This derivation
based on statistical mechanics was originally provided by Volmer and Mahnert in 1925. The partition function of the finite number of adsorbents adsorbed on a surface, in a canonical ensemble, is given by
where  is the partition function of a single adsorbed molecule,  is the number of adsorption sites (both occupied and unoccupied), and  is the number of adsorbed molecules which should be less than or equal to . The terms in the bracket give the total partition function of the  adsorbed molecules by taking a product of the individual partition functions (refer to Partition function of subsystems). The  factor accounts for the overcounting arising due to the indistinguishable nature of the adsorbates. The grand canonical partition function is given by
 is the chemical potential of an adsorbed molecule. As it has the form of binomial series, the summation is reduced to
where 
The grand canonical potential is
based on which the average number of occupied sites is calculated
which gives the coverage 
Now, invoking the condition that the system is in equilibrium, that is, the chemical potential of the adsorbed molecules is equal to that of the molecules in gas phase, we have
The chemical potential of an ideal gas is
where  is the Helmholtz free energy of an ideal gas with its partition function
 is the partition function of a single particle in the volume of  (only consider the translational freedom here).
We thus have , where we use Stirling's approximation.
Plugging  to the expression of , we have
which gives the coverage
By defining
and using the identity , finally, we have
It is plotted in the figure alongside demonstrating that the surface coverage increases quite rapidly with the partial pressure of the adsorbants, but levels off after P reaches P.