A student investigates the gas-phase dimerization of nitrogen dioxide, an elementary reaction forming N₂O₄. The student collects kinetic data at two temperatures, 298 K and 318 K, but does not have access to graphing software. Instead, the student is asked to perform the following tasks.
At 298 K, the most probable kinetic energy of NO₂ molecules is 37 kJ mol⁻¹. At 318 K, it is 39 kJ mol⁻¹. The activation energy for dimerization is 110 kJ mol⁻¹.
(a) On appropriately labeled axes, sketch qualitative Maxwell–Boltzmann distributions for NO₂ molecules at the two temperatures. Indicate clearly: (i) the two most probable kinetic energies, (ii) the activation energy threshold, and (iii) the high-energy tails.
(b) Using the sketches, estimate which temperature has the larger fraction of molecules with E≥110kJ mol−1 and justify your answer by referring to areas under the curves.
(c) Assuming that the frequency of collisions and orientation factors are identical at the two temperatures, derive an expression for the ratio k298k318 of the rate constants in terms of activation energy, the gas constant, and the two temperatures.
(d) Calculate the numerical value of the ratio obtained in part (c) using R=8.314J mol−1K−1. Express your answer to two significant figures.
(e) Critique the assumption made in part (b) that the orientation factor is independent of temperature. Provide one microscopic argument supporting the assumption and one argument challenging it.
a. Part (a)
b. Part (b)
c. Part (c)
d. Part (d)
e. Part (e)