Gibbs Energy as the Criterion of Spontaneity

Consider a beaker sitting open to the atmosphere, thermostatted in a water bath. Both pressure and temperature are pinned by the surroundings — this is the constant-$T$, constant-$p$ world where essentially all wet chemistry, biochemistry, and industrial synthesis happens. The second law says a process is spontaneous only if it increases the total entropy of the universe: system plus surroundings. The trouble is that "surroundings entropy" is inconvenient. It depends on how much heat leaks out and at what temperature the bath absorbs it. Josiah Willard Gibbs's insight, formalized in the 1870s, was to notice that at constant $T$ and $p$ the surroundings' entropy change can be written using only system quantities: heat leaving the system at temperature $T$ becomes $-q_p/T$ entering the surroundings, and $q_p$ at constant pressure is just $\Delta H$ of the system. Substituting this back into the total entropy inequality produces a new function, $G = H - TS$, whose sign change alone tells you whether the universe's entropy went up. Physically, $G$ is a compromise between two competing tendencies: systems want to lower their enthalpy (settle into stronger bonds, more stable arrangements) and simultaneously want to raise their own entropy (spread energy and matter more widely). At constant $T$ and $p$, $\Delta G = \Delta H - T\Delta S$ weighs these two drives on a common footing, with temperature setting the exchange rate between them.

A second picture worth holding in mind is how $G$ behaves as a reaction proceeds from pure reactants to pure products. Plotting $G$ against the extent of reaction $\xi$ gives a curve with a minimum — not necessarily at $\xi = 1$ (complete conversion), but wherever the slope $\partial G/\partial \xi$ first hits zero. That minimum, not "all reactants consumed," is chemical equilibrium.

where: $H$ is enthalpy, $T$ is absolute temperature, $S$ is entropy, all of the system.

where: the second law demands total entropy never decreases for any real process.

where: this identity is the bridge — it converts the universal criterion into a system-only one.

where: equality holds only at equilibrium; strict inequality marks a spontaneous direction.

where: the working formula for computing spontaneity from tabulated thermodynamic data.

Example 1 — Foundational: is ice melting at 263 K and 298 K spontaneous? For the fusion of ice, $\mathrm{H_2O(s)} \rightarrow \mathrm{H_2O(l)}$, standard data at 273 K give

(These are consistent: $\Delta H_{\text{fus}}/T_{\text{fus}} = 6010\ \text{J mol}^{-1} / 273\ \text{K} = 22.0\ \text{J K}^{-1}\text{mol}^{-1}$, confirming equilibrium exactly at the melting point.) Treat $\Delta H$ and $\Delta S$ as roughly temperature-independent over a narrow range near 273 K, and evaluate $\Delta G = \Delta H - T\Delta S$ at two temperatures. At $T = 263$ K (−10 °C, supercooled liquid region / normal freezing conditions):

$\Delta G > 0$: melting is non-spontaneous at 263 K; the reverse process, freezing, is spontaneous. This matches everyday experience — water freezes below 0 °C. At $T = 298$ K (25 °C):

$\Delta G < 0$: melting is spontaneous at room temperature — ice melts, as observed. At exactly $T = 273.0$ K,

the small residual reflecting rounding in the tabulated $\Delta S$; physically this is the coexistence point, $\Delta G = 0$, solid and liquid in equilibrium. The sign flip between 263 K and 298 K, straddling 273 K, is exactly the expected physical behavior — a strong sanity check.

Example 2 — Applied: does calcium carbonate decompose spontaneously at 800 °C and 1 bar? Consider $\mathrm{CaCO_3(s)} \rightarrow \mathrm{CaO(s)} + \mathrm{CO_2(g)}$, important in cement manufacture (calcination of limestone). Standard thermodynamic data at 298 K:

Step 1: check at 298 K.

Strongly positive: limestone does not decompose spontaneously at room temperature, consistent with limestone buildings surviving centuries.

Step 2: check at 800 °C = 1073 K, the industrial calcination temperature, assuming $\Delta H^{\circ}$ and $\Delta S^{\circ}$ stay roughly constant (a simplification revisited when temperature-dependence of $G$ is treated later):

Still slightly positive at standard pressure of CO₂ (1 bar) — decomposition is not yet spontaneous under standard conditions, though the driving force has shrunk by a factor of 22.

Step 3: find the temperature where ΔG° = 0 (the "onset" temperature under standard conditions):

Above roughly 837 °C, $\Delta G^{\circ}$ turns negative and calcination becomes spontaneous at standard 1 bar CO₂ pressure — matching the well-known industrial kiln temperatures (900–1000 °C, used partly for reaction rate and partly to push well past this threshold, and because real kilns keep CO₂ partial pressure below 1 bar, lowering the threshold further via the reaction quotient). The 837 °C figure is a benchmark every cement chemistry course reproduces.

Josiah Willard Gibbs, working largely in isolation at Yale in the 1870s, published "On the Equilibrium of Heterogeneous Substances" — a paper so dense and so far ahead of experimental chemistry that it took decades for chemists to fully absorb it. He introduced the free energy function, chemical potential, and the phase rule in one extended memoir, essentially inventing chemical thermodynamics as a predictive discipline rather than a bookkeeping exercise for engines.

Ask the tutor: How does ΔG connect to the equilibrium constant K, and why does that relationship change with temperature?