# Grand canonical partition function of 2d lattice

Assume we put a gas, such as a simple atomic system (Argon), into contact with a two dimensional surface so that this gas acts as a reservoir of particles. Assume that the chemical potential of this reservoir is $\mu$. Furthermore, assume that on this surface there are $M$ lattice sites, arranged on a triangular lattice, which can adsorb molecules from the gas with a binding energy $\epsilon_B$.

Assume that the particles on the surface do not interact at all, i.e. multiple patricles can share a lattice site at no cost.

I would like to find the grand canonical partition function for the 2d surface.

I thought the following:

$$\Xi(\mu, V, T) = \sum^\infty_{N=0}\exp[-\mu \beta N]\binom{M}{N}\exp[-\beta \epsilon_B N]$$ However, I'm not entirely sure this is correct, since there shouldn't be a state in which $N=0$ (no particle is adsorbed), right? Shouldn't this be the grand canonical partition function if two particles CAN'T be on the same site, but all are adsorbed?

Any clarification is very welcome

## 1 Answer

The state in which $N=0$ is certainly a valid and necessary part of the partition function. While it would have a low Boltzmann weight compared to the occupied states, it contributes nonetheless.

Yes, as written, this sum would terminate once all sites are occupied. However, there isn't a need for a binomial coefficient at all. Since there are no interactions and particles can occupy the same site, the multiplicity of a given state is simply given by $M^N$. There's no difference energetically between one site and another, so each particle is independent of the others and has $M$ options for where to go. So, your partition function becomes

\begin{equation} \Xi(\mu,V,T) = \sum_{N=0}^{\infty} e^{-\beta \mu N} M^N e^{-\beta \epsilon_B N} = \sum_{N=0}^{\infty} (Me^{-\beta (\mu + \epsilon_B)})^N = \frac{1}{1-Me^{-\beta (\mu + \epsilon_B)}}, \end{equation}

provided that $Me^{-\beta (\mu + \epsilon_B)} < 1$. Since $\epsilon_B <0$, this means that $\mu$ needs to be positive and large enough to overcome the binding energy, or else the sum will diverge. The physical interpretation of this divergence is that without a large enough chemical potential that would push particles away and establish an equilibrium, the system will accumulate adsorbed particles without limit.