University of Pennsylvania - CBE 618618 18f HW_5_solutions

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CBE 618 Homework 5 Due November 5, 2018 1. Consider a two-level system, where a single particle is able to occupy one of two states: a ground state with energy E0, and an excited state energy E1 = ... E0 + ∆E. Answer the following questions: (a) Show that the microstate probabilities P0 and P1 are independent of E0. (b) Derive the expression for the heat capacity of the system, Cv(T), defining a characteristic temperature θ2 = ∆E=kB along the way. (c) Set θ2 = 10K and plot Cv(T) from T = 0K to T ≈ 75K (d) Why is limT!0 Cv(T) = 0? Similarly, why is limT!1 Cv(T) also 0? It may be useful to examine hE(T)i and hE2(T)i to guide your answer. 2. Prove that the Gibbs formula for the entropy S = −kB X i Pi log Pi; (1) where the sum is over all microstates of the system, is consistent with the Boltzmann formula for the entropy S = kB log Ω in the microcanonical ensemble. 3. Show that the partition function of a mixture of two monatomic, ideal gases is given by Q = q1N1q2N2 N1!N2!: (1) Also show that E = 3 2 (N1 + N2)kT (2) and S = N1k ln V e5=2 Λ3 1N1 ! + N2k ln V e Λ3 2N5=22 ! [Show More]

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