# Can the same isothermal operation increase free energy in one system but decrease it in another? [closed]

Consider two different thermodynamic systems, S1 and S2, with the following properties:

1. Both are in contact with a heat bath.
2. Both are macroscopically uniform (e.g., a gas with uniform density throughout).
3. Both have identical constraints at the initial state.

Suppose we perform the same reversible (quasi-static) isothermal operation on both systems.

Is it possible for this operation to:

• Increase the Helmholtz free energy of S1 (requiring work to be done on S1)
• Decrease the Helmholtz free energy of S2 (allowing work to be extracted from S2)

In other words, can the same isothermal process extract work from one system while requiring work input for another?

Edit 2024/07/31

Only one heat bath is used.

• Not if only one heat bath is used. Commented Jul 30 at 10:37
• @ChetMiller Yes. There is only one heat bath used. Why not in this case? Does the second law of thermodynamics make it impossible? Commented Jul 30 at 21:53
• Are you referring to, say, slowly moving a partition that separates the two gases? The expanding gas isothermally does work on the partition, and the partition isothermally does work on the gas being compressed. Commented Jul 30 at 22:55
• @Chemomechanics No. I am looking into the general case. I think the obvious answer is no for an ideal gas trapped in a box with a partition, but what about real gases or non-gas systems? Commented Jul 30 at 23:36
• If the reservoir is at the initial temperature of the gas, then the 2nd law tells us that $W_{REV}=-\Delta A$. Since A is a state function, and the two end states iare the same, $\Delta A$ is unique. Commented Jul 31 at 10:10