# Partition function of mixture two different gases

I have a question about statistical physics. Suppose we have a closed container with two compartments, each with volume V , in thermal contact with a heat bath at temperature T, and we discuss the problem from the perspective of a canonic ensemble. At a certain moment the separating wall is replaced by a membrane that is permeable for particles of gas A, but not for particles of gas B. What is the total partition function of the final system? My approach was to first consider a mixture between the particles in container B, $$N_B$$ and some of the particles from container A, lets say $$N_A^R$$, and calculate the partition function: $$Z_1=Z(N_B)Z(N_A^R)$$. The partition function of compartment A is: $$Z_2=Z(N_A-N_A^R)$$. And next take the partition function of two subsystems in contact with a thermal reservoir: $$Z_{tot}=Z_1Z_2$$. Is this approach correct? If yes, what do you take for the volume of $$Z(N_A^R)$$?

I hope someone can help me with this.

• Do the particles of type $A$ and $B$ interact with each other in any way? Mar 9, 2022 at 15:15
• Each component can be considered as a subsystem, that only weakly interacts with the other subsystem, i.e. the total energy is just the sum of the energies of both subsystems without any interaction contributions Mar 9, 2022 at 15:18

Your approach to the problem will not work in a straightforward way. The problem is that the $$A$$ particles can move between the two halves of the box, and so you will have to deal with statistical fluctuations in the number $$N_A^R$$. To do this you will have to work in the grand canonical ensemble and then you will have to impose the constraint that the total number of particles between both halves is fixed. This will be a complicated calculation.
There is, however, a much simpler approach. The above problems occurred because the $$A$$ particles could cross the barrier between the two halves, so simply do not consider the two halves separately. Instead separate your system into subsystems based purely on particle species. You then have a gas of $$B$$ particles in a volume $$V$$ and a gas of $$A$$ particles in a volume $$2V$$ and your partition function should factorize nicely.
• Thank you for your reply! However, it is asked to express the partition function as function of $N_R^A$: 'Calculate the partition function and the Helmholtz free energy of the total system as a function of $N_R^A$, expressed in the given quantities.' Mar 9, 2022 at 15:56