Statistical mechanics of adsorption

Statistical mechanics of adsorption

The Langumuir adsorption model can be used to model the adsorption of particles onto a surface. To derive this model we begin by noting that if we have $N < M$ atoms adsorbed on $M$ adsorption sites the canonical partition function is given by: $$ Z(N) = \frac{M!}{N!(M-N)!} \zeta^N $$ where $\zeta$ is the partition function for a single adsorbed particle. Also notice that in writing the above we have assumed that there is no interaction between the adsorbed particles. We then recall that the grand canonical partiton function can be calculated from the canonical partition function using: $$ Z(\mu) = \sum_{N=0}^M e^{\mu \beta N} Z(N) = \sum_{N=0}^M \frac{M!}{N!(M-N)!} (e^{\mu\beta} \zeta)^N = ( 1 + e^{\mu\beta} \zeta)^M $$ where $\mu$ is the chemical potential. The last result here holds because of the binomial theorem. We then recall that we can get the fraction of the sites on the surface that are occupied by gas particles by calculating: $$ \langle K \rangle = \frac{1}{M} \frac{\partial \ln Z(\mu) }{\partial (\beta \mu) } = \frac{ e^{\mu\beta} \zeta }{1 + e^{\mu\beta} \zeta } $$

Syllabus Aims