Full Text
Reports of the Academy of Sciences of the USSR
1960. Vol. 133, No. 6
PHYSICAL CHEMISTRY
R. R. DOGONADZE
THE RATE OF ELECTRON-EXCHANGE REACTIONS IN SOLUTIONS
(Presented by Academician A. N. Frumkin on 1 IV 1960)
On the basis of the adiabatic perturbation theory, in papers \((^{1,2})\) the probability of an electronic transition per unit time, \(w_{12}(R)\), was calculated between like-charged ions situated at a distance \(R\) from one another. To relate \(w_{12}\) to the specific rate constant of the reaction, it is necessary to derive the kinetic equation of the reaction. For simplicity, let us consider the reaction
\[ \mathrm{Fe}^{2+}+\mathrm{Fe}^{*3+}\to \mathrm{Fe}^{3+}+\mathrm{Fe}^{*2+}. \tag{1} \]
However, as will be seen from the calculation, analogous results can also be obtained for any electron-exchange reactions.
Let us denote the concentrations of \(\mathrm{Fe}^{2+}\), \(\mathrm{Fe}^{*2+}\), \(\mathrm{Fe}^{3+}\), \(\mathrm{Fe}^{*3+}\), respectively, by \(c_2, c_2^*, c_3, c_3^*\). The number of \(\mathrm{Fe}^{*2+}\) ions formed in unit volume during the time \(dt\) can be written in the form
\[ dc_2^*=(c_2\rho+c_2^*\rho^*)dt, \tag{2} \]
where \(\rho\) is the probability that a given \(\mathrm{Fe}^{2+}\) ion will pass into \(\mathrm{Fe}^{3+}\) per unit time, and \(\rho^*\) has an analogous meaning. Since the number of \(\mathrm{Fe}^{*3+}\) ions in a volume element \(dv\), situated at a distance \(R\) from \(\mathrm{Fe}^{2+}\), is equal to \(c_3^* e^{-\psi(R)/kT}dv\), and the probability of electron transfer per unit time is \(w_{12}\), \(\rho\) can be represented in the form
\[ \rho=\int w_{21}(R)c_3^*e^{-\psi(R)/kT}\,dv\equiv k_{12}c_3^*, \tag{3} \]
where \(\psi(R)\) is the interaction energy of the \(\mathrm{Fe}^{2+}\) and \(\mathrm{Fe}^{*3+}\) ions located at the distance \(R\). Similarly, for \(\rho^*\) we have
\[ \rho^*=\int w_{21}(R)c_3e^{-\psi(R)/kT}\,dv\equiv k_{21}c_3=k_{12}c_3. \tag{4} \]
After substituting (3) and (4) into (2), we obtain the desired equation
\[ dc_2^*/dt=k_{12}(c_2c_3^*-c_2^*c_3). \tag{5} \]
Let us write the solution of this equation under the initial condition
\[ c_2=c_{20},\qquad c_3^*=c_{30}^*,\qquad c_2^*=c_3=0\quad \text{at } t=0. \tag{6} \]
It is easy to see that at any moment of time the relations
\[ c_2^*=c_3,\qquad c_2+c_2^*=c_{20},\qquad c_3+c_3^*=c_{30}^* \tag{7} \]
hold. Consequently, (5) can be rewritten as
\[ dc_2^*/dt = k_{12}\left[(c_{20}-c_2^*)(c_{30}^*-c_2^*)-(c_2^*)^2\right] = k_{12}\left[c_{20}c_{30}^*-(c_{20}+c_{30}^*)c_2^*\right]. \tag{8} \]
The solution of equation (8) with the initial condition (6) has the form
\[ \ln\left(1-\frac{c_{20}+c_{30}^*}{c_{20}c_{30}^*}\,c_2^*\right) = -k_{12}(c_{20}+c_{30}^*)t. \tag{9} \]
As \(t \to \infty\), as is seen from (9),
\[ c_2^* \to \frac{c_{20}c_{30}^*}{c_{20}+c_{30}^*}=c_{2\infty}^*, \tag{10} \]
therefore (9) can be rewritten in the form
\[ \ln(1-c_2^*/c_{2\infty}^*)=-k_{12}(c_{20}+c_{30}^*)t. \tag{11} \]
Formula (11), as is known, was first obtained by McKay \((^3)\).
The quantity \(k_{12}\), which was determined in \((^3)\), is the specific rate constant of the reaction, determined experimentally. At low concentrations, the Debye potential can be used as \(\psi(R)\)
\[ \psi(R)=3.2\,e^2 e^{-\varkappa R}/\varepsilon_s R, \tag{12} \]
where \(\varkappa\) is the inverse Debye length, \(\varepsilon_s\) is the static dielectric constant. From physical considerations it is clear that \(1/\varkappa\) is much greater than the distance over which \(w_{12}(R)\) decreases; therefore the exponential in (12) can be replaced by unity, i.e., one may assume that \(\psi(R)\) is expressed by the formula
\[ \psi(R)=6e^2/\varepsilon_s R. \tag{13} \]
Thus,
\[ k_{12}=4\pi\int_0^\infty w_{12}(R)e^{-6e^2/\varepsilon_s RkT}R^2\,dR, \tag{14} \]
or, substituting the expression \(w_{12}\) found in \((^2)\):
\[ k_{12}=\frac{kT}{h}\int_0^\infty \left( \frac{512\pi^6}{\hbar^2\omega_0^2 kTc\int(\mathbf{D}_1-\mathbf{D}_2)^2\,dr} \right)^{1/2} |L_{21}^{(2)}|^2 e^{-\frac{\Delta F^*+6e^2/\varepsilon_s R}{kT}}R^2\,dR. \tag{15} \]
The quantity \(L_{21}^{(2)}\) is the exchange integral; \(\mathbf{D}_{1,2}\) is the induction of the electrostatic field created by the electron in the initial (or final) state, and \(c\) is expressed by the formula
\[ c=1/\varepsilon_0-1/\varepsilon_s, \tag{16} \]
where \(\varepsilon_0\) is the optical dielectric constant. In the same work \((^2)\) it was shown that
\[ \Delta F^*=\frac{c}{32\pi}\int(\mathbf{D}_1-\mathbf{D}_2)^2\,dr +kT\ln\frac{kT}{\hbar\omega_0} \equiv \Delta E^*-T\Delta S^*. \tag{17} \]
In order to compare the expression obtained for \(k_{12}\) with the known formula of the theory of absolute reaction rates,
\[ k_{12}=\frac{kT}{h}\,\varkappa e^{-\Delta F^\ddagger/kT}, \tag{18} \]
we shall use the mean-value theorem for integrals. The factor \(R^2 e^{-6e^2/\varepsilon_s RkT}\) decreases sharply with decreasing \(R\) and grows as \(R^2\) with increasing \(R\), while the remaining part of the integrand falls sharply with decreasing \(R\). The integrand reaches a maximum at some \(R=\overline{R}\) and has a rather small width \(\delta R\). Therefore the expression under the integral sign in (15) can be taken outside the integral sign at the point \(\overline{R}\) and multiplied by \(\delta R\), as a result of which we obtain a formula that coincides exactly with (18), with
\[ \Delta F^\ddagger=\Delta F^*(\overline{R})+6e^2/\varepsilon_s\overline{R}, \tag{19} \]
and the transmission coefficient is expressed by the formula
\[ \varkappa= \left( \frac{512\pi^6}{\hbar^2\omega_0^2 kTc\int|\mathbf{D}_1(\overline{R})-\mathbf{D}_2(\overline{R})|^2\,dr} \right)^{1/2} |L_{21}^{(2)}|^2\overline{R}^{\,2}\delta R. \tag{20} \]
For comparison with experiment, one should use expression (19) for the free energy of activation and expression (21) for the activation energy
\[ \Delta E^{\ne}=\Delta E^{*}(\overline{R})+\frac{6e^{2}}{\varepsilon_s \overline{R}} =\frac{c}{32\pi}\int [\mathbf{D}_1-\mathbf{D}_2]^2\,dr+\frac{6e^{2}}{\varepsilon_s \overline{R}}. \tag{21} \]
Knowing the wave function of the \(3d\)-electron of the \(\mathrm{Fe}^{2+}\) ion in water, one could calculate the integral entering into (21), and thereby determine \(\Delta E^{\ne}\) as a function of \(\overline{R}\). However, to estimate the magnitude of \(\Delta E^{\ne}\) we can use the model of rigid conducting spheres proposed by Marcus \((^4)\):
\[ \mathbf{D}_1= \begin{cases} 0, & r<a;\\ e\mathbf{r}/r^3, & r>a; \end{cases} \qquad \mathbf{D}_2= \begin{cases} 0, & |\mathbf{r}-\overline{\mathbf{R}}|<a;\\ \dfrac{e(\mathbf{r}-\overline{\mathbf{R}})}{|\mathbf{r}-\overline{\mathbf{R}}|^3}, & |\mathbf{r}-\overline{\mathbf{R}}|>a, \end{cases} \tag{22} \]
where \(a\) is the radius of the sphere. In this case
\[ \Delta E^{\ne}=\frac{6e^2}{\varepsilon_s R}+\frac{ce^2}{4}\left(\frac{1}{a}-\frac{1}{R}\right). \tag{23} \]
To determine \(\overline{R}\) without using experimental data, it is necessary to investigate the integral (15) in more detail. At the same time we shall indicate a way in which the value of \(k_{12}\) can be estimated if Slater functions are taken as the electron wave functions. Considering reaction (1) as an example, we write the electron wave function in the form
\[ \psi_1(\mathbf{r})=\psi_2(\mathbf{r}+\mathbf{R})=\psi_{3d_{z^2}} =\frac{1}{3}\alpha^3\sqrt{\frac{2\alpha}{\pi}}\,r^2 e^{-\alpha r}\cos^2\theta. \tag{24} \]
The parameter \(\alpha\), according to Slater’s method, is equal for the \(3d\)-electron of \(\mathrm{Fe}^{2+}\) to
\[ \alpha=z_{\mathrm{eff}}/n_{\mathrm{eff}}a_0\varepsilon_s=6.25/3a_0\varepsilon_s. \tag{25} \]
Following the method adopted in semiconductor theory, instead of the Bohr radius \(a_0\) we have introduced a quantity \(\varepsilon_s\) times larger, in order to take the solvent into account. The choice of the angular part of the wave function (24) is due to the fact that it is precisely the \(\psi_{3d_{z^2}}\)-functions that give the greatest overlap. To simplify the calculations, we shall use the functions (24) only in determining the exchange integral \(L^{(2)}_{21}\); for the other quantities we shall assume that approximation (22) is valid.
We write the exchange integral \(L^{(2)}_{21}\) in the form
\[ L^{(2)}_{21}=\int \psi_2^*\,\frac{3e^2}{\varepsilon_s|\mathbf{r}-\mathbf{R}|}\,\psi_1\,d\mathbf{r}. \tag{26} \]
After simple but lengthy calculations (the integral should be taken in elliptic coordinates), the formula obtained was
\[ L^{(2)}_{21}=\frac{\alpha e^2 e^{-\alpha R}}{24\varepsilon_s} \left(48+48\alpha R+20\alpha^2R^2+4\alpha^3R^3+\alpha^4R^4+\frac{11}{15}\alpha^5R^5\right) \equiv P_5(R)e^{-\alpha R}. \tag{27} \]
Since the square root standing under the integral in (15) changes very slowly, it may be taken outside the integration sign and \(k_{12}\) written in the form
\[ k_{12}=\frac{4\pi}{\hbar e}\sqrt{\frac{\pi a}{ckT}}\, e^{-ce^2/4akT} \int_0^\infty P_{12}(R)\,e^{-2\alpha R-(6/\varepsilon_s-c/4)e^2/kT\,R}\,dR, \tag{28} \]
where the polynomial of the 12th degree \(R^2P_5^2(R)\) has been denoted by \(P_{12}(R)\).
If it is assumed that \(\varepsilon_s\) does not depend on \(R\), the integral in (28) can be evaluated exactly. Indeed, denoting \((6/\varepsilon_s-c/4)e^2/kT\) by \(\gamma\), the desired integral can be represented as the sum \(\sum_{n=2}^{12} A_n I_n\), where\(^5\)
\[ I_n=\int_0^\infty R^n e^{-2\alpha R-\gamma/R}\,dR =2(2\alpha)^{n-1}(2\alpha\gamma)^{(n+1)/2}K_{n+1}(\sqrt{8\alpha\gamma}). \tag{29} \]
Thus, integral (28) is expressed in terms of Macdonald functions. As is seen from (29), such a calculation is valid for electron exchange between isotopes with charges \(z\) and \(z+1\) only under the condition that \(\gamma>0\), since \(\alpha\) is always positive. This means that \(z(z+1)/\varepsilon_s-c/4>0\), or, after simple transformations,
\[ \varepsilon_s<\varepsilon_0(2z+1)^2. \tag{30} \]
The calculation presented is sufficiently rigorous if the integrand in (28) reaches its maximum at such an \(R_M\) that \(\varepsilon_s\) can already be regarded as independent of \(R\). One can estimate \(R_M\) by setting equal to zero the derivative of the exponent in (28), assuming \(\varepsilon_s\) independent of \(R\):
\[ -2\alpha+\gamma/R_M^0{}^2=0,\qquad R_M^0=\sqrt{\gamma/2\alpha}. \tag{31} \]
However, because \(P_{12}(R)\) is present under the integral, \(R_M\) will be somewhat larger than \(R_M^0\).
For reaction (1), condition (30) requires that \(\varepsilon_s<45\). Unfortunately, the dependence of \(\varepsilon_s\) on \(R\) is difficult to investigate theoretically. However, the data of Laidler\(^6\) show that \(\varepsilon_s<45\) for \(R<5\) Å. It follows from this that the most probable distance at which electron transfer occurs in reaction (1) is less than 4 Å. When considering exchange reactions between ions with charges 3 and 4, condition (30) is satisfied automatically, and equation (31) can be written as
\[ R_M^{(0)}=13.9\sqrt{n_{\mathrm{eff}}/z_{\mathrm{eff}}}\ \text{Å}. \tag{32} \]
Therefore, one may expect that \(R_M\) is sufficiently large to make it possible to neglect the dependence of \(\varepsilon_s\) on \(R\). For example, for the electron-exchange reaction in the system \(\mathrm{Ce}^{3+}-\mathrm{Ce}^{4+}\), \(R_M^0\) is found to be of the order of 8.2 Å (\(n_{\mathrm{eff}}=4\) and \(z_{\mathrm{eff}}=11.4\)); i.e., the scheme developed above for calculating \(k_{12}\) can be used with a good degree of accuracy.
It is interesting to note that the expression for the activation energy for an electron-exchange reaction between identical ions differing only in valence state, obtained in the present work (23), coincides with the free energy of activation calculated by Marcus\(^4\). For reaction (1), Marcus obtained the value \(\Delta F^{\ddagger}=9.8\) kcal/mol; therefore, according to our theory, \(\Delta E^{\ddagger}=9.8\) kcal/mol, which agrees very well with the experimental value \(\Delta E_{\mathrm{exp}}^{\ddagger}=9.9\) kcal/mol.
Institute of Electrochemistry
Academy of Sciences of the USSR
Received
28 III 1960
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