For a reaction taking place in four different steps, the rate constants follow a relationship \( \displaystyle K=\frac{K_2K_4}{K_1K_3} \). If the energies of activation E1, E2, E3 and E4 are 10 kJ/mol, 30 kJ/mol, 40 kJ/mol and 20 kJ/mol respectively, then the overall energy of activation would be:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → Zero kJ/mol
By the Arrhenius equation, every rate constant has the form k = A e−Ea/RT. When rate constants are multiplied or divided, the exponential terms combine, so their activation energies are added or subtracted in the same way.
\( \displaystyle K=\frac{K_2K_4}{K_1K_3}=\frac{A_2A_4}{A_1A_3}\;e^{-(E_2+E_4-E_1-E_3)/RT} \)
Comparing with K = A e−Ea/RT, the activation energy of the whole reaction is:
\( \displaystyle E_a = E_2 + E_4 - E_1 - E_3 \)
\( \displaystyle E_a = 30 + 20 - 10 - 40 = 0\ \mathrm{kJ\,mol^{-1}} \)
So the energy of activation of the reaction is zero.
Shortcut: Constants in the numerator add their Ea, constants in the denominator subtract theirs: (30 + 20) − (10 + 40) = 0.