PHYSICAL CHEMISTRY

A. K. PIKAEV

A METHOD FOR THE APPROXIMATE ESTIMATION OF THE ABSOLUTE VALUES OF RATE CONSTANTS OF RADIATION REACTIONS IN AQUEOUS SOLUTIONS

(Presented by Academician V. I. Spitsyn, January 10, 1964)

Earlier, in work \((^{1})\), carried out with the participation of the author of the present article, the absolute values of the rate constants of the reactions \(\mathrm{H}+\mathrm{OH}\), \(\mathrm{H}+\mathrm{O}_2\), and \(\mathrm{Fe}^{2+}+\mathrm{OH}\) were approximately estimated. The method for calculating them consisted in using two independent methods of kinetic treatment of experimental data obtained in the study of the pulse radiolysis of aqueous ferrosulfate solutions containing oxygen. The first method is based on the application of the method of stationary concentrations, and the second on the assumption that reactions in the bulk of the solution do not occur during the action of the electron pulse (duration \(5\cdot 10^{-6}\) sec).

In the present work this method is used to calculate the rate constants of a number of other reactions. For this purpose, experimental data obtained in the study of the pulse radiolysis of the following aqueous systems were treated: \(\mathrm{Ce}^{4+}—\mathrm{Ce}^{3+}\) \((^{2,3})\), \(\mathrm{Br}^-—\mathrm{O}_2^*\), \(\mathrm{NO}_3^-—\mathrm{NO}_2\) \((^{4})\), and \(\mathrm{H}_2\mathrm{O}_2\) \((^{5})\). It was found that in all cases the method under consideration gives realistic values of the constants.

As is known \((^{1-13})\), the yields of radiolytic transformations at sufficiently high absorbed-dose rates produced by pulsed electron radiation, owing to competition in the bulk of the solution between radical–radical and radical–dissolved-substance reactions, change substantially in comparison with low absorbed-dose rates. In the simplest case this change in yields is due to competition between the reactions:

\[ \mathrm{R}+\mathrm{R}\to \mathrm{R}_2, \tag{1} \]

\[ \mathrm{R}+\mathrm{S}\to \text{product}, \tag{2} \]

where \(\mathrm{R}\) is a radical, \(\mathrm{R}_2\) is a molecular product, and \(\mathrm{S}\) is the dissolved substance. Then the following equations can be obtained \((^{5,7,11})\):

\[ [\mathrm{R}]= \frac{K_2[\mathrm{S}][\mathrm{R}]_0} {(K_2[\mathrm{S}]+2K_1[\mathrm{R}]_0)e^{K_2[\mathrm{S}]t}-2K_1[\mathrm{R}]_0}, \tag{3} \]

\[ f=\frac{K_2[\mathrm{S}]}{2K_1[\mathrm{R}]_0} \ln\left(1+\frac{2K_1[\mathrm{R}]_0}{K_2[\mathrm{S}]}\right), \tag{4} \]

where \([\mathrm{R}]\) is the concentration of \(\mathrm{R}\) at any time \(t\) after the pulse; \([\mathrm{S}]\) is the concentration of \(\mathrm{S}\); \([\mathrm{R}]_0\) is the concentration of \(\mathrm{R}\) immediately after the pulse (it is assumed that during the action of the pulse no reactions occur in the bulk of the solution), \(K_1\) and \(K_2\) are the rate constants of reactions (1) and (2); \(f\) is the fraction of radicals \(\mathrm{R}\) reacting with \(\mathrm{S}\). Obviously,

\[ [\mathrm{R}]_0=G_R\frac{D}{100N}, \tag{5} \]

where \(G_R\) is the yield of \(\mathrm{R}\), \(D\) is the dose per pulse (in eV/l), and \(N\) is Avogadro’s number. Thus, with the aid of equation (4), it is possible, on the basis of experimental data, to calculate \(K_1/K_2\).

On the other hand, the method of stationary concentrations as applied to reactions (1) and (2) gives, in the general form, the following equation:

\[ \frac{I\sqrt{2G}}{(G_R-G)^{1/2}} = \frac{K_2}{K_1^{1/2}}[\mathrm{S}] \left(\frac{100N}{I}\right)^{1/2}, \tag{6} \]

* Experimental data on the radiolysis of this system, obtained with the participation of the author of the present article, will be set forth in detail in a subsequent publication.

where \(G\) is the yield of the product of reaction (2), and \(I\) is the absorbed dose rate (in eV/l·sec). Thus, knowing \(G\) at various \(I\) or \([S]\), one can find \(K_2/K_1^{1/2}\) from equation (6). By comparing \(K_2/K_1\) with \(K_2/K_1^{1/2}\), it is evidently possible to determine \(K_2\) and \(K_1\).

For the systems indicated above, the yields in the absence of dissolved substances capable of interacting with radicals \(R\) (reaction (2)) are higher under pulse irradiation conditions than at low dose rates. In the presence of sufficient amounts of such a substance, the yields decrease.

The increase in \(G(\mathrm{Ce}^{3+})\) for a \(\mathrm{Ce}^{4+}\) solution, \(G(\mathrm{H_2O_2})\) for an \(\mathrm{O_2}\) solution, and \(G(\mathrm{NO_2^-})\) for a \(\mathrm{NO_3^-}\) solution is mainly explained by the increased role of recombination of OH radicals \((^{2-4})\):

\[ \mathrm{OH}+\mathrm{OH}\to \mathrm{H_2O_2}. \tag{7} \]

Fig. 1. Dependence of \(G(\mathrm{Ce}^{3+})\) on the concentration of \(\mathrm{Ce}^{3+}\) in a 0.4 \(M\) sulfuric acid solution of \(\mathrm{Ce}^{4+}\) \(([\mathrm{Ce}^{4+}]=2\cdot10^{-4}\ M,\) dose rate \(1.5\cdot10^{23}\) eV/ml·sec). Points—experimental data \((^3)\); curve calculated at \(K_9/K_7=3\cdot10^{-2}\).

Fig. 2. Dependence of \(G(\mathrm{H_2O_2})\) on the concentration of KBr in a 0.4 \(M\) \(\mathrm{H_2SO_4}\) solution saturated with air (dose rate \(2.9\cdot10^{23}\) eV/ml·sec). Points—experimental data; curve calculated at \(K_{10}/K_7=2.5\).

The increase in \(G(\mathrm{H_2})\) for deaerated acidic water is due chiefly to recombination of H atoms \((^{5,11})\):*

\[ \mathrm{H}+\mathrm{H}\leftarrow \mathrm{H_2}. \tag{8} \]

Addition to the indicated solutions of \(\mathrm{Ce}^{3+}\), \(\mathrm{Br^-}\), \(\mathrm{NO_2^-}\), and \(\mathrm{H_2O_2}\), as a result of the occurrence of the reactions:

\[ \mathrm{Ce}^{3+}+\mathrm{OH}\to \mathrm{Ce}^{4+}+\mathrm{OH^-}, \tag{9} \]

\[ \mathrm{Br^-}+\mathrm{OH}\to \mathrm{Br}+\mathrm{OH^-}, \tag{10} \]

\[ \mathrm{NO_2^-}+\mathrm{OH}\to \mathrm{NO_2}+\mathrm{OH^-}, \tag{11} \]

\[ \mathrm{H}+\mathrm{H_2O_2}\to \mathrm{OH}+\mathrm{H_2O} \tag{12} \]

leads, respectively, to a decrease in \(G(\mathrm{Ce}^{3+})\), \(G(\mathrm{H_2O_2})\), \(G(\mathrm{NO_2^-})\), and \(G(\mathrm{H_2})\).

In these cases the yields \(G\) of the products at a given concentration \(S\) and dose rate are related to the quantity \(f\) by the following relation:

\[ G=(1-f)(G_0-G_0')+G_0', \tag{13} \]

where \(G_0\) is the product yield at the given high dose rate in the absence of \(S\), and \(G_0'\) is the yield of the same product at a low dose rate, likewise in the absence of \(S\). In Figs. 1–3 the experimental values and the values calculated from equations (4) and (13) of \(G(\mathrm{Ce}^{3+})\), \(G(\mathrm{H_2O_2})\), and \(G(\mathrm{NO_2^-})\) are compared at \(K_9/K_7=3\cdot10^{-2}\), \(K_{10}/K_7=2.5\), and \(K_{11}/K_7=1.5\), respectively. As can be seen, the agreement is satisfactory. In the \(\mathrm{Ce}^{4+}—\mathrm{Ce}^{3+}\) system, as was shown

* In sufficiently acidic solutions, hydrated electrons are converted into H atoms:

\[ e_{\mathrm{aq}}^-+\mathrm{H^+}\to \mathrm{H}. \]

in work (³), the products of the interaction of OH radicals with sulfuric acid play an important role. Therefore, a more accurate value of \(K_9/K_7\) can be obtained at sufficiently high concentrations of \(\mathrm{Ce}^{3+}\), when the reaction of OH with \(\mathrm{HSO_4^-}\) is suppressed. For this concentration range of \(\mathrm{Ce}^{3+}\) (\([\mathrm{Ce}^{3+}] \geqslant 5\cdot10^{-3}\ M\)), \(K_9/K_7 = 4\cdot10^{-2}\). The value of the constant \(K_{12}/K_8\) was calculated in a similar way by J. Thomas and E. Hart (⁵). It is equal to \(8\cdot10^{-3}\).

The relative constants \(K_9/K_7^{1/2}\) and \(K_{10}/K_7^{1/2}\), as it turned out, are equal to \(3.6\cdot10^3\ (\mathrm{l/mol\cdot sec})^{1/2}\) and \(2\cdot10^5\ (\mathrm{l/mol\cdot sec})^{1/2}\), respectively. The constant \(K_{11}/K_7^{1/2}\) can be calculated from the experimental data obtained

![Figure 3 and Figure 4 graphs shown on the page.]

Fig. 3. Dependence of \(G(\mathrm{NO_2^-})\) on the concentration of \(\mathrm{NO_2^-}\) in \(5\cdot10^{-3}\ M\) \(\mathrm{NaNO_3}\) solution saturated with air (dose rate \(1.5\cdot10^{23}\ \mathrm{eV/ml\cdot sec}\)). Points—experimental data (⁴); curve calculated for \(K_{11}/K_7 = 1.5\)

Fig. 4. Graphical solution of equation (20) using the data of work (⁵)

in work (⁴). If it is assumed that the mechanism of radiolysis of \(\mathrm{NO_3^-}\) solutions at high absorbed-dose rates includes reactions (7) and (11), as well as the reactions:

\[ \mathrm{NO_3^- + H \to NO_2 + OH^-}, \tag{14} \]

\[ \mathrm{NO_3^- + e_{aq}^{-} \to NO_3^{2-}}, \tag{15} \]

\[ 2\mathrm{NO_2} + \mathrm{H_2O} \to \mathrm{NO_2^- + NO_3^- + 2H^+}, \tag{16} \]

\[ 2\mathrm{NO_3^{2-}} + \mathrm{H_2O} \to \mathrm{NO_2^- + NO_3^- + 2OH^-}, \tag{17} \]

then, using the steady-state concentration method, one can derive the equation*:

\[ \frac{G_{\mathrm{red}} - G(\mathrm{NO_2^-})} {\left[G(\mathrm{NO_2^-}) + G_{\mathrm{H_2}} - G_{\mathrm{H_2O_2}}\right]^{1/2}} = \frac{G_{\mathrm{OH}} + 2G_{\mathrm{H_2O_2}} - 2G(\mathrm{H_2O_2})} {\left[G(\mathrm{H_2O_2}) - G_{\mathrm{H_2O_2}}\right]^{1/2}} = \frac{K_{11}}{K_7^{1/2}}[\mathrm{NO_2^-}] \left(\frac{100N}{I}\right)^{1/2}. \tag{18} \]

From the data presented in work (⁴), it follows that

\[ K_{11}/K_7^{1/2} = (1.3 \pm 0.7)\cdot10^5\ (\mathrm{l/mol\cdot sec})^{1/2} **. \]

For calculating the relative constant \(K_{12}/K_8^{1/2}\), the results obtained by J. Thomas and E. Hart (⁵) were used. It was assumed that the value \(G(\mathrm{H_2})\) is determined chiefly by reactions (8) and (12), and also by the reaction:

\[ \mathrm{H + OH \to H_2O}. \tag{19} \]

* Here and below the symbol \(G_{\mathrm{red}}\) denotes the total yield of \(\mathrm{e_{aq}^-}\) and H.

** The calculation of \(K_{11}/K_7^{1/2}\) was carried out on the basis of the values \(G(\mathrm{NO_2^-})\) and \(G(\mathrm{H_2O_2})\) for a \(5\cdot10^{-3}\ M\) \(\mathrm{NaNO_3}\) solution at dose rates up to \(10^{22}\ \mathrm{eV/ml\cdot sec}\). In the calculations, dose-rate values averaged over the irradiated volume and arithmetic mean values of the concentrations of the \(\mathrm{NO_2^-}\) and \(\mathrm{H_2O_2}\) formed were used.

Applying the method of stationary concentrations, we find:

\[ \frac{G_{\text{red}}-G_{\mathrm{OH}}-G(\mathrm{H}_2)+G_{\mathrm{H}_2}} {\sqrt{2}\,[G(\mathrm{H}_2)-G_{\mathrm{H}_2}]^{1/2}} = \frac{K_{12}}{K_8^{\prime\,1/2}}[\mathrm{H}_2\mathrm{O}_2]\left(\frac{100N}{I}\right)^{1/2}. \tag{20} \]

Solving equation (20) for \(K_{12}/K_8^{\prime\,1/2}\) using all the data obtained in work (5) is not possible. In that work the values of \(G(\mathrm{H}_2)\) are given as a function of \([\mathrm{H}_2\mathrm{O}_2]\) at various doses per pulse, the dose being expressed as the number of micromoles of \(\mathrm{Fe}^{3+}\). It is impossible to calculate the dose rate (in eV/ml·sec) from these data, since in each separate case the pulse duration is not indicated. However, it follows from work (5) that the mechanism including the competition of reactions (8) and (12) operates at dose rates up to \(10^{23}\) eV/ml·sec. Therefore it may be assumed that one of the curves given in Fig. 2 of the cited work, namely the curve plotted at the maximum dose rate at which this mechanism still operates, was obtained at a dose rate of \(10^{23}\) eV/ml·sec. Figure 4 shows the graphical solution of equation (20) using the data corresponding to this curve. In this figure

\[ \varphi(G)= \frac{G_{\text{red}}-G_{\mathrm{OH}}-G(\mathrm{H}_2)+G_{\mathrm{H}_2}} {\sqrt{2}\,[G(\mathrm{H}_2)-G_{\mathrm{H}_2}]^{1/2}}. \]

The tangent of the angle of inclination of the straight line in Fig. 4 is equal to

\[ \frac{K_{12}}{K_8^{\prime\,1/2}}\left(\frac{100N}{I}\right)^{1/2}. \]

Then \(K_{12}/K_8^{\prime\,1/2}=9.8\cdot 10^2\) \((\mathrm{l/mol\cdot sec})^{1/2}\).

Table 1

Rate constants of radiation reactions in aqueous solutions

* Calculated in work (1).
* From data on the radiolysis of the \(\mathrm{Ce}^{4+}-\mathrm{Ce}^{3+}\) system (at \(K_9/K_7=4\cdot 10^{-2}\)).
* From data on the radiolysis of the \(\mathrm{Br}^- - \mathrm{O}_2\) system.
*** From data on the radiolysis of the \(\mathrm{NO}_3^- - \mathrm{NO}_2^-\) system.

Table 1 gives the values of the constants found by comparing the obtained values of the ratios \(K_2/K_1\) and \(K_2/K_1^{\prime\,1/2}\). It is of interest to compare the constants calculated in the present work with those known in the literature. According to (14), \(K_7=4\cdot 10^9\) l/mol·sec, \(K_9=7.2\cdot 10^7\) l/mol·sec and \(K_{11}=2.5\cdot 10^9\) l/mol·sec. As follows from work (5), \(K_8=5\cdot 10^9\) l/mol·sec. According to data (15), \(K_{\mathrm{Br}^-+\mathrm{OH}}/K_{\mathrm{H}_2+\mathrm{OH}}=830\). Since \(K_{\mathrm{H}_2+\mathrm{OH}}=4.5\cdot 10^7\) l/mol·sec (14), then \(K_{10}=3.7\cdot 10^{10}\) l/mol·sec. As can be seen, the agreement is satisfactory.

It should be noted that the proposed method for estimating the absolute values of rate constants of radiation reactions is approximate. Obviously, it is most applicable in those cases where the reactions partly proceed during the action of the pulse and end comparatively quickly after its introduction into the irradiated system.

Institute of Physical Chemistry
Academy of Sciences of the USSR

Received
26 XII 1963

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