Subtract $(i)$ $\ln \, k_1 = - \frac{E_a}{R T_1} + \ln A$ and $(ii)$ $\ln \, k_2 = - \frac{E_a}{R T_2} + \ln A$ and write the resulting equation.
Given equations are:
$(i) \ln k_1 = - \frac{E_a}{R T_1} + \ln A$
$(ii) \ln k_2 = - \frac{E_a}{R T_2} + \ln A$
Subtracting equation $(i)$ from equation $(ii)$:
$\ln k_2 - \ln k_1 = (- \frac{E_a}{R T_2} + \ln A) - (- \frac{E_a}{R T_1} + \ln A)$
$\ln \left( \frac{k_2}{k_1} \right) = - \frac{E_a}{R T_2} + \frac{E_a}{R T_1}$
$\ln \left( \frac{k_2}{k_1} \right) = \frac{E_a}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right)$
$(i) \ln k_1 = - \frac{E_a}{R T_1} + \ln A$
$(ii) \ln k_2 = - \frac{E_a}{R T_2} + \ln A$
Subtracting equation $(i)$ from equation $(ii)$:
$\ln k_2 - \ln k_1 = (- \frac{E_a}{R T_2} + \ln A) - (- \frac{E_a}{R T_1} + \ln A)$
$\ln \left( \frac{k_2}{k_1} \right) = - \frac{E_a}{R T_2} + \frac{E_a}{R T_1}$
$\ln \left( \frac{k_2}{k_1} \right) = \frac{E_a}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right)$
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