Consider a complex reaction taking place in three steps with rate constants \(\mathrm{k}_{1}, \mathrm{k}_{2}\) and \(\mathrm{k}_{3}\) respectively. The overall rate constant k is given by the expression \(\mathrm{k}=\sqrt{\frac{\mathrm{k}_{1} \mathrm{k}_{3}}{\mathrm{k}_{2}}}\). If the activation energies of the three steps are 60,30 and \(10 \mathrm{~kJ} \mathrm{~mol}^{-1}\) respectively, then the overall energy of activation in \(\mathrm{kJ} \mathrm{mol}^{-1}\) is \_\_\_\_ . (Nearest integer)

JEE Main · 2025 · 24 Jan 2025 · Shift 2 · Chemical Kinetics

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 = \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)

Answer: 20

For the sequential three-step mechanism, each rate constant obeys the Arrhenius equation

k_{i} = A_{i} e^{- E_{a_{i}} / R T}

. Substituting into the overall expression

k = k_{1} k_{3} / k_{2}

A e^{- E_{a} / R T} = \frac{A _{1} e ^{- E_{a_{1}} / R T} \cdot A _{3} e ^{- E_{a_{3}} / R T}}{A _{2} e ^{- E_{a_{2}} / R T}}

Comparing the exponents on both sides:

\frac{E _{a}}{R T} = \frac{1}{2} (\frac{E _{a_{1}} + E _{a_{3}} - E _{a_{2}}}{R T}) ⟹ E_{a} = \frac{E _{a_{1}} + E _{a_{3}} - E _{a_{2}}}{2}

Substituting the given activation energies:

E_{a} = \frac{60 + 10 - 30}{2} = \frac{40}{2} = 20 kJ mo l^{- 1}

JEE Main · 2025 · 24 Jan 2025 · Shift 2 · Integer answer