Suppose we have an adiabatic and isothermal free expansion of an ideal gas at temperature $T_0$ so that the gas incurs an increase in entropy $$\Delta S_{sys}=Nk_B \ln\left(\frac{V_f}{V_i}\right)>0$$
The expansion occurs in adiabatic conditions so clearly $Q=0$. Also, the work done by the gas is $W=0$ because we are examining a free expansion into a vacuum. Thus we have that $\Delta U_{sys}=0$. This is all clear to me. The trouble begins when I calculate the change in helmholtz free energy $\Delta F$. The change in helmholtz free energy for this process is calculated as follows: $$\Delta F=\Delta U -T\Delta S$$ $$\Rightarrow \Delta F_{sys}=0-T_0\cdot Nk_B \ln\left(\frac{V_f}{V_i}\right) $$ $$\therefore \Delta F_{sys}<0$$
Now one of the uses of $F$ is that the negative of the change in $F$ at constant temperature tells us "the maximum work that can be extracted when a process occurs at constant temperature". We can see this from the equation $-\Delta F\geq W_{by\,the\,system}$. But if this interpretation is true, then the maximum work that can be extracted from a free expansion should be
$$W_{max\, possible}=T_0\cdot Nk_B \ln\left(\frac{V_f}{V_i}\right)$$
To be perfectly honest, this result sounds like gibberish to me. How can we possibly extract work from a free expansion into a vaccuum at constant temperature? Is my understanding of free expansions incorrect, my understanding of Helmholtz free energy incorrect, or perhaps both? Any help on this issue would be most appreciated!
