Derivation of availability

Consider a system in thermal contact with its surrounding at $T_0$, and at the pressure of the surroundings $P_0$. What spontaneous changes will occur? Taking the surroundings and the system together, the entropy will be maximised. What does that mean for the system?

Imagine a spontaneous change during which $Q$ is absorbed by the system, and the volume change $\Delta V$, so that work $W=-P_0\Delta V$ is done on the system. Then from the first law, $Q=\Delta E+P_0\Delta V$. The total change in entropy has two parts, $\Delta S$ for the system and $-Q/T_0$ for the surroundings. So

$$\begin{eqnarray*} \Delta S_{\rm tot}=\Delta S-{Q\over T_0}={T_0\Delta S-\Delta E... ...ightarrow\quad\Delta (E-T_0 S+P_0 V )\!\!\!&\le&\!\!\!0\nonumber \end{eqnarray*}$$

The quantity $A=E-T_0 S+P_0 V$ is called the availability, and it always decreases during a spontaneous change.