Question

A solid surface is in diffusive and thermal equilibrium with a monatomic ideal gas at temperature t and chemical potential μ. An adsorption site on the solid surface can be either empty (energy ε0 = 0), singly occupied (ε1 = ε), or doubly occupied (ε2 = 2ε) by an atom from the gas. If two atoms adsorb onto the same site, they also interact by a vibrational mode of frequency w. Thus, the energy of a doubly occupied site can take on the values ε2 + n!w, where n = 0, 1, 2,…

a) Prove that the grand partition function can be written in the form

ℨ = 1 + ??^-ε/? + (?)^2??^-2ε/?

where ? = ?μ/? is the absolute activity of the gas and ? is the vibrational partition

function. Show all work including evaluation of ?.

b) Use the Gibbs factor to determine the ratio of probabilities for single versus

empty occupancy and double versus single occupancy.

c) What are the limiting probability ratios when the vibrational frequency is large

such that ℏ? ≫ ??

d) Prove that when the vibrational frequency is very small (ℏ? ≪ ?), double

occupancy becomes strongly favored.

Answer 1