Fluid Dynamics Problem: Gas Diffusion into a Container

I am trying to solve for the temperature/pressure profile (w/ respect to time) of a gas diffusing into a vacuum. This is analogous to a partitioned container (punctured wall in the middle) with the first chamber being held at constant temp/pressure (e.g. sensors + valves introduce heated gas).

Essentially, I believe that the pressure and temp. will increase w/ a decaying rate towards the eventual steady state.

I have attached my first attempt at solving this problem along w/ a diagram illustrating the scenario. I believe that it is wrong b/c at eqn.4 I assume that Temp. is constant. How can I resolve this?

Any help would be appreciated. Alternatively, are they any other fluid dynamics eqns that can can be used?

(A=area of hole, k=boltzman's constant, Na= Avogadro's constant, Z=effusion flux, M=molar mass, N=# molecules, subscripts 1 and 2 refer to containers 1(steady temp/pressure) and 2 (vacuum), respectively. I am using PV=NkT)

solution attempt with diagram.doc