A certain physiologically important first-order reaction has activation energy equal to 45.0kJmol- 1 at a normal body temperature (37 ° C). Without a catalyst, the rate constant for the reaction is 5⋅ 0× 10- 4s- 1 . To be effective in the human body, where the reaction is catalysed by an enzyme, the rate constant must be at least 2⋅ 0× 10- 2s- 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.

Question

A certain physiologically important first-order reaction has activation energy equal to 45.0kJmol- 1 at a normal body temperature (37 ° C). Without a catalyst, the rate constant for the reaction is 5⋅ 0× 10- 4s- 1 . To be effective in the human body, where the reaction is catalysed by an enzyme, the rate constant must be at least 2⋅ 0× 10- 2s- 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. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

The process is first-order:
Activation energy without catalyst

(E_{a}) = 45 k J m o l^{- 1}

Activation energy with catalyst

= E_{a l}

Temperature

= 273 + 37 = 310 K

Rate constant without catalyst

(K_{a}) = 5 \times 1 0^{- 4} s^{- 1}

Rate constant with catalyst

(K_{a 1}) = 2 \times 1 0^{- 2} s^{- 1}

Arrhenius equation:

K_{a} = A e^{- E_{a} / RT}

A = 5 \times 1 0^{- 4} \times e^{45000/ (8.314 \times 310)}

K_{a 1} = A e^{- E_{a 1} / RT} 2 \times 1 0^{- 2} = 5 \times 1 0^{- 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 k J m o l e^{- 1}

Answer:

(= 10 \times 9.51 ≃ 95)