Question

A certain physiologically important first-order reaction has activation energy equal to $45.0kJmol^{- 1}$ at a normal body temperature $\left(37 ^\circ C\right).$ Without a catalyst, the rate constant for the reaction is $5\cdot 0\times 10^{- 4}s^{- 1}$ . To be effective in the human body, where the reaction is catalysed by an enzyme, the rate constant must be at least $2\cdot 0\times 10^{- 2}s^{- 1}$ . If the activation energy is the only factor affected by the presence of the enzyme, by how much must the enzyme lower the activation energy of the reaction to achieve the desired rate? Give answer to the nearest integer value after multiplying with 10.

Answer 1

The process is first-order:
Activation energy without catalyst $\left( E _{ a }\right)=45 kJ mol ^{-1}$
Activation energy with catalyst $= E _{ al }$
Temperature $=273+37=310 K$
Rate constant without catalyst $\left( K _{ a }\right)=5 \times 10^{-4} s ^{-1}$
Rate constant with catalyst $\left( K _{ a 1}\right)=2 \times 10^{-2} s ^{-1}$
Arrhenius equation:
$K _{ a }= A e ^{- E _{ a } / RT }$
$A =5 \times 10^{-4} \times e ^{45000 /(8.314 \times 310)}$
$K _{ a 1}= Ae ^{- E _{ a 1} / RT } 2 \times 10^{-2}=5 \times 10^{-4} \times e ^{\frac{45000}{(8.314 \times 310)}} \times e ^{\frac{- E _{ a 1}}{(8.314 \times 310)}} e ^{-\frac{ E _{ a 1}}{(8.314 \times 3}}$
$E _{ a }- E _{ a 1}=(45-35.49)=9.51 kJ mole ^{-1}$
Answer: $(=10 \times 9.51 \simeq 95)$