And now what we can do is we can write down the single particle

partition function of an adsorbed molecule as simply the product

of the one dimensional harmonic oscillator in each direction.

Now each adsorbed molecule occupies a lattice site

with a distinct spatial location.

So are these particles distinguishable or indistinguishable?

Now because of their localized location, they are indeed distinguishable particles.

Let's take the case that the lattice has M sites and containing N adsorbed molecules.

What is the degeneracy of this configuration?

Well, this is simply given by the combinatorial factor M choose N.

Now the canonical partition function for

the localized adsorption is given by the product of the degeneracy factor

times the single molecule partitition raised to the power N.

Now the chemical potential of the adsorbate can be evaluated by taking

a partial derivative of the logarithm of the partition function

with respect to the number of adsorbed molecules.

Now, let's assume that the molecules are adsorbing from the gas phase.

Now this implies that there is an equilibrium between the gas phase and

the chemical potential of the adsorb state.

Remember for

an ideal gas, the chemical potential can be written as a sum of two parts.

One that depends only on temperature and

then a pressure dependent part that scales as the logarithm of the pressure.

Now equality condition from the equilibrium

leads to the famous Langmuir Adsorption Isotherm.