Questions on Stat Mech Physical Adsorption Problem

[Silviu]

Hello! I have a question about the first problem in Stat Mech (Physical Adsorption) from http://web.mit.edu/physics/current/graduate/exams/gen2_F01.pdf. The solution to it can be found http://web.mit.edu/physics/current/graduate/exams/gen2sol_F01.pdf. I understand the logic they use for the solution, but I am not sure I understand why do they use the partition function of the canonical ensemble and not the grand canonical one. If I understand the problem well, the particles can go from the wall of the container to the inside of the container and the other way around, so the number of particles in not fixed (but the average is). Also, there is no way to exchange energy here, other than particles moving from wall to the inside of the container. So why don't we use the GC ensemble partition function (i.e. adding the term $e^{-\mu N/KT}$)? Thank you!

 

[DrClaude]

The problem considers a closed box, with a fixed number of particles N. While the particles can be in two "states," in the bulk or adsorbed, their number is fixed. That's why the canonical ensemble is the right one.

For the grand canonical ensemble, you would have a reservoir of particles at fixed μ, and the number of particles N would not be fixed.

 

[Silviu]

The problem considers a closed box, with a fixed number of particles N. While the particles can be in two "states," in the bulk or adsorbed, their number is fixed. That's why the canonical ensemble is the right one.

For the grand canonical ensemble, you would have a reservoir of particles at fixed μ, and the number of particles N would not be fixed.

I understand what you mean, but wouldn't this be the case if you write the partition function for the whole system (N is fixed for $N=N_{bulk} + N_{surface}$) so for that I would use the canonical ensemble. But he writes the partition functions for $N_{bulk}$ and $N_{surface}$ separately, and these are not fixed.