Consider a container divided into two compartments $A$ and $B$ by a barrier. The walls of the container as well as the barrier are adiabatic.
Initially both compartments contain some gas (not necessarily the same gas in both compartments), at temperatures $T_A$ and $T_B$ respectively. Let's say $T_A>T_B$.
Question 1:
If we consider the system $S$ consisting of the gas in compartment $A$ in conjuction with the gas in compartment $B$, would it be correct to say that $S$ is initially NOT in thermodynamic equilibrium?
I'm a little bit confused about the definition of thermodynamic equilibrium, but as far as I understand,a system in thermodynamic equilibrium must in particular be in thermal equilibrium, i.e it has a well-defined temperature, i.e the temperature of any subsystem is the same (if we measure the mean kinetic energies of any subsystem of the gas particles, they should be equal). This is clearly not the case in our example.
The barrier between the compartments is now suddenly broken, allowing the gases to mix.
Question 2:
Is it true that after some time the system $S$ will reach thermal equilibrium, i.e the mean kinetic energy of the gas particles initially in compartment $A$ will be equal to that of the gas particles initially in $B$? According to what I read in the Feynman lectures, the answer is yes, but I found his explanation to be rather hand wavy.
Question 3:
Assuming the answer to the previous question is yes, is it possible to calculate the new equilibrium temperature of the system?
Best Answer
If there is a barrier present, then each sub-system is individually at thermodynamic equilibrium. It is true that, at some time after the barrier is removed, the combined system will reach thermodynamic equilibrium. This means that there will be a final equilibrium temperature. The final equilibrium temperature can be calculated by taking into account the fact that, with no work being done on the boundary of S and no heat transferring through the boundary of S, the change in internal energy between the initial and final states of S is zero. So, for an ideal gas, you would have that $$(m_A+m_B)u(T_f)=m_Au(T_A)+m_Bu(T_B)$$where u(T) is the internal energy per unit mass at temperature T.