A defect on the surface of an oxide ceramic binds at most one oxygen molecule
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 grand canonical ensemble, ensemble equivalence and fluctuations
Problem
A defect on the surface of an oxide ceramic binds at most one oxygen molecule. The empty site has energy zero; the single occupied state has energy −ε0 with ε0=0.80eV. The defect is weakly coupled to the surrounding oxygen, which is a reservoir at T=500K with chemical potential μ=−0.70eV. In the same oxygen (T=500K, pressure p=1.0Pa, treated as an ideal gas) consider a fixed volume V=1.0mm3 marked out inside the gas, with imaginary walls so that it is open to its surroundings.
((a)) the grand partition function Z of the defect site;
((b)) its mean occupation ⟨N⟩;
((c)) its grand potential ΦG, in joules;
((d)) the mean number of molecules in the marked volume;
((e)) the relative number fluctuation σN/⟨N⟩ there. Take kB=1.381×10−23JK−1 and 1eV=1.602×10−19J.