A monatomic particle at rest can be in either of two energy levels, separated by an energy ε. Consider a dilute gas of N such particles at a fixed temperature T. a) Write down the probability for a single particle to be moving with momentum |k| and in the excited energy level. b) Derive the canonical partition function of a single particle. [You may use the Gaussian integral: 2 22 dx x²e-x²y² dxxe 0 πT 4y³ and approximate the density of states in three dimensions as g(k)dk ≈ V k²dk] 2πT c) Hence, or otherwise, find the internal energy per particle of this gas and comment on how it behaves at large temperature d) Plot the heat capacity Cy of this gas as a function of temperature, assuming that the energy level separation is ε/kB ≈ 1 K.

A monatomic particle at rest can be in either of two energy levels, separated by an energy ε. Consider a dilute gas of N such particles at a fixed temperature T. a) Write down the probability for a single particle to be moving with momentum |k| and in the excited energy level. b) Derive the canonical partition function of a single particle. [You may use the Gaussian integral: 2 22 dx x²e-x²y² dxxe 0 πT 4y³ and approximate the density of states in three dimensions as g(k)dk ≈ V k²dk] 2πT c) Hence, or otherwise, find the internal energy per particle of this gas and comment on how it behaves at large temperature d) Plot the heat capacity Cy of this gas as a function of temperature, assuming that the energy level separation is ε/kB ≈ 1 K.

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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