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Equilibrium and Spontaneity Criteria in Thermodynamics

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In thermodynamics, a spontaneous process is one that can happen on its own without any continuous push from outside — like ice melting in a warm room or iron rusting in moist air. Spontaneity says nothing about speed; it only means the change is thermodynamically favoured once it gets started.

Chemists decide spontaneity and equilibrium with two related tools: the Gibbs free energy (G) and the total entropy change of the universe. When Gibbs energy reaches its minimum value, the system is at equilibrium and no further net change is possible in either direction.

Key Concepts

  • ΔG = ΔH − TΔS, where T is the absolute temperature in kelvin.
  • ΔG < 0: spontaneous (product-favoured). ΔG = 0: equilibrium. ΔG > 0: non-spontaneous as written (the reverse reaction is spontaneous).
  • ΔS(universe) = ΔS(system) + ΔS(surroundings): positive for every spontaneous process — this is the second law of thermodynamics.
  • At constant T and P the Gibbs criterion applies; at constant T and V, the Helmholtz energy is used instead (ΔA < 0).

Formulas

  • ΔG = ΔH − TΔS: ΔH is the enthalpy change (J/mol), ΔS the entropy change (J/mol·K), T in kelvin. Assumes constant T and P, and that ΔH and ΔS do not change much with temperature.
  • ΔG = ΔG° + RT ln Q: connects free energy to the reaction quotient Q at any moment.
  • ΔG° = −RT ln K: at equilibrium Q = K and ΔG = 0, linking standard free energy to the equilibrium constant.
  • Sign combinations: (ΔH < 0, ΔS > 0) → spontaneous at all temperatures; (ΔH > 0, ΔS < 0) → never spontaneous; (ΔH < 0, ΔS < 0) → spontaneous only at low temperature; (ΔH > 0, ΔS > 0) → spontaneous only at high temperature.

Worked Example

For the Haber process, N2 + 3H2 → 2NH3: ΔH° = −92.4 kJ/mol and ΔS° = −198.7 J/mol·K = −0.1987 kJ/mol·K. At 298 K:

  1. Compute TΔS° = 298 × (−0.1987) = −59.2 kJ/mol.
  2. ΔG° = ΔH° − TΔS° = −92.4 − (−59.2) = −33.2 kJ/mol.
  3. Since ΔG° < 0, the reaction is spontaneous at 298 K.
  4. Find K: K = e^(−ΔG°/RT) = e^(33200/(8.314 × 298)) = e^13.40 ≈ 6.6 × 10^5, a very large equilibrium constant favouring ammonia.

Industry still runs it near 700 K with an iron catalyst: the higher temperature speeds the slow kinetics, and high pressure favours the side with fewer gas moles.

Common Misconceptions

  • "Spontaneous means instant." Diamond converting to graphite has ΔG < 0, yet takes geological time. Thermodynamics gives the direction; kinetics gives the speed.
  • "A negative ΔG° means the reaction goes to completion." Completion depends on K = e^(−ΔG°/RT). A small negative ΔG° gives a modest K and a genuine mixture at equilibrium.
  • "The system's entropy must always increase." Only the total entropy of the universe must increase. When water freezes, the system's entropy falls, but the heat released raises the surroundings' entropy by even more.

Real-World Applications

  • Metallurgy: Ellingham diagrams plot ΔG° against temperature to choose which reducing agent (like carbon or aluminium) can extract a metal from its ore.
  • Batteries: a discharging cell is a spontaneous redox reaction; its voltage comes directly from ΔG = −nFE.

Gibbs energy is the chemist's compass: ΔG < 0 points toward products, ΔG = 0 marks equilibrium, and ΔG° = −RT ln K links energy to the equilibrium constant. Always pair it with kinetics — thermodynamics says what can happen, while rate laws say how fast.

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