An idealised isothermal column of pure nitrogen (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

An idealised isothermal column of pure nitrogen (molecular mass 28 u28\,\mathrm{u}), an ideal gas at T=300 KT = 300\,\mathrm{K} at every height and in diffusive equilibrium in the uniform field g=9.81 m s−2g = 9.81\,\mathrm{m\,s^{-2}}, has pressure p0=150 Pap_0 = 150\,\mathrm{Pa} at its base. Adsorption sites on the wall each hold at most one nitrogen molecule; a site is empty with energy 00 or occupied with energy −ε0-\varepsilon_0, ε0=0.60 eV\varepsilon_0 = 0.60\,\mathrm{eV}, measured from a free molecule at the same height. Sites are independent and weakly coupled to the gas, which is the reservoir. ((a)) From the grand partition function of one site, and the ideal-gas chemical potential, obtain the coverage in the form θ=p/(p+p1/2)\theta = p/(p+p_{1/2}) and evaluate p1/2p_{1/2} at this temperature. ((b)) What is the coverage of a site at the base? ((c)) At what height above the base is a site occupied half the time? You may use μ=kBTln⁡(nλT3)\mu = k_BT\ln(n\lambda_T^3) with λT=h/2πmkBT\lambda_T = h/\sqrt{2\pi mk_BT} (Chapter 14) and the uniformity of the total chemical potential, internal plus mgzmgz, in a uniform field (Chapter 8). Constants: kB=1.381×10−23 J K−1k_B = 1.381\times10^{-23}\,\mathrm{J\,K^{-1}}, h=6.626×10−34 J sh = 6.626\times10^{-34}\,\mathrm{J\,s}, 1 u=1.661×10−27 kg1\,\mathrm{u} = 1.661\times10^{-27}\,\mathrm{kg}, 1 eV=1.602×10−19 J1\,\mathrm{eV} = 1.602\times10^{-19}\,\mathrm{J}.

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