§ C-29 Gibbs Free Energy ΔG = ΔH − TΔS, Entropy & Reaction Spontaneity (Grade 10–11 · Chemical Thermodynamics)
Competition between enthalpy ΔH and entropy TΔS, inversion temperature T* = ΔH/ΔS, and equilibrium K. Nature strives simultaneously toward two opposing goals: minimum enthalpy (ΔH 0) at high temperatures.
Theory Summary & Scientific Mechanism
At constant temperature T and pressure P, chemical spontaneity is governed by the Gibbs free energy Δ G = Δ H - TΔ S, which balances the enthalpic driving force (Δ H, heat of bond rearrangement) against the entropic driving force (TΔ S, disorder and microstate multiplicity).
A reaction is spontaneous when Δ G 0, and at chemical equilibrium when Δ G = 0. The standard free energy change determines the equilibrium constant through the van ’t Hoff isotherm: Δ G^° = -RTln Kₑq.
When Δ H and Δ S share the same sign, spontaneity reverses across the inversion temperature T^* = (Δ H^°)/(Δ S^°): an endothermic reaction with positive entropy change (Δ H > 0, Δ S > 0) becomes spontaneous at high temperatures T > T^*.
Everyday Analogy
Nature loves rolling downhill in energy (Δ H 0). Gibbs free energy Δ G = Δ H - TΔ S is the referee: at high temperature T, the entropy term TΔ S wins and drives endothermic reactions forward!
Interactive Question & Derivation
Under what thermodynamic condition is an endothermic chemical reaction (Δ H > 0) spontaneous (Δ G < 0) at constant temperature and pressure?
- A: When the entropy change of the system is positive (Δ S > 0) and the absolute temperature exceeds the inversion temperature T > T^* = (Δ H)/(Δ S), so that TΔ S > Δ H
- B: Endothermic reactions (Δ H > 0) can never occur spontaneously at any temperature
- C: When Δ S < 0 and the temperature approaches absolute zero (T → 0 K)
Study Hint: An endothermic reaction (Δ H > 0) is spontaneous (Δ G 0 and temperature exceeds T^* = Δ H / Δ S.
Proof & Explanation: At constant T and P, the criterion for spontaneity is the Gibbs free energy change Δ G = Δ H - TΔ S 0: spontaneous at all T; 2) Δ H > 0, Δ S 0, Δ S > 0 (entropy-driven endothermic processes): spontaneous above T > T^* = Δ H / Δ S.
Key Equations & 3D Simulation Parameters
Δ G^° = Δ H^° - TΔ S^° = -RTln Kₑq, Tᵢnv^* = (Δ H^°)/(Δ S^°)
- Reaction enthalpy ΔH° (− exo, + endo): -120 … 120 kJ/mol (default: 60 kJ/mol)
- Reaction entropy ΔS°: -200 … 250 J/(mol·K) (default: 150 J/(mol·K))
- Absolute temperature T: 200 … 1000 K (default: 500 K)
- System pressure P: 0.5 … 5 atm (default: 1 atm)