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

[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, E_s is the corresponding energy gain and N_s is the number of occupied sites, then E_s must be equal to -N_sW. However, I am not 100% certain that defining N_s 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 N_s=1, state s=2 has 2 occupied site, so N_s=2, etc. so that I can replace N_s 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 N_s 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.

 

[mark726]

Thank you for reaching out for help on this homework problem. As a scientist with expertise in statistical physics, I would be happy to offer some guidance on how to approach this problem and provide some clarification on the concepts involved.

Firstly, your expression for the grand canonical partition function is correct. It is a sum over all possible states of the system, where each state has a corresponding energy and number of particles. In this case, the particles are gas molecules adsorbed onto the substrate, and the energy gain is -W when this happens. So, your expression is correct in terms of the variables used.

To address your concerns about the definition of Ns, it is correct to define it as the number of occupied sites in a given state s. This means that different states s could have the same number of occupied sites, but they would still be considered different states because they have different arrangements of occupied and vacant sites. In your example, each arrangement would be considered a different state.

Now, to express Z in terms of \mu_{ad}, we need to consider the relationship between chemical potential and the number of particles in a grand canonical ensemble. In this case, \mu_{ad} represents the chemical potential for the adsorbed layer, which is related to the number of adsorbed particles by the equation \mu_{ad} = -k_{B}T ln(N_{ad}/N), where N_{ad} is the number of adsorbed particles and N is the total number of particles in the system. So, you can substitute this relationship into your expression for Z to get it in terms of \mu_{ad}.

The grand canonical partition function tells us about the statistical properties of the system, such as the average number of particles and the fluctuations in that number. It also allows us to calculate other thermodynamic quantities, such as the Helmholtz free energy and entropy. It is a fundamental concept in statistical physics and has many applications in understanding different physical systems.

I hope this helps clarify some of your concerns and gives you a better understanding of the problem. If you have any further questions or need more guidance, please don't hesitate to ask. Best of luck with your homework!