Example of available work

What is the maximum amount of work theoretically available from one mole of monatomic ideal gas, cooled at constant volume from an initial temperature $T$ to the temperature of the surroundings $T_0$?

$$W^{\rm max}=-\Delta A=-\Delta E+T_0\Delta S$$

Now $\Delta E=-C_V(T-T_0)$ for an ideal gas and $\Delta S=-C_V\ln(T/T_0)$ (see here.) So

$$W^{\rm max}=C_V\left(T-T_0-T_0\ln{T\over T_0}\right)$$

For 1 mole, $C_V=3R/2$.

The rest of the energy is discarded to the surroundings to ensure no overall decrease in entropy. Of course this is simple enough that we could have done the calculation directly from entropy considerations, taking the surroundings into account explicitly. Compare with this example.