# Chem potential for adsorbed molecule from partition function

1. Feb 4, 2016

### magnesium12

1. The problem statement, all variables and given/known data
Consider a gas in equilibrium with a surface. The surface can adsorb the gas molecules onto any of M independent, distinguishable sites. The molecular partition function for an adsorbed molecule is q(T) ≡ exp[−β A_surface].
a) Assume that the adsorbed molecules are independent and use the canonical ensemble for N adsorbed molecules to compute the chemical potential for the molecules on the surface. Remember to include the number of ways to adsorb the molecules onto the surface. Also remember that the molecules themselves are indistinguishable.

2. Relevant equations
q(T) ≡ exp[−βAsurface]
Q=∑exp[-βε]
Q=qN
β=1/kbT
μ = (dA/dN)T,V

3. The attempt at a solution
Since the gas is at equilibrium with the surface, the partition functions should be equal so
q(T) = Q
exp[-βA] = ∑exp[-βε]
Take ln of both sides to get:
A = -[∑-βε]/β
But since we are only summing over ε, we can pull β out of the sum so that it cancels out with the one on the bottom and I'm left with A=∑ε and I don't think there's an expression for ε, so this leaves me nowhere.

Am I right in thinking that I have to find an expression for A by equating q and Q?
To get μ from there, I think I just take the derivative with respect to T and correct for indistinguishability and number of ways to adsorb onto the surface by multiplying by M!/N! .

Could someone point me in the right direction if this is totally off base?
Thanks!

2. Feb 9, 2016