PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Statistical foundations and the equilibrium ensembles · Statistical foundations and the equilibrium ensembles
Thermal Physics, Ch The Boltzmann factor, the canonical ensemble and the partition function
Problem
A crystal contains N=1.0×1020 identical defect sites. Each site is fixed at its own place in the lattice, so the sites are distinguishable, and they do not interact with one another; each is weakly coupled to the crystal, which acts as a reservoir at temperature T. Each site has exactly two states, one of energy 0 and one of energy ε=4.00×10−21J.
((a)) Write the single-site partition function Z1 and the partition function ZN of the whole set of sites, and hence the Helmholtz free energy F(T). Evaluate F at T=290K.
((b)) Using the thermodynamic relation S=−(∂F/∂T)V of Chapter 7, obtain S(T) in closed form and evaluate it at T=290K.
((c)) As T→∞ every state of every site becomes equally likely, so the multiplicity of Chapter 11 is Ω=2N. Evaluate the entropy S=kBlnΩ this gives, and quote the ratio of your answer to (b) to it. Take kB=1.381×10−23JK−1.