The analysis indicates that early dissociation and association steps favor the light isotope in ozone, while ozone decomposition enriches the remaining ozone in the heavy isotope. The resulting model identifies decomposition as the controlling stage and gives satisfactory agreement with the reported temperature dependence of the experimental separation coefficient.
Full Text
ON THE NATURE OF THE ISOTOPE EFFECT IN THE ELECTROSYNTHESIS OF OZONE
V. N. MELENEVSKII, I. A. TIKHOMIROV
(Presented by the scientific seminar of the Faculty of Physics and Technology)
In works [1, 2, 3, 4] it was established that, in the synthesis of ozone in an electric discharge, enrichment of ozone with the heavy oxygen isotope \(O^{18}\) occurs, and the supposition was made [1] that separation of the oxygen isotopes occurs as a result of exchange between oxygen and ozone according to the reaction:
\[ 2O^{16}_{3}+3O^{18}_{2}\rightleftarrows 2O^{18}_{3}+3O^{16}_{2}, \tag{1} \]
However, as is evident from the statistical calculation carried out in work [5], the experimental separation coefficient does not coincide with that calculated for reaction (1); moreover, the resulting dependence proved to be negative, whereas experiment shows that, with an increase in temperature from \(254^\circ\mathrm{K}\) to \(365^\circ\mathrm{K}\), the separation coefficient increases from 1.03 to 1.14, respectively. This shows that the overall isotope effect observed in electrosynthesis is not due to equilibrium isotope effects of exchange reactions of type (1), but is determined, in all probability, by isotope effects of the individual stages of ozone formation and decomposition, proceeding through the breaking of old bonds and the formation of new bonds in oxygen and ozone molecules.
In the present article a calculation is given of the overall separation coefficient for the reaction of ozone formation from oxygen in an electric discharge.
In the calculation we proceeded from the generally accepted scheme of the mechanism of ozone formation [6, 7], according to which the first act of ozone formation is the process of oxygen dissociation as a result of electron impact into excited and unexcited atoms:
\[ O_{2}+e\to O(p)+O(^{1}D)+e. \tag{2} \]
The oxygen atoms arising as a result of dissociation form ozone analogously to Lindemann’s scheme according to the following reactions:
\[ O_{2}+O\to O^{*}_{3}, \tag{3} \]
\[ O^{*}_{3}\to O_{2}+O, \tag{4} \]
\[ O^{*}_{3}+M\to O_{3}+M, \tag{5} \]
where \(O^{*}_{3}\) denotes an excited ozone molecule, and \(M\) is an oxygen molecule or the wall of the ozonizer.
Ozone decomposition may have either a purely thermal character,
\[ \mathrm{O}_3+\mathrm{O}\to 2\mathrm{O}_2, \tag{6} \]
or proceed through collision with an electron
\[ \mathrm{O}_3+e\to \mathrm{O}_2+\mathrm{O}+e. \tag{7} \]
To calculate the kinetic isotope effects of the individual stages, the formula for the ratio of the rate constants of two isotopic substances [8] was used:
\[ \frac{k_1}{k_2}\cdot \frac{s_2}{s_1}\cdot \frac{S_1^{+}}{S_2^{+}} = \frac{\vartheta^{+}_{L(1)}}{\vartheta^{+}_{L(2)}}\cdot \left( 1-\sum^{3n^{+}-7} G(u_i^{+})\,\Delta u_i^{+} +\sum^{3n-6} G(u_i)\,\Delta u_i \right), \tag{8} \]
where subscript 1 refers to the heavier isotope, and subscript 2 to the lighter isotope; the parameters of the transition complex are denoted by the superscript \(+\).
The initial data for the calculation, such as the fundamental vibrational frequencies of the isotopic molecules of oxygen and ozone, as well as the values of the Bigeleisen function \(G(u)\), were taken from works [5] and [9], respectively. Since information on the configuration and force constants of the transition complexes of reactions (2)—(7) is lacking, as a simplification we used the approximation that each individual elementary reaction corresponds to a certain average kinetic isotope effect
\[ \alpha=\frac{\alpha_{\max}+\alpha_{\min}}{2}, \tag{9} \]
where
\[ \alpha_{\max}= \frac{\vartheta^{+}_{L(1)}}{\vartheta^{+}_{L(2)}}\cdot \left(1+\sum^{3n-6}G(u_i)\,\Delta u_i\right) \tag{10} \]
corresponds to the maximum kinetic effect, i.e., to a transition complex containing the atom of the isotope in the free state and
\[ \sum^{3n-7}G(u_i^{+})\,\Delta u_i^{+}=0, \]
whereas
\[ \alpha_{\min}= \frac{\vartheta^{+}_{L(1)}}{\vartheta^{+}_{L(2)}} \tag{11} \]
corresponds to the case
\[ \sum^{3n^{+}-7}G(u_i^{+})\,\Delta u_i^{+} = \sum^{3n-6}G(u_i)\,\Delta u_i, \]
i.e., to a transition complex whose vibrational part of the function does not change.
The temperature-independent factor, according to work [8], was calculated from the equation
\[ \frac{\vartheta^{+}_{L(1)}}{\vartheta^{+}_{L(2)}}= \left(\frac{m_2^{*}}{m_1^{*}}\right)^{1/2} = \left[ \frac{\dfrac{1}{M_1'}+\dfrac{1}{M_1''}} {\dfrac{1}{M_2'}+\dfrac{1}{M_2''}} \right]^{1/2}, \tag{12} \]
where \(m^{*}\) denotes the reduced mass of two fragments of the molecule connected by the bond that is broken in the course of the reaction, and \(M'\) and \(M''\) are the masses of the fragments,
Using the above-listed equations and literature data, one can calculate the values of the kinetic isotope effects \(\alpha_1, \alpha_2, \alpha_3\) for the following pairs of isotope reactions:
\[ \begin{aligned} &\mathrm{O}^{16}\mathrm{O}^{18}+e \to \mathrm{O}^{16}+\mathrm{O}^{18}+e\\ &\mathrm{O}^{16}\mathrm{O}^{16}+e \to \mathrm{O}^{16}+\mathrm{O}^{16}+e \end{aligned} \tag{2a} \]
\[ \begin{aligned} &\mathrm{O}^{16}\mathrm{O}^{18}+\mathrm{O}^{16}\to \mathrm{O}^{16}\mathrm{O}^{16}\mathrm{O}^{18*}\\ &\mathrm{O}^{16}\mathrm{O}^{16}+\mathrm{O}^{16}\to \mathrm{O}_3^{16*} \end{aligned} \tag{3a} \]
\[ \begin{aligned} &\mathrm{O}^{16}\mathrm{O}^{16}\mathrm{O}^{18}\to \mathrm{O}_2^{16}+\mathrm{O}^{18}\\ &\mathrm{O}^{16}\mathrm{O}^{16}\mathrm{O}^{16}\to \mathrm{O}_2^{16}+\mathrm{O}^{16} \end{aligned} \tag{13} \]
The results of the calculations are summarized in Table 1; it also gives the values of the overall experimental separation coefficient, taken from [1]. It should be noted that the kinetic isotope effects of the elementary reactions (4), (6), and (7) were assumed to be identical and equal to the isotope effect of a reaction of type (13), owing to the absence, as already mentioned above, of information on the transition complexes.
Table 1
| \(T^\circ\mathrm{K}\) | 254 | 283 | 293 | 313 | 352 | 365 |
|---|---|---|---|---|---|---|
| \(a_{\mathrm{exp}}\) | 1,03 | 1,07 | 1,08 | 1,10 | 1,12 | 1,14 |
| \(x_1\alpha_2\) | 0,938 | 0,949 | 0,951 | 0,953 | 0,956 | 0,957 |
| \(x_3\) | 0,924 | 0,927 | 0,931 | 0,935 | 0,940 | 0,941 |
As is seen from Table 1, in the first two stages, (2) and (3), the light oxygen isotope \(\mathrm{O}^{16}\) will predominantly accumulate in ozone molecules, and with increasing temperature the kinetic isotope effect decreases. In the process of thermal decomposition and decomposition as a result of electron collision (according to reactions (4), (6), and (7)), the ozone residue will be enriched in heavy oxygen \(\mathrm{O}^{18}\). The total separation of oxygen isotopes \(q\) in the decomposition stage [8] can be expressed by the equation
\[ q=(1-x)^{\alpha_3-1}, \tag{14} \]
where \(x\) is the degree of conversion, or, for our case, the degree of decomposition of the excited \(\mathrm{O}_3^*\) molecules first into stable ones and then of the latter during thermal decomposition and electron collision, i.e.,
\[ x=1-\frac{[\mathrm{O}_3]}{[\mathrm{O}_3^*]}, \tag{15} \]
where brackets \([\,]\) denote stationary concentrations. The value of \(x\) is very difficult to estimate theoretically, since for this one must know the macroscopic rate constants (2)—(7). If the overall separation coefficient for the ozone synthesis reaction is represented as
\[ \alpha=\alpha_1\cdot \alpha_2\cdot q, \tag{16} \]
then from Fig. 1 one can graphically find \(\lg q\) and, knowing \(\alpha_3\), determine \((1-x)\).
Calculations show that in the temperature interval from \(254^\circ\mathrm{K}\) to \(365^\circ\mathrm{K}\), \((1-x)\) changes from 0,13 to 0,07. This is quite permissible if one takes into account that, with increasing temperature, the stationary concentration of ozone de-
decreases, since the total decomposition constant of ozone increases with temperature, whereas the formation constant does not depend on temperature [6]. Thus, with increasing temperature, despite a decrease in the kinetic isotope effect $\alpha_3$, the overall separation $q$ will increase owing to the more rapid decrease in the value of $(1-x)$. The separation of oxygen isotopes in the purely thermal decomposition of ozone was investigated in [10], where it was shown that the change in the concentration of $\mathrm{O}^{18}$ in the undecomposed ozone residue obeys a Rayleigh equation analogous to formula (14).
Fig. 1. Dependence of the coefficient of separation of oxygen isotopes in the electrosynthesis of ozone on temperature.
Consequently, summarizing the foregoing, the separation of oxygen isotopes during ozone synthesis may be represented as follows: as a result of the first two steps (2), (3), a kind of twofold enrichment of the excited ozone molecules $\mathrm{O}_3^*$ in the isotope $\mathrm{O}^{18}$ takes place—first, owing to the process of dissociation of $\mathrm{O}_2$, and, second, owing to the interaction of atoms with oxygen molecules; the resulting enrichment of ozone in $\mathrm{O}^{18}$ arises during the decomposition of $\mathrm{O}_3$ in reactions (4), (6), and (7). This model explains the experimentally observed dependence of the overall separation coefficient on temperature.
Conclusions
On the basis of the generally accepted scheme for the formation of ozone in an electric discharge, a theoretical analysis has been carried out of the separation of oxygen isotopes during ozonation. It is shown that the controlling stage in the isotope-separation process is ozone decomposition. The results of the work are in satisfactory agreement with the experimental data.
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