The Helmholtz free energy A is defined as
\[ A=U-T S.\]
As when we first defined enthalpy, this doesn't seem to mean much until we write it in differential form:
\[d A=d U-T d S-S d T.\]
On substitution from equation 12.1.6 (dU = TdS − PdV + ∑XdY), this becomes
\[ d A=-S d T-P d V+\sum X d Y.\]
This tells us that in an isothermal process (in which dT = 0), the increase in the Helmholtz function of a system is equal to all the reversible work (−PdV + ∑XdY) done on it. Conversely, if a machine does any reversible work at constant temperature, the Helmholtz function decreases, and the decrease in the Helmholtz function is equal (if the temperature is held constant) to the reversible work (of all types) done by the machine. It is in this sense that the Helmholtz function is called the “free energy”. It is the energy, so to speak, that is free for the performance of external reversible (i.e. useful) work.