# Does helmholtz free energy necessarily increase if work added to a fixed temp and volume system?

This is for an isochoric system held at a temperature $$T$$ by its surroundings, and that heat $$dQ$$ is added to the system by its surroundings.

According to chapter 16.5 in Concepts in Thermal Physics by Blundell and Blundell, for a system held at a constant temperature and volume, the inequality $$dw \geq dF$$ shows that Helmholtz free energy of the system increases if work is added to the system. But I don't see why the RHS of the inequality is more than 0 if $$dw$$ is positive.

Note that it is likely that, due to my misunderstandings, I am possibly not representing what the book said correctly.

• To learn how to format, click the question mark on the edit bar of the post, then click on "Advanced Help" that opens underneath. To get math in a line put them between two dollar signs, to make a centered equation enclose them between two dollar signs. May 1, 2023 at 11:13

The definition of F tells us that, at constant temperature, $$\Delta F=\Delta U-T\Delta S=Q+W-T\Delta S$$ where W is the work done by the surroundings on the system. For a system in contact with a reservoir at constant temperature T, the 2nd law tells us that $$\Delta S=\frac{Q}{T}+\sigma$$where $$\sigma$$ is the (positive) generated entropy. So if we combine these equations, we obtain: $$\Delta F=W-T\sigma$$So,if the process is reversible, F will increase if non-PV work is done on the system, but, if irreversibilites are present to a large enough extent, F can decrease.
I think your confusion arises because you assume that $$dF$$ is always negative. It is negative for a spontaneous process, in which case the system has the ability to produce work, i.e. $$dw<0$$.
If we must add work to the system, the process is not spontaneous and its $$dF$$ is positive, as is $$dw$$.
• @Tarnish3d "surely it can still be less than 0 regardless of dw". What is your reason for saying this? As a state function, if $dF$ is negative for a process, it is positive for the reverse process. May 1, 2023 at 15:23