Determining temperature of the gas in the container

Question

Solve the following problem:

A thick-walled insulated chamber contains n_i moles of helium at high pressure P_i. It is connected through a valve with a large, almost empty container of helium at constant pressure P_o, very nearly atmospheric. The valve is opened slightly, and the helium flows slowly and adiabatically into the container until the pressures on the two sides of the valve are equal.

Assuming the helium to behave like an ideal gas with constant heat capacities, show that:

(a) The final temperature of the gas in the chamber is

T_f = T_i(P_f/P_i)^(γ-1)γ

(b) The number of moles left in the chamber is

n_f = n_i(P_f/P_i)^1/γ

(c) The final temperature of the gas in the container is

T_f = (T_i/γ) 1 - P_f/P_i / 1 - (P_f/P_i)^1/γ