When Should You Use the Grand Partition Function in Statistical Mechanics?

[secret2]

I am taking a statictical mechanics course, and one thing bothers me. I am not sure when we should use the normal partition function (Z) and when the grand partition function (twisty Z). In particular, why can we not use grand partition function when we are considering the following system:

He atoms may be adsorbed from vapor phase onto a solid surface, forming a 2-dimensional gas. If the adsorption energy is e, then by treating the vapor as a reservoir, find the density of He atom on the surface.

Shouldn't we always use grand partition function whenever the number of particle N is not constant? Or is there no limitation on which one we Have to use, just which is more convenient?

 

[Galileo]

For the adsorption problem, you should use the grand canonical ensemble (which uses the grand partition function), since the number of particles on the surface is clearly not constant.

 

[GSXR750]

The (Grand) Canonical Ensemble is a statistical mechanical model used to describe the behavior of a system in equilibrium with a reservoir, where the number of particles can fluctuate. In this ensemble, the system is allowed to exchange both energy and particles with the reservoir, while the volume and temperature are kept constant.

The main difference between the normal partition function (Z) and the grand partition function (Ξ) is that Z only takes into account the energy of the system, while Ξ also considers the number of particles. So, in cases where the number of particles is not constant, it is more appropriate to use the grand partition function.

In the example you mentioned, where He atoms are adsorbed from a vapor onto a solid surface, the number of particles on the surface can fluctuate as they are adsorbed or desorbed. Therefore, it would be more accurate to use the grand partition function in this case.

However, there may be cases where using the normal partition function is more convenient, even if the number of particles is not constant. It ultimately depends on the specific system and the information that is needed to be calculated.

In summary, the grand partition function should be used when the number of particles is not constant, but there may be situations where using the normal partition function is also acceptable. It is important to carefully consider the system and the information needed before deciding which one to use.