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

Question

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.

Answer 1

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.

Answer 2

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 $dw$.