Consider a complex reaction taking place in three steps with rate constants , , and respectively. The overall rate constant is given by the expression . If the activation energies of the three steps are 60, 30, and 10 kJ respectively, then the overa

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Solution

Q.
Consider a complex reaction taking place in three steps with rate constants $k_{1}$ , $k_{2}$ , and $k_{3}$ respectively.

The overall rate constant $k$ is given by the expression $k = \sqrt{\frac{k_{1} k_{3}}{k_{2}}}$ . If the activation energies of the three steps are 60, 30, and 10 kJ ${mol}^{- 1}$ respectively, then the overall energy of activation in kJ ${mol}^{- 1}$ is _________. (Nearest integer) [2025]

Ans.
(20)

$K = \sqrt{\frac{K_{1} K_{3}}{K_{2}}}$

$A e^{- E_{a} / R T} = \sqrt{\frac{A_{1} e^{- E_{a 1} / R T} \times A_{3} e^{- E_{a 3} / R T}}{A_{2} e^{- E_{a 2} / R T}}}$

$A e^{- E_{a} / R T} = \sqrt{\frac{A_{1} A_{3}}{A_{2}}} \sqrt{e^{(- E_{a 1} - E_{a 3} + E_{a 2}) / R T}}$
$A e^{- E_{a} / R T} = \sqrt{\frac{A_{1} A_{3}}{A_{2}}} e^{\frac{- (E_{a 1} + E_{a 3} - E_{a 2})}{2 R T}}$

$\Rightarrow E_{a} = \frac{(E_{a 1} + E_{a 3} - E_{a 2})}{2}$

$= \frac{(60 + 10 - 30)}{2} {kJ mol}^{- 1}$
$= 20 {kJ mol}^{- 1}$

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