Entropy change with identical and different gases

[Antti]

Homework Statement

There's a thermally insulated container with a removable wall. So there are two compartments inside, one with volume V (i call it left box) and the other with volume V + B (right box) so the total volume is 2V + B. Both boxes have N molecules of an ideal gas and are in thermal equilibrium with each other. The compartment with volume V has initial pressure p.

Find

a) the final pressure of the mixed gas when the wall is removed
b) the total change in entropy if the gases in the left and right box are different
c) the total change in entropy if the gases are identical

I've done a). I just imagine that the gas in the left box expands into vacuum and then i do the same for the right box and add the final pressures of both boxes to get the final total pressure.

Using the idel gas law:

p_left = NkT/(2V+B) = p/(2+B/V)
p_right = p/(2+B/V)

The pressures turn out the same because the same number of molecules occupy the same volume at the same temperature.

p_tot = 2p/(2+B/V)

But what to do with the two other questions? How and why does the entropy change when the boxes contain different gases?

 

[jbsteele]

Homework Equations pV = NkTdS = dQ/TThe Attempt at a Solution For b) the total change in entropy when the gases in the left and right box are different:I've been thinking about this for a while but can't come up with an answer. I know that entropy is a measure of the disorder of a system, so when the wall is removed the entropy should increase because the gases mix together and become more disordered. But I'm not sure how to calculate this change in entropy. Any help would be appreciated.For c) the total change in entropy if the gases are identical:I'm also not sure how to solve this one. Since the gases are identical, the entropy should remain the same after the wall is removed. But again, I'm not sure how to calculate this change in entropy.

 

[thomate1]

And how does the entropy change when the gases are identical?

I would approach this problem by first defining the concept of entropy and its relationship to gases. Entropy is a measure of the disorder or randomness in a system, and it plays a crucial role in thermodynamics. In simple terms, entropy is a measure of the number of ways in which a system can be arranged, and the higher the entropy, the more disordered the system is.

In the given scenario, the initial state of the system is one of thermal equilibrium, where both compartments have the same temperature and number of molecules. When the wall is removed, the gases will mix and reach a new equilibrium state. This process is spontaneous and irreversible, and it results in an increase in entropy.

Now, let's consider the two different cases: when the gases in the left and right boxes are different and when they are identical.

a) When the gases in the two boxes are different, the final pressure will depend on the nature of the gases and their individual properties, such as molecular weight and intermolecular forces. This will result in a change in entropy as the gases mix and reach a new equilibrium state. The total change in entropy will be the sum of the individual changes in entropy for each gas.

b) On the other hand, when the gases in the two boxes are identical, the final pressure will be the same for both compartments, as seen in the solution for part a). In this case, there will be no change in entropy as the gases mix, since they are already identical and there is no increase in disorder or randomness.

In conclusion, the change in entropy in this scenario will depend on the nature of the gases in the two compartments. If they are different, there will be an increase in entropy due to the mixing process, while if they are identical, there will be no change in entropy. This highlights the importance of considering the individual properties of gases when studying thermodynamic processes.