Question

A reaction occurs in three rate determining steps having rate constants ${K_1}$, ${K_2}$ and ${K_3}$ respectively and arrhenius factor ${A_1}$, ${A_2}$ and ${A_3}$ respectively. The overall rate constant $K = \dfrac{{{K_1}{K_2}}}{{{K_3}}}$. If energy of activations for each step is ${E_{a1}}$, ${E_{a2}}$ and ${E_{a3}}$ respectively, the overall energy of activation is:
(A) ${E_{a1}} + {E_{a2}} + {E_{a3}}$
(B) $\dfrac{{{E_{a1}} \times {E_{a2}}}}{{{E_{a3}}}}$
(C) ${E_{a1}} + {E_{a2}} - {E_{a3}}$
(D) ${K_3}\left[ {\dfrac{{{E_{a1}}}}{{{K_1}}} + \dfrac{{{E_{a2}}}}{{{K_2}}} + \dfrac{{{K_3}{E_{a3}}}}{{{K_1}{K_2}}}} \right]$

Answer 1

To solve this we must know the Arrhenius equation. The Arrhenius equation gives the relationship between the activation energy and the rate constant. Using the given expression for the overall rate constant and the Arrhenius equation, derive the relationship between the overall activation energy and the activation energy of the individual steps.

Formula Used:
$K = A \times {e^{\left( { - \dfrac{{{E_a}}}{{R \times T}}} \right)}}$

Complete step-by-step solution: We know the expression for the relationship between the rate constant of a first order reaction and the activation energy is as follows:
$K = A \times {e^{\left( { - \dfrac{{{E_a}}}{{R \times T}}} \right)}}$
Where $K$ is the rate constant of a first order reaction,
$A$ is the pre-exponential factor,
${E_a}$ is the energy of activation,
$R$ is the universal gas constant,
$T$ is the temperature.
Thus,
${K_1} = {A_1} \times {e^{\left( { - \dfrac{{{E_{a1}}}}{{R \times T}}} \right)}}$
${K_2} = {A_2} \times {e^{\left( { - \dfrac{{{E_{a2}}}}{{R \times T}}} \right)}}$
${K_3} = {A_3} \times {e^{\left( { - \dfrac{{{E_{a3}}}}{{R \times T}}} \right)}}$
We are given the expression for overall rate constant as follows:
$K = \dfrac{{{K_1}{K_2}}}{{{K_3}}}$
Thus,
$A \times {e^{\left( { - \dfrac{{{E_a}}}{{R \times T}}} \right)}} = \dfrac{{{A_1} \times {e^{\left( { - \dfrac{{{E_{a1}}}}{{R \times T}}} \right)}} \times {A_2} \times {e^{\left( { - \dfrac{{{E_{a2}}}}{{R \times T}}} \right)}}}}{{{A_3} \times {e^{\left( { - \dfrac{{{E_{a3}}}}{{R \times T}}} \right)}}}}$
Assume that the Arrhenius factor are equal i.e. $A = {A_1} = {A_2} = {A_3}$. Thus,
${E_a} = {E_{a1}} + {E_{a2}} - {E_{a3}}$
Thus, the overall energy of activation is ${E_{a1}} + {E_{a2}} - {E_{a3}}$.

Thus, the correct option is (c) ${E_{a1}} + {E_{a2}} - {E_{a3}}$.

Note: The minimum amount of energy that the reacting species must possess to undergo a specific reaction is known as the activation energy. The rate of any chemical reaction is inversely proportional to the activation energy. Higher the activation energy, slower is the chemical reaction. Reactions with very high activation energy do not proceed unless energy is supplied to them. For reactions that have a high activation energy, catalysts are used for the reaction to proceed. The catalyst reduces the activation energy and thus, the rate of the reaction increases. The catalyst forms a transition state. The transition state forms because energy is released.