04 // Interactive Formula Compendium // CHEMISTRY-THERMOΔG = ΔH − T · ΔS, ΔG° = −R · T · ln(K)
Gibbs Free Energy & Equilibrium Constant Formula
Gibbs free energy change ΔG = ΔH − TΔS combines enthalpy ΔH and entropy ΔS at temperature T; processes at constant temperature and pressure are spontaneous when ΔG < 0.
Formula Reference
Main Formula
ΔG = ΔH − T · ΔS, ΔG° = −R · T · ln(K)
Rearranged Forms
K_eq = exp(−ΔG° / (R · T))
T_crossover = ΔH / ΔS
Variables & Units
| Symbol | Name | Unit |
|---|---|---|
| dG | Gibbs Free Energy ΔG | kJ/mol |
| dH | Enthalpy Change ΔH | kJ/mol |
| T | Absolute Temperature | K |
| dS | Entropy Change ΔS | J/(mol·K) |
Inputs & Parameters
[kJ/mol]
[K]
[J/(mol·K)]
Calculated Result (Gibbs Free Energy ΔG)
dG =-32.9576kJ/mol
Formula Notes & Background
- From the Second Law ΔS_univ = ΔS_sys + ΔS_surr ≥ 0 with ΔS_surr = −ΔH_sys/T at constant P, multiplying by −T yields ΔG = ΔH − TΔS ≤ 0.
- Note unit consistency: ΔH is conventionally tabulated in kJ/mol while ΔS is in J/(mol·K) (divide TΔS by 1,000).
Step-by-Step Examples
#1: Haber ammonia synthesis N₂ + 3H₂ ⇌ 2NH₃ at 298.15 K (ΔH° = −92.2 kJ/mol, ΔS° = −198.7 J/(mol·K)).
ΔH° = −92.2 kJ/molΔS° = −198.7 J/(mol·K)T = 298.15 K
ΔG° = −92.2 − (298.15 × −0.1987) = −92.2 + 59.24
→ ΔG° = −32.96 kJ/mol (Spontaneous at 298 K; crossover at T = 464 K)
#2: Endothermic ice melting at +10 °C (283.15 K) with ΔH_fus = +6.01 kJ/mol and ΔS_fus = +22.0 J/(mol·K).
ΔH = +6.01 kJ/molΔS = +22.0 J/(mol·K)T = 283.15 K
ΔG = 6.01 − (283.15 × 0.0220) = 6.01 − 6.23
→ ΔG = −0.22 kJ/mol (Spontaneous melting above 273.15 K)
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Frequently Asked Questions
Does a strongly negative ΔG guarantee a fast reaction?+
No — ΔG measures thermodynamic driving force (equilibrium position), while reaction speed is controlled kinetically by activation barrier E_a (diamond → graphite has ΔG° < 0 yet is kinetically frozen).
When is a reaction spontaneous at all temperatures?+
When it is both exothermic (ΔH < 0) and entropy-increasing (ΔS > 0), making ΔG = ΔH − TΔS negative for every T > 0.
What is the physical meaning of |ΔG|?+
At constant T and P, −ΔG equals the maximum reversible non-PV work (such as electrical work in a fuel cell) extractable from the process.