Langmuir Equation - Statistical Derivation

Statistical Derivation

Langmuir isotherm can also be derived using statistical mechanics with the following assumptions:
1. Suppose there are M active sites to which N particles bind.
2. An active site can be occupied only by one particle.
3. Active sites are independent. Probability of one site being occupied is not dependent on the status of adjacent sites.

The partition function for a system of N particles adsorbed to M sites (under the assumption that there are more sites than the particles) is:

with being the distribution function for one particle:

and .

If we allow the number of particles to increase so that all sites are occupied, the partition function becomes:

We can see that this partition function of a single active state can be expressed as

.

The average number of occupied spaces can now be easily calculated.

Rearranging yields

And finally: