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# phys4146 - (a(2 points At very low temperatures the entropy...

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1 PHYS 414: Homework 7 This homework due Wednesday March 21 1. A large number N of independent fermions are in a box of volume V at temperature T . (a) (2 points) Express the pressure P exerted by this gas as a function of V and the total energy U . (b) (2 points) This result differs from that for black-body radiation. Explain the origin of this difference. (c) (2 points) In the limit of small V at fixed temperature it is approximately true that PV γ T α . Evaluate γ and α for this system. (d) (2 points) In the limit of large V at fixed temperature it is approximately true that PV γ T α . Evaluate γ and α for this system. (e) (2 points) For an adiabatic change, PV γ = constant. Evaluate γ for this system. 2. A non-interacting Fermi gas of N independent particles in D dimensions has a density of states that varies with energy E as E ( D/ 2 - 1) . Answer the following questions for each of the three cases of systems in one, two, and three dimensions.
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Unformatted text preview: (a) (2 points) At very low temperatures the entropy of the system varies with the temperature T as T ν . What is the numerical value of ν ? (b) (2 points) At T = 0 the energy of the system is E 1 . At a temperature τ such that k B τ ± E 1 /N the energy is E 2 . What is the approximate energy at temperature 2 τ ? (c) (2 points) If more particles are added, but the temperature is held constant, does the entropy increase or decrease? Why? (d) (2 points) If the temperature is raised, but the number of particles is held constant, does the chemical potential increase or decrease? Why? (e) (2 points) Do the answers to parts ( d ) and ( e ) (increase or decrease) have to be the same or do they have to be diﬀerent? Why?...
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