One adsorption site on a nickel surface at is exposed to a mixture of carbon monoxide…

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Thermodynamic potentials and the formal structure of thermodynamics · Statistical foundations and the equilibrium ensembles

Thermal Physics, Ch The grand canonical ensemble, ensemble equivalence and fluctuations

Problem

One adsorption site on a nickel surface at

T = 300\,\mathrm{K}

is exposed to a mixture of carbon monoxide and nitrogen. The site is empty, or holds exactly one CO molecule, or holds exactly one

\mathrm{N_2}

molecule; it can never hold two molecules. The empty site has energy zero; a bound molecule has energy

-\varepsilon_0

, with binding energies

\varepsilon_{\rm CO} = 1.20\,\mathrm{eV}

and

\varepsilon_{\rm N_2} = 0.80\,\mathrm{eV}

. Under the conditions of the experiment the two gases have chemical potentials

\mu_{\rm CO} = -0.55\,\mathrm{eV}

and

\mu_{\rm N_2} = -0.40\,\mathrm{eV}

. The site is weakly coupled to the gas mixture, which acts as a reservoir fixing

T

and both chemical potentials. Take

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

. ((a)) Write the grand partition function

\mathcal Z

of the site as a sum of three terms and evaluate it numerically. ((b)) Derive expressions for the mean occupations (coverages)

\theta_{\rm CO}

and

\theta_{\rm N_2}

of the site by each species, and evaluate them. ((c)) Show that

\theta_{\rm CO}/\theta_{\rm N_2} = \exp\{[(\varepsilon_{\rm CO}+\mu_{\rm CO}) - (\varepsilon_{\rm N_2}+\mu_{\rm N_2})]/k_BT\}

, so that the species with the larger

\varepsilon_0 + \mu

takes the site, and evaluate this ratio.

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