This section provides a kinetic derivation for a single-adsorbate case. The kinetic derivation applies to gas-phase adsorption. However, it has been mistakenly applied to solutions. The multiple-adsorbate case is covered in the competitive adsorption sub-section.
The model assumes adsorption and desorption as being elementary processes, where the rate of adsorption r and the rate of desorption r are given by
where p is the partial pressure of A over the surface, [S] is the concentration of free sites in number/m, [A] is the surface concentration of A in molecules/m (concentration of occupied sites), and k and k are constants of forward adsorption reaction and backward desorption reaction in the above reactions.
At equilibrium, the rate of adsorption equals the rate of desorption. Setting r = r and rearranging, we obtain
The concentration of sites is given by dividing the total number of sites (S) covering the whole surface by the area of the adsorbent (a):
We can then calculate the concentration of all sites by summing the concentration of free sites [S] and occupied sites:
Combining this with the equilibrium equation, we get
We define now the fraction of the surface sites covered with A as
This, applied to the previous equation that combined site balance and equilibrium, yields the Langmuir adsorption isotherm: