Chem potential for adsorbed molecule from partition function

[magnesium12]

Homework Statement

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.

Homework Equations

q(T) ≡ exp[−βA_surface]
Q=∑exp[-βε]
Q=q^N
β=1/k_bT
μ = (dA/dN)_T,V

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!

 

[mark726]

Hi there,

Your approach is on the right track, but there are a few things to consider. First, the partition function for the gas molecules in the canonical ensemble would be Q_gas = ∑exp[-βε], where the sum is over all possible energy levels of the gas molecules. This is different from the partition function for the adsorbed molecules, q(T) = exp[-βA_surface], which only considers the energy of the adsorbed molecule on the surface.

To find the chemical potential for the adsorbed molecules, you can use the relation μ = (dA/dN)_T,V, as you mentioned. However, instead of taking the derivative with respect to T, you should take the derivative with respect to N. This will give you the chemical potential for a single adsorbed molecule. Then, you can multiply by M!/N! to account for the indistinguishability and the number of ways to adsorb onto the surface.

Hope this helps! Keep up the good work on your research.