Question

A reaction takes place in three steps; the rate constants are $K_1,K_2$ and $K_3$ respectively. The overall rate constant $K=\frac{K_1^{3/2}{K}_3^{2/3}}{K_2^{1/4}}$. If activation energy of each steps are $50,500$ and $90kJ$ respectively, then overall activation energy of the reaction is:

• A. $10$
• B. $270$
• C. $300$
• D. $620$

Answer 1

The correct option is A $10$

Given,

Activation energy for step 1 = 50 KJ

Activation energy for step 2 = 500 KJ

Activation energy for step 3 = 90 KJ

As we know that,

$K=A{e}^{\frac{-Ea}{RT}}$

Where,

$K=$ rate constant

$A=$ Pre-exponential factor

$Ea=$ activation energy

$R=$ gas constant

$T=$ temperature

Taking 'ln' on both the sides, we get

$lnK=\frac{-Ea}{RT}$

The overall rate constant is,

$K=\frac{{k_1}^{3/2}\times (k_3{)}^{2/3}}{{k_2}^{1/4}}$

Similarly, we are taking 'ln' on the sides in this expression, we get

$Ea=\frac{3}{2}\times 50+\frac{2}{3}\times 90-\frac{1}{4}\times 500$

Now put all the given values in this expression, we get

$Ea=\frac{3}{2}E{a}_1+\frac{2}{3}E{a}_3-\frac{1}{4}E{a}_2$

$Ea=10KJ$

So, option $A$ is correct.