Why Limestone Only Breaks Down Inside a Red-Hot Kiln

Iron rusts, ice melts on a warm day and salt dissolves in water, all without any push — yet some of these processes absorb heat. Releasing energy is not the whole story; the spreading out of energy and matter, measured as entropy, matters too, and Gibbs energy combines the two into a single test.

This lesson covers entropy, Gibbs energy and spontaneity, and the link between Gibbs energy and equilibrium.

Entropy (S) is a state function that measures the disorder or randomness of a system, and it increases when matter or energy spreads out more — as when a solid melts, a liquid evaporates or a reaction produces more gas molecules.

Entropy change:

$$\Delta S=\frac{q_{\text{rev}}}{T}$$

**Predicting the sign of $\Delta S$:

- Increases — melting, boiling, sublimation, dissolving a solid, heating a substance, and reactions that increase the moles of gas
- Decreases — freezing, condensation, and reactions that reduce the moles of gas
- For one substance, entropy order is gas > liquid > solid

Worked example 1 — vaporisation of water.** ${\Delta }_{\text{vap}}H=40.7$ kJ mol${}^{-1}$ at 373 K:

$${\Delta }_{\text{vap}}S=\frac{40700}{373}=109{\text{ J K}}^{-1}{\text{ mol}}^{-1}$$

Worked example 2 — melting of ice. ${\Delta }_{\text{fus}}H=6.01$ kJ mol${}^{-1}$ at 273 K:

$${\Delta }_{\text{fus}}S=\frac{6010}{273}=22.0{\text{ J K}}^{-1}{\text{ mol}}^{-1}$$

Worked example 3 — a reaction. In $2{\mathrm{H}}_2(\mathrm{g})+{\mathrm{O}}_2(\mathrm{g})\to 2{\mathrm{H}}_2\mathrm{O}(\mathrm{l})$, three moles of gas become liquid, so $\Delta S$ is negative.

Second law of thermodynamics. In a spontaneous process, the total entropy of the system and its surroundings increases: $\Delta S_{\text{total}}>0$.

An everyday example. The scent of rose water spreading through a room is entropy increasing — the molecules spread out and never gather back by themselves.

The substance. Vaporisation raises entropy far more than melting — molecules gain much more freedom going from liquid to gas than from solid to liquid.

**Gibbs energy change, $\Delta G=\Delta H-T\Delta S$, combines enthalpy and entropy: a process at constant temperature and pressure is spontaneous when $\Delta G$ is negative, non-spontaneous when it is positive, and at equilibrium when it is zero.**

$$\Delta G=\Delta H-T\Delta S$$

**Spontaneity from the signs of $\Delta H$ and $\Delta S$:**

- $\Delta H<0$ and $\Delta S>0$ — spontaneous at all temperatures
- $\Delta H>0$ and $\Delta S<0$ — never spontaneous
- $\Delta H<0$ and $\Delta S<0$ — spontaneous at low temperatures
- $\Delta H>0$ and $\Delta S>0$ — spontaneous at high temperatures

Worked example — decomposition of limestone. For $\mathrm{CaC}{\mathrm{O}}_3(\mathrm{s})\to \mathrm{CaO}(\mathrm{s})+\mathrm{C}{\mathrm{O}}_2(\mathrm{g})$, $\Delta H=+178$ kJ mol${}^{-1}$ and $\Delta S=+160$ J K${}^{-1}$ mol${}^{-1}$. At 298 K:

$$\Delta G=178-298\times 0.160=178-47.7=+130.3{\text{ kJ mol}}^{-1}$$

so the reaction is not spontaneous at room temperature. It becomes spontaneous above the temperature where $\Delta G=0$:

$$T=\frac{\Delta H}{\Delta S}=\frac{178000}{160}=1112.5\text{ K}$$

An everyday example. Lime kilns in Rajasthan and Gujarat heat limestone to very high temperatures, because its decomposition becomes spontaneous only once $T\Delta S$ outweighs $\Delta H$.

The substance. **A negative $\Delta G$ says a reaction can happen, not how fast** — diamond turning into graphite is spontaneous but far too slow to notice.