# The grand canonical partition function and a gas/2D solid?

1. Apr 28, 2010

### the_phys

Hi,
I've got this homework problem on my statistical physics module and I'm really unsure about it as this stuff is all new to me. I have an "atomically flat" solid substrate in contact with a gas of molecular mass m, and the two are in thermal equilibrium. The substrate has a total of M sites where a single gas molecule can be adsorbed onto the surface. The energy gain is -W when this happens, and I am supposed to calculate "the grand canonical partition function of the adsorbed layer, in terms of the chemical potential $$\mu_a_d$$."

So, I've been through my notes and got an expression for the grand canonical partition function, which is $$Z = \sum_s e^{(\mu N_{s} - E_{s})/k_{B}T}$$ (where the sum is over s).
I've started by saying that at any given time, some of the sites will be occupied, and the others vacant. The maximum number of occupied sites possible is M, so the maximum energy gain is -MW, although I'm not sure if that's going to be useful yet.

If s refers to some state of the system, Es is the corresponding energy gain and Ns is the number of occupied sites, then Es must be equal to -NsW. However, I am not 100% certain that defining Ns the way I have done is correct - if anyone recognises this as being incorrect, please tell me.

I have then said that, say, the state s=1 has 1 occupied site, so Ns=1, state s=2 has 2 occupied site, so Ns=2, etc. so that I can replace Ns with just s. Then I get $$Z = \sum e^{(s\mu_{ad} - -sW)/k_{B}T} = \sum e^{s(\mu_{ad} +W)/k_{B}T}$$, and since 0 </= s </= M, the sum is between s=0 and s=M.

However, I am unconvinced of my answer for a couple of reasons. One is that for the state s, I am not sure whether I should take that to mean every different s means a different number of occupied sites and a different energy, or whether different states s could mean the same number of sites occupied but just different ones. Take this example (+ = occupied, - = vacant) for M = 4, and forget about setting Ns equal to s for now:

+ -
- -

- +
- -

- -
+ -

- -
- +

Is that different examples of the same s, or is that 4 different s values?

The other thing that I was worried about was that I am supposed to express Z in terms of $$\mu_{ad}$$. All I have done in my answer is set $$\mu = \mu_{ad}$$ from the expression in my notes. Is this justified or is there some complicated relationship between $$\mu$$ and $$\mu_{ad}$$?

I'm really not confident with this stuff right now, so any help at all would be appreciated. If I'm honest I'm not even sure what the "grand canonical partition function" is supposed to tell us. Please speak up if you feel like you can contribute anything at all.
Thanks.