How does the gas flow?

Here is an argument involving entropy. Let's suppose only 6 molecules of a gas were introduced into the system. There are many ways that these atoms can be arranged in each flask. For example, there could be three in A, two in B, and one in C. Let's say that at some moment, there are $a$ atoms in A, $b$ atoms in B, and $c$ atoms in C. The total number of energetically equivalent ways that the gas molecules can be arranged in this microstate is equal to

$\dfrac{6!}{a!b!c!}$

by combinatorics.

Now, we turn to the Second Law of Thermodynamics, which states that for any process, the change in the entropy of the universe must increase. The entropy change of the universe is equal to the sum of the entropy change of a system plus the entropy change of its surroundings. However, in this case, we know that the entropy change of the surroundings is exactly zero, since no work is being done on the system to influence the movement of the gas. Therefore, the entropy change of the system must be positive.

What this means for the situation is that the gas molecules will always arrange themselves in the state that gives them the greatest possible entropy. If it is not in this state, then it will arrange its gas molecules until it does reach that state. Thus, we find the macrostate that has the greatest number of microstates. We can use our formula derived above to find that the macrostate we seek occurs when each flask has the same number of gas molecules i.e. $(a, b, c) = (2, 2, 2).$ At this state, no more movement will increase the entropy, so the process is at equilibrium.

The key point here is that equilibrium is reached when the amount of gas in each flask is the same . Anything else, and the system will change it back.

Applying this concept of entropy change to the original problem, we see that there will be $\boxed{2}$ moles of gas in flask C once the volumes in each flask stop changing.