Assume that the probability of occupying a given energy state is given by the relationship provided in Problem P29.27.

In problem 29.27

In a subsequent chapter we will encounter the energy distribution P (ε) = Ae^−ε/kT, where P (ε) is the probability of a molecule occupying a given energy state, ε is the energy of the state, k is a constant equal to 1.38 × 10⁻²³ J K⁻¹, and T is temperature. Imagine that there are three energy states at 0, 100, and 500 J mol⁻¹.

a. Consider a collection of three total states with the first state located at ε = 0 and others at kT and 2kT, respectively, relative to this first state. What is the normalization constant for the probability distribution?

b. How would your answer change if there are five states with ε = kT, in addition to the single states at ε = 0 and ε = 2kT?

c. Determine the probability of occupying the energy level ε = kT for the cases in which one and five states exist at this energy.