An Einstein solid consists of identical quantised oscillators

PHYSICS · Subjective · Fermi University Physics · Thermal, Statistical & Complex Systems · Statistical foundations and the equilibrium ensembles · Statistical foundations and the equilibrium ensembles

Thermal Physics, Ch Microstates, multiplicity and the statistical entropy

Problem

An Einstein solid consists of N=60N = 60 identical quantised oscillators, weakly coupled to one another and isolated from everything else. Each oscillator can hold only whole units of energy of size hν=4.0×10−21 Jh\nu = 4.0\times10^{-21}\,\mathrm{J}, measured from the lowest level of each oscillator, and every accessible microstate of the solid is equally likely. The multiplicity of the macrostate with qq units shared among the NN oscillators is Ω(N,q)=(q+N−1q)\Omega(N,q) = \binom{q+N-1}{q}. The solid holds q=30q = 30 units. Estimate its temperature by treating the entropy slope (∂S/∂U)V,N(\partial S/\partial U)_{V,N} as the change in S=kBln⁡ΩS = k_B\ln\Omega when a single extra unit of energy is added, the volume and NN being held fixed. Give the answer to two significant figures.

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