(a) For an electron donor/acceptor pair at 298 K, $H_{a}(d)=0.04\ \mathrm{cm}^{-1}, \Delta_{r} G^{\ominus}=-0.185\ \mathrm{eV},$ and $k_{n}=37.5\ \mathrm{s}^{-1}.$ Use mathematical software to estimate the value of the reoryanization energy. (b) For an electron donor/acceptor pair at 298 K, $k_{a}=2.02 \times 10^{3}\ \mathrm{s}^{-1}$ and $\Delta_{r} G^{\ominus}=-0.665\ \mathrm{eV}.$ The standard reaction Gibbs energy changes to $\Delta_{r} G^{\ominus}= -0.975\ \mathrm{eV}$ when a substituent is added to the electron acceptor and the rate constant for electron transfer changes to $k_{n}=3.33 \times 10^{6}\ \mathrm{s}^{-1}.$ Assume that the distance between donor and acceptor is the same in both experiments and estimate the values of $H_{\mathrm{n}}(d)$ and $\Delta E_{\mathrm{R}}.$

Solution

Verified**(a):**

In this question, we have an electron transfer reaction which occurs at 298 K. In this given reaction, by using any mathematical software, we need to calculate the reorganisational energy of this reaction.

Given data:

Hamiltonian operator, $H_{et} (d) = 0.04$ cm$^{-1}$

Standard Gibbs energy for the electron transfer process, $\Delta_r G^{\phi} = -0.185$ eV

Rate constant for the electron-transfer reaction, $k_{et} = 37.5$ s$^{-1}$

T = 298 K

Boltzmann Constant $k$ = $1.38 \times 10^{-23}$ J/K = $\dfrac{1.38 \times 10^{-23}}{1.602\times10^{-19}}$ eV

Reorganisation energy, $\Delta E_R$ = ?