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We know that closed adiabatic systems at constant volume tend to evolve toward their lowest energy state (i.e., $dU=0$ at equilibrium, where $dU=T\,dS-P\,dV$). But we often deal with systems at constant temperature rather than adiabatic systems. Therefore, we perform a so-called Legendre transform by writing $F=U-TS$; this produces $dF=-S\,dT-P\,dV$, which tells us that constant-temperature systems at constant volume tend to minimize $F$ and satisfy $dF=0$ at equilibrium. So that's the quick story of where the $TS$ comes from. Does this make sense?