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(C) The overall rate constant is given by $k = (k_1 k_3 / k_2)^{1/2}$.Using the Arrhenius equation $k = A \cdot e^{-E_a / RT}$,we substitute the expre

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$20$

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(C) The overall rate constant is given by $k = (k_1 k_3 / k_2)^{1/2}$.Using the Arrhenius equation $k = A \cdot e^{-E_a / RT}$,we substitute the expressions for each rate constant:$A \cdot e^{-E_a / RT} = \left( \frac{A_1 e^{-E_{a_1} / RT} \cdot A_3 e^{-E_{a_3} / RT}}{A_2 e^{-E_{a_2} / RT}} \right)^{1/2}$.Comparing the exponential terms:$-E_a / RT = \frac{1}{2} (-E_{a_1} / RT - E_{a_3} / RT + E_{a_2} / RT)$.Multiplying by $-RT$,we get $E_a = \frac{1}{2} (E_{a_1} + E_{a_3} - E_{a_2})$.Substituting the given values $E_{a_1} = 60 \ kJ \ mol^{-1}$,$E_{a_2} = 30 \ kJ \ mol^{-1}$,and $E_{a_3} = 10 \ kJ \ mol^{-1}$:$E_a = \frac{1}{2} (60 + 10 - 30) = \frac{40}{2} = 20 \ kJ \ mol^{-1}$.

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