homework and exercises - Change in entropy of two isolated systems merged into one system

From Statistical Physics, 2nd Edition by F. Mandl:

Two vessels contain the same number $N$ molecules of the same perfect gas. Initially the two vessels are isolated from each other, the gases being at the same temperature $T$ but at different pressures $P_1$ and $P_2$. The partition separating the two gases is removed. Find the change in entropy of the system when equilibrium has been re-established, in terms of the initial pressures $P_1$ and $P_2$. Show that this entropy change is non-negative.

I'm a little confused about a few things.

1. Is there a temperature change in this process? Intuitively, I would say no because $(T+T)/2=T$. My other guess would be that the temperature must change because we now have a third pressure, $P_3$, that is different from the pressure of the other two and also because we have increased the volume.


2. I believe this is an irreversible process, correct? Because you can't realistically separate the gasses into that which came from vessel A and that which came from vessel B.



3. 
   

   Can the change in volume simply be called $V_A+V_B$? I thought it would be that simple but thinking more about it I feel as though the change in pressure and possible change in temperature might change things.

   
   


4. 
   

   When the partition is removed, is there a heat exchange between the 2 gases? My intuition says no because heat can only flow when there is a temperature difference and in this case both vessels are at temperature $T$.

   
   

My attempt:

So all in all I need to solve $\Delta S= \int \frac{dQ}{T}$

$$\Delta S= \int \frac{dQ}{T}$$ $$=\int \frac{dE+dW_{by}}{T}$$ We know that $dE=0$ and that $dW=PdV$

$$=\frac{1}{T}\int PdV$$ $$=\frac{P\Delta V}{T}$$

This is where I'm stuck - I don't think there is a valid thing to put in for $\Delta V$ because there were 2 systems that formed into 1 bigger system. If the final system is $V_1+V_2$, then what was its previous size? $V_1$ or $V_2$? Or can I say that it was $\frac{V_1+V_2}{2}$?

Answer

The total volume of the two rigid containers does not change, so the combined system does no work W on the surroundings. The two containers are presumably insulated, so no heat Q is exchanged with the surroundings. So, from the first law of thermodynamics, the change in internal energy of the combined system is zero. Since, for an ideal gas, internal energy is a function only of temperature, the final temperature of the combined system is equal to the initial temperature of the separate systems.

The process is irreversible, but not for the reason you gave. Since the same gas is present in both containers, the system can be returned to its original state, but not without incurring a change in the surroundings, involving heat transfer.

Quarky Quanta's intuition was correct with regard to the final equilibrium pressure of the combined system, provided n is the total number of moles of gas in the two original containers.

COMPLETION OF PROBLEM SOLUTION: $$V_1=\frac{NkT}{P_1}$$ $$V_2=\frac{NkT}{P_2}$$ $$V_1+V_2=\frac{NkT(P_1+P_2)}{P_1P_2}$$ $$P_F=\frac{2NkT}{(V_1+V_2)}=\frac{2P_1P_2}{(P_1+P_2)}$$ $$\Delta S=Nk\ln{\frac{P_1}{P_F}}+Nk\ln{\frac{P_2}{P_F}}=2Nk\ln{\left[\frac{(P_1+P_2)/2}{\sqrt{P_1P_2}}\right]}$$ So the change in entropy is determined by the ratio of the arithmetic mean of the initial pressures to their geometric mean (a ratio which is always greater than 1).