Question

A reaction takes place in three steps. The rate constant of the three steps is $K_1,K_2$ and $K_3$ respectively. The overall rate constant $K=\frac{K_1{K}_3}{K_2}$. If the energy of activation for the three steps is $40,30$ and $20KJ$ respectively, then :

• A. overall energy of activation is $10KJ$
• B. overall energy of activation is $30KJ$
• C. the reaction mechanism is $2A\rightleftharpoons A^{\ast }+A;A^{\ast }\underset{rds}{\overset{K_3}{\to }}$ product and overall order is one
• D. the reaction mechanism is $A\stackrel{K_1}{\to }B;B\stackrel{K_2}{\to }C;C\stackrel{K_3}{\to }product$ and overall order is one

Answer 1

Answer: (B), (C)

The correct options are
B overall energy of activation is $30KJ$
C the reaction mechanism is $2A\rightleftharpoons A^{\ast }+A;A^{\ast }\underset{rds}{\overset{K_3}{\to }}$ product and overall order is one

The energy of activation for three steps are 40, 30 and 20 kJ respectively. The overall energy of activation is $40-30+20=30kJ$.
Thus the energy of activation is 30 kJ and not 10 kJ. Thus option A is incorrect.
The expression for the overall rate constant is $K=\frac{K_1{K}_3}{K_2}$. The terms $K_1$ and $K_3$ appear in the numerator. Hence, the first step and the third step are the forward reactions.
The term $K_2$ appears in the denominator. Hence, second step is a backward reaction. Thus first step and second step combine to form an equilibrium reaction
$2A\rightleftharpoons A^{\ast }+A$
The third step is the rate determining step.
$A^{\ast }\underset{rds}{\overset{K_3}{\to }}$.
Thus the mechanism of the reaction is $2A\rightleftharpoons A^{\ast }+A;A^{\ast }\underset{rds}{\overset{K_3}{\to }}$.
The expresion for the rate of the overall reaction is $rate=K_3[A\ast ]$
But the expression for the equilibrium constant is $\frac{K_1}{K_2}=\frac{[A\ast ]\left[A\right]}{[A{]}^2}$ or $[A\ast ]=\frac{K_1}{K_2}[A{]}^2$
Thus the expression for the rate of the overall reaction becomes $rate=\frac{K_1{K}_3}{K_2}\left[A\right]$
Thu the reaction has overall order of 1.
Option D is incorrect as all the steps are forward reactions. Hence, the term $K_2$ for overall rate constant will not appear in the denominator.