**3.3 Langmuir Isotherm - derivation from kinetic considerations**

The equilibrium that may exist between gas adsorbed on a surface and molecules in the gas phase is a dynamic state, i.e. the equilibrium represents a state in which the rate of adsorption of molecules onto the surface is exactly counterbalanced by the rate of desorption of molecules back into the gas phase. It should therefore be possible to derive an isotherm for the adsorption process simply by considering and equating the rates for these two processes.

Expressions for the rate of adsorption and rate of desorption have been derived in Sections 2.3 & 2.6 respectively : specifically ,

Equating these two rates yields an equation of the form :

[1] |

where the terms *f*(θ) & *f* '(θ) contain the pre-exponential surface coverage dependence of the rates of adsorption and desorption respectively and all other factors have been taken over to the right hand side to give a temperature-dependent "constant" characteristic of this particular adsorption process, *C*(*T*) .

We now need to make certain simplifying assumptions ... the first is one of the key assumptions of the Langmuir isotherm

Adsorption takes place only at specific localized sites on the surface and the saturation coverage corresponds to complete occupancy of these sites.

Let us initially further restrict our consideration to a simple case of **reversible molecular adsorption**. i.e.

**S -**** + M**_{ (g)} = **S - M**

where **S -** represents a vacant surface site and **S - M** the adsorption complex.

Under these circumstances it is reasonable to assume coverage dependencies for rates of the two processes of the form :

Adsorption, |
i.e. proportional to the fraction of sites that are unoccupied. |

Desorption, |
i.e. proportional to the fraction of sites which are occupied by adsorbed molecules. |

where θ is the fraction of sites occupied at equilibrium.

**Note** : these coverage dependencies are exactly what would be predicted by noting that the forward and reverse processes are elementary reaction steps , in which case it follows from standard chemical kinetic theory that

*the forward adsorption process will exhibit kinetics having a first order dependence on the concentration of vacant surface sites.**the reverse desorption process will exhibit kinetics having a first order dependence on the concentration of adsorbed molecules.*

Substitution into equation [1] then yields

where *B*(*T*) = (*c*'/*c*).*C*(*T*) . After rearrangement this gives the Langmuir Isotherm expression for the surface coverage

where *b* ( = 1/*B*(*T*) ) is a function of temperature and contains an exponential term of the form

** b = **......

Consequently, *b* can only be regarded as a constant with respect to coverage if the enthalpy of adsorption is itself independent of coverage - this is the second major assumption of the Langmuir Isotherm.

Further study : What is the Langmuir Isotherm for dissociative adsorption ?