Gibbs Free Energy: The Equation That Decides

What 'free' really means

Not all of a system's energy change is usable. When a reaction proceeds, some energy inevitably disperses as heat into the surroundings — the entropy tax the universe always collects. Gibbs free energy, $G$, is what remains: the portion of the energy change available to do useful work at constant temperature and pressure. The word 'free' is a technical term meaning 'available,' not 'costless.'

Chemists care about the change, $\Delta G$, because its sign is the master switch of spontaneity. A negative $\Delta G$ means the process can happen on its own — the system has free energy to spend. A positive $\Delta G$ means the process cannot happen without an energy input; it would need work done on it. This single number distills the tug-of-war between enthalpy (which favors low energy) and entropy (which favors spreading out) into one verdict.

The equation and its two drives

The defining equation is $\Delta G = \Delta H - T\Delta S$. Each term has a role. $\Delta H$ is the enthalpy drive: negative (exothermic) pushes $\Delta G$ down toward spontaneity. $-T\Delta S$ is the entropy drive: positive $\Delta S$ (more disorder) also pushes $\Delta G$ down, but its strength scales with temperature. At low temperatures the $\Delta H$ term dominates; at high temperatures the $T\Delta S$ term takes over.

This is secretly the second law in disguise. $\Delta G < 0$ is mathematically equivalent to $\Delta S_{universe} > 0$ at constant temperature and pressure — the equation just repackages 'total entropy increases' into quantities measured on the system alone, which is far more convenient. Instead of tracking the entire universe's entropy, chemists measure the system's $\Delta H$ and $\Delta S$ and let the equation do the universe's bookkeeping.

Units: the kelvin and kilojoule traps

The equation demands consistent units, and this is where most mistakes happen. $\Delta H$ is typically in kJ/mol while $\Delta S$ is in J/(mol·K) — a factor of 1000 apart. Convert $\Delta S$ to kJ/(mol·K) before multiplying by $T$, or convert $\Delta H$ to joules. And $T$ must be in kelvin, always: the equation is built on absolute temperature, so 25°C enters as 298 K.

Work an example. A reaction has $\Delta H = -120$ kJ/mol and $\Delta S = -150$ J/(mol·K) at $T = 298$ K. Convert: $\Delta S = -0.150$ kJ/(mol·K). Then $T\Delta S = 298 \times (-0.150) = -44.7$ kJ/mol. So $\Delta G = -120 - (-44.7) = -75.3$ kJ/mol. Negative — spontaneous at room temperature. The entropy term opposed spontaneity (negative $\Delta S$), but the large exothermic $\Delta H$ won. Had you forgotten the J-to-kJ conversion, you would have computed $298 \times (-150) = -44{,}700$ — nonsense units, nonsense answer.

Worked example: temperature flips the verdict

Now watch temperature flip a verdict. Take a reaction with $\Delta H = +50$ kJ/mol (endothermic) and $\Delta S = +200$ J/(mol·K) (entropy-increasing). At 298 K: convert $\Delta S$ to $0.200$ kJ/(mol·K), so $T\Delta S = 298 \times 0.200 = 59.6$ kJ/mol, and $\Delta G = 50 - 59.6 = -9.6$ kJ/mol — spontaneous, driven by entropy.

Cool the system to 200 K and recompute: $T\Delta S = 200 \times 0.200 = 40.0$ kJ/mol, giving $\Delta G = 50 - 40.0 = +10.0$ kJ/mol — non-spontaneous. Same reaction, opposite verdict, changed only by temperature. The crossover point where $\Delta G = 0$ is $T = \Delta H / \Delta S = 50{,}000 / 200 = 250$ K. This temperature dependence is why some reactions need heat to run (lime kilns, metal smelting) while others run only when cold — the $T\Delta S$ term is a dial, and temperature turns it.

What ΔG doesn't tell you: speed and conditions

A word of caution: $\Delta G$'s sign says nothing about speed. The reaction may be thermodynamically downhill yet kinetically frozen behind a huge activation barrier — diamond turning to graphite is spontaneous ($\Delta G < 0$) but imperceptibly slow at room temperature. Conversely, explosives have enormously negative $\Delta G$ values and tiny barriers, which is why they detonate.

Also distinguish $\Delta G$ from $\Delta G^\circ$. The ° marks standard conditions (1 M concentrations, 1 bar pressure): $\Delta G^\circ$ is a fixed property of the reaction. The actual $\Delta G$ shifts with real concentrations — a reaction with positive $\Delta G^\circ$ can still proceed if products are swept away as they form, because removing products lowers the real $\Delta G$. Living cells exploit this constantly, coupling unfavorable steps to favorable ones. Thermodynamics sets the rules; conditions and catalysts decide how the game plays out.

Common mistakes

Real-world applications

• Chemical manufacturing: screening reactions for thermodynamic feasibility before scale-up.

• Biochemistry: coupling unfavorable reactions to ATP hydrolysis (ΔG negative).

• Electrochemistry: linking ΔG to cell voltage via ΔG = −nFE.

• Geochemistry: predicting mineral stability over geological time.

• Drug design: estimating binding free energies for candidate molecules.