# Two gases seperated by semi-permeable membrane

The problem is as follows: Two ideal gases are contained adiabatically and separated by an insulating, fixed piston that blocks the molecules of gas 2 but allows the molecules of gas 1 through(in both directions). The initial pressures, volumes, temperatures and number of molecules on each side is given. What is the equilibrium state?

What I want to know is: am I correct in assuming that (adiabatic expansion) $$P_{1f}V_1^{\gamma} = P_{1i}(\frac{n_{1rf}}{n_1}V_1)^\gamma$$

Where $P_1$ is the pressure of gas 1 on the right, $V_1$ is the volume of the right chamber, $n_1$ is the total number of moles of gas 1, $n_{1rf}$ is the final number of moles of gas 1 remaining in the right chamber at equilibrium. (I'm assuming that gas 1 starts on the right side and gas 2 on the left side).

• Yes, you are correct. Aug 3, 2017 at 12:32

Start with the enthalpy $H$ to obtain

$dH = VdP+TdS+\mu dn$

with $H=\frac{5}{2}nRT$ ,$V=\frac{nRT}{P}$ and $\mu = RTln(P/P_0)$ (chemical potential $\mu$ can be obtained from the Maxwell relation $(\frac{\partial \mu}{\partial P})_S=(\frac{\partial V}{\partial n})_S$).

Adiabatic means $dS=0$ and therefore, Integration of above equation yields:

$H_f-H_i = n_fRT_fln(P_f)-n_iRT_iln(P_i)$

and clearly $S_f=S_i$ as a second condition, where you can eliminate the variable $T_f$.

Yes, you are correct. The gas that has remained in the chamber after the system has equilibrated has expanded adiabatically and reversibly to push the gas ahead of it through the semipermeable piston. This same kind of physical situation is discussed in a problem presented in Fundamentals of Engineering Thermodynamics by Moran et al for a gas escaping through a valve from a pressurized tank. If you would like, I can provide the exact reference.