Consider a complex reaction taking place in three steps with rate constants k1, k2 and k3 respectively. The overall rate constant k is given by the expression k=k1k3k2.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)

K=K1K3K2A.e−Ea/RT=A1e−Ea1/RT×A3e−Ea3/RTA2e−Ea2/RTBy comparinig exponential termEaRT=12×Ea1RT+Ea3RT−Ea2RTEa=Ea1+Ea3−Ea2/2Ea=(60+10−30)/2=20kJmol−1

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Consider a complex reaction taking place in three steps with rate constants k₁, k₂ and k₃ 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)

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answer is 20.

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Detailed Solution

$\begin{array}{l} K = \sqrt{\frac{K_{1} K_{3}}{K_{2}}} \\ A . e^{- Ea / RT} = \sqrt{\frac{A_{1} e^{- {Ea}_{1} / RT} \times A_{3} e^{- {Ea}_{3} / RT}}{A_{2} e^{- {Ea}_{2} / RT}}} \end{array}$

By comparinig exponential term
$\begin{array}{l} \frac{E_{a}}{RT} = \frac{1}{2} \times (\frac{E_{a_{1}}}{RT} + \frac{E_{a_{3}}}{RT} - \frac{E_{a_{2}}}{RT}) \\ E_{a} = (E_{a_{1}} + E_{a_{3}} - E_{a_{2}}) / 2 \\ E_{a} = (60 + 10 - 30) / 2 = 20 {kJmol}^{- 1} \end{array}$