I am trying to prove that the pressure/temperature profiles in the second compartment of a partitioned container follow a typical growth/logistics function (i.e. are sigmoidal) with respect to time. Essentially, the partitioned container is a single container with a punctured wall in the center. The first compartment is kept at steady pressure/temp., the second is a vacuum.

I tried solving this by setting dp/dt = dp/dt(in) + dp/dt(out), where dp/dt (in)= constant (constant volume, temp. and pressure in 1st container) and dp/dt(out)=kT / V * dN/t, where dN/dt= PANa/ (2piMRT)1/2 (effusion equation) and then solving the subsequent differential equation. (k=boltzman's const. N=number molecules, A=area of hole, Na=avogadro's constant, M=molar mass)

I imagine I went wrong by setting Temperature in dp/dt(in) constant.

How can I fix this? Alternatively, are there any fluid dynamics equations that I can use? Any help would be appreciated.