A tall closed column contains pure methane (molecular mass )
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 tall closed column contains pure methane (molecular mass 16u), treated as an ideal gas held at the single temperature T=250K throughout and in diffusive equilibrium in the uniform gravitational field g=9.81ms−2. At the base of the column the pressure is p0=3.0Pa. Identical adsorption sites are fixed to the inner wall at various heights z above the base. Each site is either empty (energy 0, no molecule) or holds exactly one methane molecule, whose energy is then −ε0 with ε0=0.55eV, measured from a free gas molecule at the same height. Sites do not interact with one another and are weakly coupled to the gas, which acts as the reservoir fixing T and the local chemical potential.
((a)) Write the grand partition function Z of one site and show that its mean occupation is θ=p/(p+p1/2), where p is the local pressure and p1/2=(kBT/λT3)e−ε0/kBT. Evaluate p1/2.
((b)) Find the coverage θ of a site at the base.
((c)) Find the coverage of a site z=8.0km above the base. You may use the ideal-gas results of Chapter 14, μ=kBTln(nλT3) with λT=h/2πmkBT, and Chapter 8's statement that in a uniform field it is the total chemical potential, internal plus mgz, that is uniform at equilibrium. Constants: kB=1.381×10−23JK−1, h=6.626×10−34Js, 1u=1.661×10−27kg, 1eV=1.602×10−19J.