To find the ratio between the rate constants for the catalysed
(k2) and uncatalysed
(k1) reactions, we can utilize the Arrhenius equation. The Arrhenius equation is given by:
k=Ae−‌where:
k is the rate constant.
A is the pre-exponential factor.
Ea is the activation energy.
R is the gas constant (approximately
8.314J∕(mol⋅K) ).
T is the temperature in Kelvin.
Given:
The reduction in Activation Energy is
100J∕mol.
Temperature is
27∘C, which is 300 K .
Let's denote the activation energy of the uncatalysed reaction as
Ea1 and the activation energy of the catalysed reaction as
Ea2. Since the activation energy is reduced by
100J∕mol due to the catalyst:
Ea2=Ea1−100 We want to find the ratio
‌. Using the Arrhenius equation for both the catalysed and uncatalysed reactions, we get:
‌k1=Ae−‌‌k2=Ae−‌ Combining these equations gives:
‌=‌=e−‌e‌=e‌ Substituting
Ea2=Ea1−100, we get:
‌=e‌ Now, substitute
R=8.314J∕(mol⋅K) and
T=300K :
‌=e‌ Calculating the exponent:
‌≈0.0401So:
‌=e0.0401≈1.041Hence, the ratio between the rate constants for the catalysed and uncatalysed reactions is approximately 1.04 . The correct answer is:
Option D: 1.04