Stat-Mech problem: pressure from a partition function

[TroyElliott]

Homework Statement

A vessel having a volume $V$ initially contains $N$ atoms of dilute (ideal) helium gas in thermal equilibrium with the surroundings at a temperature $T$, with initial pressure $P_{i} (T ,V ) = \frac{NRT}{V}$ . After some time, a number of helium atoms adhere to the walls of the vessel, each occupying one of $N_{0}$ available surface states having binding energy $\Delta$, where $N_{0}>>N.$ When $M$ atoms are adsorbed on the surface, the partition function for the system is given by

$$Z = \frac{q^{N-M}(N_{0}e^{\beta \Delta})^{M}}{M!(N-M)!}$$

where $q = V\sqrt{\frac{mkT}{2\pi\hbar^{2}}}.$

Show that the final pressure is $P_{f} = P_{i}(1+\frac{N_{0}}{q}e^{\beta \Delta})^{-1},$

after equilibrium is reached between the gas and the surface.

Homework Equations

$P_{final} = \frac{1}{\beta}\frac{\partial \ln{Z}}{\partial V}$

The Attempt at a Solution

After using $P = \frac{1}{\beta}\frac{\partial \ln{Z}}{\partial V},$ I get $P = \frac{(N-M)}{\beta V}.$ I am not seeing how I can write this such that the answer appears like $P_{f} = P_{i}(1+\frac{N_{0}}{q}e^{\beta \Delta})^{-1}.$ Any ideas on where to go from here?

Edit: From here I thought that I could relate $N-M$ to the following $$<M> = \frac{Ne^{\beta \Delta}}{Z},$$

or

$$<N-M> = \frac{N}{Z}$$

 

[TSny]

$$Z = \frac{q^{N-M}(N_{0}e^{\beta \Delta})^{M}}{M!(N-M)!}$$
where $q = V\sqrt{\frac{mkT}{2\pi\hbar^{2}}}.$

I hope I'm not leading you astray. But, I believe the actual partition function for the system would be obtained by summing the above expression for $Z$ over all allowable values of M. The sum can be carried out without much difficulty and the resultant $Z$ appears to lead to the given expression for the final pressure.

The reason for summing over M is that M will vary in time as molecules in the gas phase randomly attach to the wall and then get kicked off the wall by collisions from other molecules in the gas phase. So, the possible micro-states of the system will include states with various values of M.

 

[TroyElliott]

I hope I'm not leading you astray. But, I believe the actual partition function for the system would be obtained by summing the above expression for $Z$ over all allowable values of M. The sum can be carried out without much difficulty and the resultant $Z$ appears to lead to the given expression for the final pressure.

The reason for summing over M is that M will vary in time as molecules in the gas phase randomly attach to the wall and then get kicked off the wall by collisions from other molecules in the gas phase. So, the possible micro-states of the system will include states with various values of M.

I hope I'm not leading you astray. But, I believe the actual partition function for the system would be obtained by summing the above expression for $Z$ over all allowable values of M. The sum can be carried out without much difficulty and the resultant $Z$ appears to lead to the given expression for the final pressure.

The reason for summing over M is that M will vary in time as molecules in the gas phase randomly attach to the wall and then get kicked off the wall by collisions from other molecules in the gas phase. So, the possible micro-states of the system will include states with various values of M.

Thank you for the suggestion. I think it would be strange for the writer of the problem to purposely call $Z$ the partition function, but then not include a summation over the states. I will take a look at your suggestion and see if it leads to the correct answer. Thanks!

 

[robphy]

Possibly helpful: This document has the problem statement.
https://physics.unm.edu/pandaweb/graduate/prelims/SM_F16.pdf

 

[TroyElliott]

Possibly helpful: This document has the problem statement.
https://physics.unm.edu/pandaweb/graduate/prelims/SM_F16.pdf

Thanks for the link. The problem seems to be identically worded, so I don't think it will provide any new insight.