Statistical mechanics- Stirling's Approximation and Particle Configurations

[aurora14421]

Homework Statement

N weakly interacting distinguishable particles are in a box of volume V. A particle can lie on one of the M possible locations on the surface of the box and the number of states available to each particle not on the surface (in the gas phase) is aV, for some constant a.

1. What is the number of configurations for n particles on the surface?

2. What is the number of configurations for the remaining N-n particles in the gas phase (i.e. not on the surface)?

3. Show the entropy, S, of the configuration of n particles on the surface and N-n particles in the gas phase is given by:

S = k[M ln(M) - n ln(n)-(M-n) ln(M-n) + (N-n) ln(aV)]

k is Boltzmann's constant.

Homework Equations

$$S=k ln (\Omega)$$

ln (N!) = N ln(N) - N (Stirling's approximation)

The Attempt at a Solution

1. You have n atoms and M possible locations, so number of configurations is:

$$\binom{M}{n}$$

2. You have aV possible locations and N-n atoms so number of configurations is:

$$\binom{aV}{N-n}$$

3. $$\Omega = \binom{M}{n}\binom{aV}{N-n}$$ and use Stirling's approximation in the expression for entropy.

I can't get the algebra to work in this question, which makes me think that I've got part 1 or 2 (or both) wrong.

Any help would be appreciated. Thanks.

 

[illwerral]

In part 1 it seems you are neglecting that each location has aV configurations...

 

[aurora14421]

Sorry, I mistyped the question. It's fixed now. So there are only M possible sites on the surface and aV states when it's not on the surface.