A single adsorption site on a tungsten surface can be either empty or hold exactly…

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 single adsorption site on a tungsten surface can be either empty or hold exactly one nitrogen molecule. Take the energy of the empty site as zero; when the site holds a molecule its energy is

-\varepsilon_0

with

\varepsilon_0 = 0.42\,\mathrm{eV}

. The site is weakly coupled to the surrounding nitrogen gas, which acts as a reservoir at temperature

T = 300\,\mathrm{K}

and chemical potential

\mu = -0.35\,\mathrm{eV}

. There is only one occupied state and one empty state. Separately, in the same gas (

T = 300\,\mathrm{K}

, pressure

p = 1.00\times10^{5}\,\mathrm{Pa}

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

0.500\,\mathrm{\mu m}

whose walls are imaginary, so the cell is freely open to the gas around it. ((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

\langle N\rangle

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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