Q.For the reaction at 298 K: $2A + B \rightarrow C$, $\Delta H = 400$ KJ mol$^{-1}$; $\Delta S = 0.2$ KJK$^{-1}$mol$^{-1}$. Determine the temperature at which the reaction would be spontaneous.

Exothermicity favours spontaneity but doesn't guarantee it (ammonium nitrate dissolves spontaneously despite being endothermic), and entropy increase alone doesn't guarantee it either -- predicting spontaneity needs a function combining both. Gibbs free energy, $G=H-TS$ (Josiah Willard Gibbs, 1870s -- originally the "available energy" to do work), is extensive and a single-valued state function; at constant $T$, $\Delta G=\Delta H-T\Delta S$. Combining the first and second laws shows that for a SPONTANEOUS process $\Delta G<0$ (equivalently $\Delta H-T\Delta S<0$); at equilibrium $\Delta G=0$; for a non-spontaneous process $\Delta G>0$. $T\Delta S$ is energy NOT available for useful work, so $\Delta G$ is the net energy that IS available -- hence "free" energy; more precisely, $-\Delta G=w-P\Delta V$ at constant $T,P$, meaning the decrease in free energy equals the MAXIMUM work obtainable other than ordinary expansion work. …