Predicting Spontaneity: The Four Cases of ΔH and ΔS

What 'spontaneous' really means

In everyday language, 'spontaneous' suggests sudden and unplanned. In thermodynamics it means something precise and undramatic: a process with $\Delta G < 0$, one that can proceed without a continuous input of energy. Rust forming on iron is spontaneous. Water flowing downhill is spontaneous. Neither is fast — rust takes months — and neither needs a push to keep going once started.

The critical separation is between thermodynamics (can it happen?) and kinetics (how fast?). Thermodynamics is the map: it tells you whether the destination lies downhill. Kinetics is the vehicle: the activation barrier decides whether you walk, drive, or never arrive. A spontaneous reaction with a huge barrier sits unchanged for centuries — thermodynamically favored, kinetically frozen. Confusing the two is the costliest error in this topic: $\Delta G$ is a verdict on possibility, never a speedometer.

The four cases: a table worth memorizing

Because $\Delta G = \Delta H - T\Delta S$, the signs of $\Delta H$ and $\Delta S$ create exactly four cases:

• $\Delta H < 0$, $\Delta S > 0$ → $\Delta G$ always negative → spontaneous at all temperatures. Both drives agree. Example: combustion — exothermic and entropy-increasing.
• $\Delta H > 0$, $\Delta S < 0$ → $\Delta G$ always positive → never spontaneous. Both drives oppose. Example: forcing water uphill into ice at room temperature.
• $\Delta H < 0$, $\Delta S < 0$ → spontaneous at low temperatures. Enthalpy wins when $T$ is small. Example: freezing water, rusting iron.
• $\Delta H > 0$, $\Delta S > 0$ → spontaneous at high temperatures. Entropy wins when $T$ is large. Example: melting ice, decomposing limestone.

Memorize the logic, not just the table: when the signs agree, temperature cannot change the verdict; when they disagree, temperature is the tiebreaker.

The crossover temperature: ice as proof

In the two mixed cases, spontaneity flips at the temperature where $\Delta G = 0$ — the crossover temperature, $T = \Delta H / \Delta S$ (with $\Delta H$ and $\Delta S$ in consistent units). Below it one drive wins; above it the other does.

Ice melting is the perfect calibration. For $\mathrm{H_2O(s) \rightarrow H_2O(l)}$: $\Delta H = +6.01$ kJ/mol (endothermic) and $\Delta S = +22.0$ J/(mol·K) (entropy-increasing) — the high-temperature case. Compute $T = 6010 / 22.0 = 273$ K, which is 0°C. Above 273 K, $T\Delta S$ outweighs $\Delta H$ and melting is spontaneous; below it, freezing wins. The equation predicts the familiar melting point of ice from two thermodynamic numbers — a striking demonstration that the four-case table is not a mnemonic trick but real physics.

Kilns and rust: the mixed cases in industry

The mixed cases explain major industrial choices. Limestone decomposition — $\mathrm{CaCO_3(s) \rightarrow CaO(s) + CO_2(g)}$ — is endothermic ($\Delta H > 0$) but produces gas ($\Delta S > 0$): spontaneous only at high temperature. That is why lime kilns run near 900°C; at room temperature the reaction's $\Delta G$ is positive and limestone sits unchanged for millennia.

Rusting flips the pattern: $\mathrm{4Fe + 3O_2 \rightarrow 2Fe_2O_3}$ is strongly exothermic ($\Delta H < 0$) but consumes gas ($\Delta S < 0$): spontaneous at low temperatures — which includes room temperature, unfortunately for car owners. Heat it enough and the verdict would reverse, though iron has other problems long before that. Whenever a process demands heating (smelting, kilns) or cooling (refrigeration, cryogenics), ask which mixed case you are in — the temperature requirement is thermodynamics talking, not habit.

Spontaneous is necessary, not sufficient

Return to the kinetics warning, because it is where predictions meet reality. A reaction can be spontaneous at every temperature ($\Delta H < 0$, $\Delta S > 0$) and still never visibly happen: hydrogen and oxygen gases coexist peacefully in a balloon — combustion is enormously spontaneous — until a spark supplies the activation energy. Thermodynamics said 'downhill'; kinetics said 'cliff in the way.'

This is why chemists keep two toolkits. The $\Delta G$ analysis answers: is this process allowed, and at what temperature does the verdict change? The kinetics analysis answers: is there a viable path, and can a catalyst or heat open it? Industrial ammonia synthesis is the classic compromise: low temperatures favor spontaneity ($\Delta H < 0$, $\Delta S < 0$ case) but the reaction is too slow there, so plants run hot with a catalyst — sacrificing some thermodynamic favor for workable speed. Spontaneity is necessary; it is never sufficient.

Common mistakes

Real-world applications

• Industrial chemistry: choosing operating temperatures for ammonia synthesis and smelting.

• Food storage: why refrigeration slows spoilage (kinetics) and freezing preserves (thermodynamics).

• Geology: predicting which minerals are stable at Earth's surface temperatures.

• Battery design: selecting cell reactions with large negative ΔG for maximum voltage.

• Environmental engineering: determining whether pollutants break down spontaneously.