Gibbs Free Energy (G)

A spontaneous reaction is one that runs to completion under a given set of conditions without needing any external driving force; if it cannot, it is called non-spontaneous. Everyday spontaneous processes: a waterfall runs downhill, but never spontaneously uphill; a lump of sugar dissolves spontaneously in coffee, but never spontaneously reappears in its original solid form; heat flows spontaneously from a hotter object to a colder one, never the reverse; and a gas expands spontaneously into an evacuated bulb, while the reverse -- all its molecules spontaneously gathering back into one bulb -- never happens. In every one of these cases, the process that occurs spontaneously in one direction simply cannot occur spontaneously in the opposite direction.

A great many EXOTHERMIC reactions are also spontaneous -- methane combustion, $CH_4+2O_2\rightarrow CO_2+2H_2O$, $\Delta H^0=-890.4$ kJ mol$^{-1}$, and acid-base neutralisation, $H^++OH^-\rightarrow H_2O$, $\Delta H^0=-57.32$ kJ mol$^{-1}$, are both examples. But some ENDOTHERMIC processes are ALSO spontaneous -- ammonium nitrate dissolving in water, $NH_4NO_3\rightarrow NH_4^++NO_3^-$, $\Delta H^0=+25$ kJ mol$^{-1}$, is a spontaneous dissolution despite being endothermic. So exothermicity FAVOURS spontaneity, but does not GUARANTEE it -- energy changes alone cannot decide whether a reaction will be spontaneous. From the second law we also know a spontaneous process increases entropy -- but not every entropy-increasing process turns out to be spontaneous either. Predicting spontaneity genuinely needs a function that combines both quantities.

Gibbs free energy. The second law introduces exactly that combined function: Gibbs free energy, developed in the 1870s by Josiah Willard Gibbs, who originally called it the "available energy" to do work in a system -- the portion of a chemical reaction's energy that can actually be harnessed to do work. It is defined as:

$G$ is an EXTENSIVE property, and (being built entirely from state functions $H$, $T$, $S$) is itself a single-valued STATE function.

For a system changing from state 1 to state 2 at constant temperature, $G_2-G_1=(H_2-H_1)-T(S_2-S_1)$, i.e.

Connecting $\Delta G$ to spontaneity. We already know $\Delta S_{total}=\Delta S_{sys}+\Delta S_{surr}$, and that for a REVERSIBLE (equilibrium) process $\Delta S_{total}=0$ (so $\Delta S_{sys}=-\Delta S_{surr}$), and that at equilibrium $\Delta G=0$ as well. For a SPONTANEOUS process, $\Delta S_{total}>0$, i.e. $\Delta S_{sys}+\Delta S_{surr}>0$. Since $\Delta S_{surr}=-q_{surr}/T=-\Delta H_{sys}/T$ (heat lost by the surroundings is the negative of heat gained by the system), this becomes $\Delta S_{sys}-\dfrac{\Delta H_{sys}}{T}>0$, i.e. $T\Delta S_{sys}-\Delta H_{sys}>0$, i.e. $-(\Delta H_{sys}-T\Delta S_{sys})>0$, i.e. $-\Delta G>0$. Hence, for a SPONTANEOUS process:

Here, $\Delta H_{sys}$ is the reaction's enthalpy change, and $T\Delta S_{sys}$ represents energy that is NOT available to do useful work; $\Delta G$ is therefore the NET energy that IS available to do useful work -- which is exactly why it is called the reaction's "free energy." For a NON-spontaneous process, $\Delta G>0$. …