Spontaneity and Entropy

Spontaneous does not mean instant. A spontaneous process occurs on its own, without being driven continuously by an external agency, once any initial push (if needed) has been given. A hot object cooling to room temperature, ice melting above 0 degrees Celsius, and gases mixing when a partition is removed are all spontaneous. Diamond slowly turning into graphite is spontaneous too — it just takes geological time. Spontaneity is about direction, not speed; how fast a process runs is a question of kinetics, a separate Class 12 topic.

Enthalpy alone cannot decide it. Early thinking assumed exothermic reactions ($\Delta H < 0$) are always spontaneous and endothermic ones never are. This fails: ice melts above 0 degrees Celsius even though fusion absorbs heat, and ammonium chloride dissolves in water even though the solution cools down. Both are spontaneous and endothermic. Something besides enthalpy must be doing the deciding.

Entropy measures disorder. Entropy ($S$) is a state function — its value depends only on the current state of the system, not on the path taken to get there — that quantifies the degree of randomness or molecular disorder. For the same substance, gases are more disordered than liquids, which are more disordered than solids, so $S_{gas} > S_{liquid} > S_{solid}$. Entropy rises when a solid melts, a liquid vaporises, a solute dissolves, the number of moles of gas in a reaction increases, or temperature increases — in each case the molecules gain more ways to arrange themselves.

where: $\Delta S$ is the entropy change of a process, $S_{final}$ is the entropy of the final state, and $S_{initial}$ is the entropy of the initial state. Because $S$ is a state function, $\Delta S$ depends only on the initial and final states, exactly like $\Delta H$.

The second law is the real criterion. For a spontaneous process, the entropy of the universe — system plus surroundings — must increase:

where: $\Delta S_{total}$ is the total entropy change of the universe, $\Delta S_{system}$ is the entropy change of the reacting substances, and $\Delta S_{surroundings}$ is the entropy change of everything else. The inequality holds for a spontaneous (irreversible) process; equality holds only at equilibrium. This resolves the endothermic paradox: even when $\Delta H_{system} > 0$ drains entropy from the surroundings (they lose heat), the system's own entropy can rise enough that $\Delta S_{total}$ still comes out positive.

Third law gives a reference zero. The entropy of a perfectly crystalline substance is taken as zero at absolute zero (0 K), qualitatively — because a perfect crystal at 0 K has only one possible arrangement of its particles, the minimum possible disorder. This reference point is what allows absolute entropies (not just entropy changes) to be tabulated for substances.

Problem. The melting of ice, $\text{H}_2\text{O}(s) \rightarrow \text{H}_2\text{O}(l)$, is spontaneous at 274 K (just above the melting point) even though $\Delta H_{fusion} > 0$. Explain why, using the second law, and state what happens at exactly 273 K.

Step 1. Melting requires the system to absorb heat from the surroundings, so $\Delta H_{system} > 0$. By the relation derived above, the surroundings lose entropy: $\Delta S_{surroundings} = -\dfrac{\Delta H_{system}}{T} < 0$.

Step 2. Melting converts an ordered solid into a more disordered liquid, so the system's own entropy rises significantly: $\Delta S_{system} > 0$.

Step 3. At 274 K, the positive $\Delta S_{system}$ outweighs the negative $\Delta S_{surroundings}$, so their sum is positive:

This is why melting proceeds spontaneously above 0 degrees Celsius even though it is enthalpically unfavourable.

Step 4. At exactly 273 K (0 degrees Celsius), ice and water coexist in equilibrium, so the two terms exactly cancel: $\Delta S_{total} = 0$ at 273 K, and the process is spontaneous (total entropy increases) at any temperature above it.

Competitive stretch

For $\text{N}_2(g) + 3\text{H}_2(g) \rightarrow 2\text{NH}_3(g)$, 4 moles of gas become 2 moles of gas, so $\Delta S_{system} < 0$ — the system becomes more ordered. For this reaction to be spontaneous at all, it must release enough heat ($\Delta H_{system}$ sufficiently negative) that $\Delta S_{surroundings} = -\Delta H_{system}/T$ is large and positive, outweighing the unfavourable system term — which is exactly why industrial ammonia synthesis is run as an exothermic, heat-releasing process.

• Assuming all exothermic reactions are spontaneous and all endothermic ones are not — spontaneity depends on total entropy change, not the sign of $\Delta H$ alone.
• Evaluating only $\Delta S_{system}$ and ignoring $\Delta S_{surroundings}$, or vice versa — the second law needs both added together.
• Counting all moles when predicting the sign of $\Delta S_{system}$ for a reaction, instead of focusing on the change in moles of gas, since solids and liquids contribute comparatively little disorder next to gases.
• Treating "spontaneous" as a claim about speed — a spontaneous process can be extremely slow; the second law says nothing about the rate at which equilibrium is approached.

Ask the tutor: Want to work through predicting the sign of $\Delta S_{system}$ for a trickier reaction, or see how this connects to Gibbs free energy next year — ask the AI tutor for guided practice.