Langmuir Adsorption Model - Derivations of The Langmuir Adsorption Isotherm - Statistical Mechanical Derivation

Statistical Mechanical Derivation

This derivation 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, are the number of sites available for adsorption. Hence, N, which is the number of molecules that can be adsorbed, can be less or equal to N_s. The first term of Z(n) accounts the total partition function of the different molecules by taking a product of the individual partition functions (Refer to Partition function of subsystems). The latter term accounts for the overcounting arising due to the indistinguishable nature of the adsorption sites. The grand canonical partition function is given by

As it has the form of binomial series, the summation is reduced to

where

The Landau free energy, which is generalized Helmholtz free energy is given by

According to the Maxwell relations regarding the change of the Helmholtz free energy with respect to the chemical potential,

which gives

Now, invoking the condition that the system is in equilibrium, the chemical potential of the adsorbates is equal to that of the gas surroundings the absorbent.

where N3D is the number of gas molecules, Z3D is the partition function of the gas molecules and A_g=-k_BT ln Z_g. Further, we get

where

Finally, we have

It is plotted in the figure alongside demonstrating the surface coverage increases quite rapidly with the partial pressure of the adsorbants but levels off after P reaches P₀.