Gas Adsorption

Consider a vapor (dilute monatomic gas) in equilibrium with a submonolayer (i.e., less than one atomic layer) of atoms adsorbed on a surface. Model the binding of atoms to the surface by a potential energy V = -ε₀. Assume there are N₀ possible sites for adsorption, and find the vapor pressure as a function of surface concentration θ = N/N₀ (N is the number of adsorbed particles).

SOLUTION

For two systems in equilibrium, the chemical potentials should be equal. Consider one of the systems as an ideal gas (vapor) in a volume, and another as a surface submonolayer film. For an ideal gas the free energy F is given by

(1)

where ε_k and Z' correspond to the energy states and statistical sum associated with the internal degrees of freedom. If the temperature is reasonably small, τ << τ_ion, where τ_ion corresponds to the ionization energy of the atoms, so that the atoms are not ionized and mostly in the ground state, and this state is nondegenerate, we can take Z' = 1, and then (1) becomes

(2)

The Gibbs free energy G is given by

(3)

where we have expressed G as a function of P and τ, using PV = Nτ. The chemical potential μ = G/N, so

(4)

Now, consider an adsorption site: we can apply a Gibbs distribution with a variable number of particles to this site:

(5)

where the possible occupational numbers of the site for a submonolayer n = 0,1 (site is empty, site is occupied), with energy E_0n = -nε₀. Performing the sums, we have

(6)

The average number of particles per site ⟨n⟩ may be written

(7)

The total number of adsorbed particles N is given by

(8)

The surface concentration θ is simply

(9)

Substituting μ for an ideal gas from (4) into (9), we have

(10)

and

(11)

where

(12)

(9) can also be derived by considering the canonical ensemble. The number of possible ways of distributing N atoms among N₀ sites is

The partition function is then

(13)

and the average number of particles

(14)

the same as (8).