A sealed cylinder contains 1.0 mol of nitrogen gas at 300 K. A simplified numerical Maxwell–Boltzmann distribution for the translational energies of the N₂ molecules at this temperature is summarized below.
Energy range (kJ mol⁻¹) | Fraction of molecules (%)
0 – 10 | 25
10 – 20 | 22
20 – 30 | 18
30 – 40 | 13
40 – 50 | 9
50 – 60 | 6
60 – 70 | 4
70 – 80 | 2
> 80 | 1
The reaction of interest has an activation energy of 55 kJ mol⁻¹.
The cylinder is then warmed to 320 K. The distribution shifts as follows (all other conditions remain identical):
Energy range (kJ mol⁻¹) | Fraction of molecules (%)
0 – 10 | 22
10 – 20 | 20
20 – 30 | 17
30 – 40 | 14
40 – 50 | 11
50 – 60 | 7
60 – 70 | 5
70 – 80 | 3
> 80 | 1
Assume that only translational kinetic energy contributes to overcoming the activation barrier and that the orientation requirement (steric factor) is unchanged by the temperature increase.
(a) Using the 300 K data, calculate the percentage of N₂ molecules that possess sufficient energy to react.
(b) Using the 320 K data, calculate the percentage of N₂ molecules that possess sufficient energy to react.
(c) Determine by what factor the population of sufficiently energetic molecules changes when the temperature is raised from 300 K to 320 K.
(d) If the initial rate of reaction at 300 K is 2.5×10−4mol L−1s−1, estimate the initial rate at 320 K, assuming concentration and orientation factors remain constant.
(e) Explain how your results support the observed temperature dependence of reaction rate and the empirical Arrhenius equation.