Gibbs Free Energy (G)
Gibbs Free Energy, symbolised G, is one of the most powerful concepts in chemical thermodynamics. Named after the American physicist Josiah Willard Gibbs, it combines two fundamental thermodynamic quantities — enthalpy (H) and entropy (S) — into a single function that predicts whether a chemical process will occur spontaneously under conditions of constant temperature and pressure. These are precisely the conditions found in most laboratory reactions and biological systems, making G extraordinarily useful in practice. The defining equation is G = H − TS, where T is the absolute temperature in Kelvin. For a chemical reaction at constant temperature, we work with changes: ΔG = ΔH − TΔS. The sign of ΔG tells us everything about spontaneity: if ΔG < 0, the reaction is spontaneous (thermodynamically favourable) and will proceed in the forward direction without requiring a continuous input of energy. If ΔG > 0, the reaction is non-spontaneous in the forward direction but would proceed spontaneously in reverse. If ΔG = 0, the system is at chemical equilibrium — the forward and reverse rates exactly balance each other. It is crucial to understand that 'spontaneous' does not mean 'fast'. A reaction can be thermodynamically spontaneous (ΔG < 0) yet proceed imperceptibly slowly because of a high activation energy barrier — the combustion of diamond in air is a classic example. Gibbs Free Energy describes the thermodynamic driving force, while kinetics describes the rate. Both perspectives are needed for a complete picture of any chemical process.
- The Two Competing Factors
- Four Possible Combinations of ΔH and ΔS
- The Crossover Temperature
- Standard Gibbs Free Energy (ΔG°)
- Non-Standard Conditions: The Reaction Quotient Q
The Two Competing Factors
Every spontaneous process seeks to do two things simultaneously: lower its enthalpy (release energy to the surroundings, ΔH < 0) and increase its entropy (increase disorder, ΔS > 0). Gibbs Free Energy captures the competition between these two tendencies. The term TΔS represents the 'entropic contribution' weighted by temperature — at high temperatures, entropy dominates; at low temperatures, enthalpy dominates.
Four Possible Combinations of ΔH and ΔS
Case 1: ΔH < 0 and ΔS > 0 — ΔG is always negative; the reaction is spontaneous at all temperatures (e.g., combustion of hydrogen: \ce{2H2 + O2 -> 2H2O}, ΔH = −572 kJ/mol, ΔS > 0). Case 2: ΔH > 0 and ΔS < 0 — ΔG is always positive; the reaction is never spontaneous (e.g., converting graphite to diamond under ambient conditions). Case 3: ΔH < 0 and ΔS < 0 — spontaneous only at low temperatures where the |ΔH| term dominates (e.g., freezing of water, \ce{H2O_{(l)} -> H2O_{(s)}}). Case 4: ΔH > 0 and ΔS > 0 — spontaneous only at high temperatures where TΔS term dominates (e.g., dissolution of ammonium nitrate, \ce{NH4NO3_{(s)} -> NH4^+_{(aq)} + NO3^-_{(aq)}}).
The Crossover Temperature
For Cases 3 and 4, there is a specific temperature at which ΔG changes sign. Setting ΔG = 0: 0 = ΔH − TΔS, so T_crossover = ΔH / ΔS. Below this temperature, Case 3 reactions are spontaneous; above it, they are not. For Case 4, the opposite applies. This crossover temperature is a powerful predictive tool — for example, ice melts spontaneously above 273 K at 1 atm because at that temperature ΔG for melting crosses from positive to negative.
Standard Gibbs Free Energy (ΔG°)
The standard Gibbs Free Energy change, ΔG°, refers to conditions where all reactants and products are in their standard states (pure solids or liquids, gases at 1 bar, solutes at 1 mol/L) at a specified temperature (usually 298 K). It can be calculated three ways: (1) ΔG° = ΔH° − TΔS°; (2) using standard Gibbs energies of formation: ΔG°_rxn = Σ ΔG°_f(products) − Σ ΔG°_f(reactants); (3) from the equilibrium constant K via ΔG° = −RT ln K, where R = 8.314 J/mol·K. This last relationship is especially powerful — it links thermodynamic data directly to measurable equilibrium concentrations.
Non-Standard Conditions: The Reaction Quotient Q
Under non-standard conditions, ΔG = ΔG° + RT ln Q, where Q is the reaction quotient. When Q < K, ln Q < ln K, so ΔG < 0 and the forward reaction is spontaneous — the system moves toward products. When Q > K, the reverse reaction is spontaneous. At equilibrium Q = K, ΔG = 0, and the system has no net driving force. This framework explains why reactions can be driven in otherwise unfavourable directions by manipulating concentrations or pressures.
The Key Equation
- \Delta G — Change in Gibbs Free Energy (J or kJ): The free energy available to do useful work. Negative value indicates a spontaneous process at constant T and P.
- \Delta H — Change in Enthalpy (J or kJ): Heat exchanged with surroundings at constant pressure. Negative for exothermic reactions, positive for endothermic.
- T — Absolute Temperature (K (Kelvin)): Must always be expressed in Kelvin. Scales the entropic contribution to spontaneity.
- \Delta S — Change in Entropy (J/K or kJ/K): Change in disorder or dispersal of energy. Positive when products are more disordered than reactants.
Worked Example
Problem. The decomposition of calcium carbonate, \ce{CaCO3_{(s)} -> CaO_{(s)} + CO2_{(g)}}, has ΔH° = +178 kJ/mol and ΔS° = +161 J/mol·K. (a) Calculate ΔG° at 298 K. (b) Is the reaction spontaneous at 298 K? (c) At what temperature does the reaction become spontaneous?
Answer. ΔG° = +130 kJ/mol at 298 K (non-spontaneous); spontaneous above approximately 1106 K (833 °C).
Origins · 1 minJosiah Willard Gibbs and how this was discovered
Gibbs developed the mathematical foundation of chemical thermodynamics between 1875 and 1878 in his landmark paper 'On the Equilibrium of Heterogeneous Substances', published in the Transactions of the Connecticut Academy of Arts and Sciences. He introduced the concepts of chemical potential, Gibbs Free Energy, and phase equilibria, essentially creating the theoretical framework that allowed chemists to predict reaction outcomes without trial and error. His work was initially published in a relatively obscure journal and was not widely recognised until Wilhelm Ostwald and Henri Le Chatelier translated and publicised it in Europe. Gibbs spent his entire academic career at Yale University, where he was a professor for 32 years — and for the first decade he received no salary, living off a family inheritance. Yale only began paying him when Johns Hopkins University offered him a position elsewhere.
Related Concepts
- Enthalpy and Hess's Law — ΔH is one of the two core components of the Gibbs equation; understanding enthalpy cycles and bond energies is essential for calculating ΔG.
- Entropy and the Second Law of Thermodynamics — ΔS is the second component of ΔG = ΔH − TΔS; predicting whether entropy increases requires understanding molecular disorder and the Second Law.
- The Equilibrium Constant K — ΔG° = −RT ln K directly connects Gibbs Free Energy to the position of equilibrium; spontaneity and equilibrium are two sides of the same coin.
- Electrochemical Cell Potential (E°) — ΔG° = −nFE° links Gibbs Free Energy to the electrical work done by a galvanic cell, showing that spontaneous reactions can do useful electrical work.