A trapping site inside a zeolite pore holds at most one methane 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 trapping site inside a zeolite pore holds at most one methane molecule. Its energy is zero when empty and

-\varepsilon_0

when occupied, with

\varepsilon_0 = 0.30\,\mathrm{eV}

, and there is a single occupied state. The site is weakly coupled to the methane gas around it, which acts as a reservoir at

T = 250\,\mathrm{K}

with chemical potential

\mu = -0.26\,\mathrm{eV}

. In the same gas (

T = 250\,\mathrm{K}

, pressure

p = 5.0\times10^{4}\,\mathrm{Pa}

, treated as ideal) consider a fixed cubical cell of edge

100\,\mathrm{nm}

bounded only by imaginary walls, so that molecules pass freely in and out. ((a)) the grand partition function

\mathcal Z

of the site; ((b)) the mean occupation

\langle N\rangle

of the site; ((c)) the grand potential

\Phi_G

of the site, in joules; ((d)) the mean number of molecules in the cell; ((e)) the relative number fluctuation

\sigma_N/\langle N\rangle

in the cell. Take

k_B = 1.381\times10^{-23}\,\mathrm{J\,K^{-1}}

and

1\,\mathrm{eV} = 1.602\times10^{-19}\,\mathrm{J}

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