Elementary Statistical Derivation of the Langmuir Isotherm

As an illustration of the statistical method at its simplest we may consider the derivation of the Langmuir isotherm. The Langmuir model (see Section 2.4) is based on the assumption of a fixed number of identical sites with no interaction between adsorbed molecules. We consider a set of M equivalent sites on which N molecules (N M) are adsorbed. The canonical partition function for the system is given simply by 

In this expression / is the molecular partition function for an individual adsorbed molecule while the factor M /N M - N) is simply the degeneracy factor or the number of ways in which the N identical molecules can be arranged on the M equivalent but physically distinguishable sites. Since both M and N are large numbers we may use Stirling s theorem (lnN = JVlnAf -N) and write 

The advantage of this derivation over the simpler kinetic derivation given in Section 2.4 is that the physical significance of the constant b is more clearly defined. 

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