The Gibbs Criterion

At constant temperature and pressure — the conditions of almost all laboratory chemistry — the Gibbs free energy $\Delta G = \Delta H - T\Delta S$ determines spontaneity. A process is spontaneous when $\Delta G < 0$, at equilibrium when $\Delta G = 0$, and non-spontaneous when $\Delta G > 0$.

The Four Cases

The sign analysis of $\Delta G = \Delta H - T\Delta S$ gives four scenarios depending on the signs of $\Delta H$ and $\Delta S$:

1. $\Delta H < 0$, $\Delta S > 0$: Both terms make $\Delta G$ negative. The reaction is spontaneous at ALL temperatures. This is the ideal combination: exothermic and increasing disorder.

2. $\Delta H > 0$, $\Delta S < 0$: Both terms make $\Delta G$ positive. The reaction is non-spontaneous at ALL temperatures — endothermic and decreasing disorder work against each other with no compensating factor.

3. $\Delta H < 0$, $\Delta S < 0$: Exothermic but decreasing disorder. $\Delta G < 0$ when $T < \Delta H/\Delta S$ (at low temperature, enthalpy dominates). Non-spontaneous above the crossover temperature.

4. $\Delta H > 0$, $\Delta S > 0$: Endothermic but increasing disorder. $\Delta G < 0$ when $T > \Delta H/\Delta S$ (at high temperature, entropy drive dominates). Spontaneous above the crossover temperature.

Crossover Temperature

When $\Delta H$ and $\Delta S$ have the same sign, there exists a crossover temperature where $\Delta G = 0$: $T_{crossover} = \frac{\Delta H}{\Delta S}$ Above this temperature, entropy wins; below it, enthalpy wins (or vice versa depending on the case).

Connecting to Equilibrium

$\Delta G^\circ = -RT\ln K$: the standard free energy change (at standard conditions) determines the equilibrium constant. Large negative $\Delta G^\circ$ → large $K$ (products favored). Large positive $\Delta G^\circ$ → small $K$ (reactants favored). At equilibrium, $\Delta G = 0$ (not $\Delta G^\circ$). $\Delta G^\circ = 0$ only when $K = 1$.