# Finding the Grand Potential for Oxygen Molecules adsorbed to carbon nanotube surface

1. Dec 8, 2012

### rmjmu507

I'm trying to follow the logic/write out the corresponding equations to something I read in a Statistical Mechanics book, but am having some trouble.

I'm trying to calculate the grand potential Ω for a carbon nanotube lattice surface, where there is no limit to how many molecules can be adsorbed on one site. In other words, the are n_i adsorbed molecules, and i varies from 1 to A, where A is the total number of sites. I'm also told that regardless of the position of the molecule, the energy gain per molecule is ALWAYS ε.

The book provides this as an exercise, and instructs me to:

- Calculate the grand potential Ω of the nanotube surface by expressing e$^{-Ω/kt}$ as a sum over all n_i values, and thus prove that Ω=Akt ln(1-e$^{(μ+ε)/kT}$.

I wrote the partition function as Z= sum over n_i of e$^{(μ+ε)/kT}$, and the grand canonical as η=Z^{n_i}. I know ln η gives the grand potential, but I do not obtain the desired value - I set up my sum incorrectly, but cannot find the mistake(s).