For a pair of distinct parallel first-order reactions (possessing an independent activation energy ) — Chemical Kinetics Chemistry Question
Question
For a pair of distinct parallel first-order reactions $A \xrightarrow{k_1} B$ (possessing an independent activation energy $E_{a1}$) and $A \xrightarrow{k_2} C$ (possessing an activation energy $E_{a2}$), the overall effective activation energy $E_a$ for the total disappearance of reactant A is correctly modeled by which of the following algebraic expressions?
💡 Solution & Explanation
For parallel reactions, the overall rate constant is $k = k_1 + k_2$. Taking the Arrhenius derivative $d(\ln k)/dT = E_a/RT^2$ for the sum $k_1+k_2$, we get $\frac{d}{dT} \ln(k_1+k_2) = \frac{1}{k_1+k_2} (\frac{dk_1}{dT} + \frac{dk_2}{dT})$. Substituting $dk_i/dT = k_i E_{ai} / RT^2$ yields the effective overall activation energy $E_a = \frac{k_1 E_{a1} + k_2 E_{a2}}{k_1 + k_2}$.