Gibbs Free Energy & Maximum Work Calculator
Enter enthalpy change ΔH, entropy change ΔS and temperature T to compute Gibbs free energy ΔG and the maximum non-expansion work W_max=−ΔG. ΔH=−100 kJ/mol, ΔS=−200 J/(mol·K), T=298K → ΔG=−40.4 kJ/mol, spontaneous.
Input Data
Results
At a glance:Gibbs free energy G=H−TS (J. Willard Gibbs, 1876) is a thermodynamic potential measuring the maximum useful (non-expansion) work a system can do at constant temperature and pressure. Criterion: ΔG=ΔH−TΔS<0 → spontaneous (forward reaction); =0 → equilibrium; >0 → non-spontaneous (needs external work). At constant T,P the maximum non-expansion work = −ΔG (W_max=−ΔG). Physical meaning: ΔH is total energy change, TΔS is the 'unavailable' energy (dispersed to surroundings as entropy), and the remainder is available to do useful work. History: Gibbs published 'On the Equilibrium of Heterogeneous Substances' in 1876, establishing chemical thermodynamics; Lewis popularized 'free energy' in 1923; the criterion is the core of chemical reaction direction. Classic example: water formation 2H₂+O₂→2H₂O, ΔH=−571.6 kJ/mol, ΔS=−326.4 J/(mol·K), at 298K ΔG=−571.6−298×(−0.3264)=−474.3 kJ/mol <0, spontaneous. Water electrolysis ΔG>0 requires electrolysis (electrical work). Applications: (1) reaction spontaneity; (2) electrochemical EMF E=−ΔG/(nF); (3) biochemical ATP hydrolysis; (4) phase equilibrium; (5) material corrosion/passivation.
Formula
ΔG = ΔH − TΔS
Maximum work: W_max = −ΔG
Equilibrium: ΔG = 0 ⟺ ΔH = TΔS
Electrochemistry: E = −ΔG / (nF)
Temperature of equilibrium: T_eq = ΔH/ΔS
$$\Delta G = \Delta H - T\Delta S, \quad W_{\max} = -\Delta G, \quad E = -\frac{\Delta G}{nF}$$How to Use
- Enter enthalpy change ΔH (J/mol), entropy change ΔS (J/(mol·K)), temperature T (K).
- The tool computes ΔG=ΔH−TΔS, spontaneity and W_max=−ΔG.
- Example: ΔH=−100 kJ/mol, ΔS=−200 J/(mol·K), T=298K → ΔG=−40.4 kJ/mol, spontaneous.
Gibbs Free Energy and Spontaneity Examples
| ΔH | ΔS | T (K) | ΔG | Spontaneous? |
|---|---|---|---|---|
| −100 kJ/mol | −200 J/(mol·K) | 298 | −40.4 kJ/mol | Yes |
| +50 kJ/mol | +100 J/(mol·K) | 298 | +20.2 kJ/mol | No |
| +50 kJ/mol | +200 J/(mol·K) | 500 | −50 kJ/mol | Yes |
| −100 kJ/mol | +100 J/(mol·K) | 298 | −129.8 kJ/mol | Yes |
| −50 kJ/mol | −100 J/(mol·K) | 800 | +30 kJ/mol | No |
ΔG=ΔH−TΔS. When ΔH<0 and ΔS>0 always spontaneous; ΔH>0 and ΔS<0 never spontaneous; mixed cases depend on temperature. T_eq=ΔH/ΔS is the crossover temperature.
Case Studies
Fuel Cell Efficiency and Maximum Work
Hydrogen fuel cell H₂+½O₂→H₂O, ΔG=−237 kJ/mol (298K). The maximum electrical work W_max=237 kJ/mol.
The reaction enthalpy ΔH=−286 kJ/mol, so the Carnot-limited efficiency = ΔG/ΔH = 237/286 = 83% (vs combustion ~40%).
Actual PEM fuel cells reach 50-60% efficiency; the gap is from overpotential and ohmic losses. This shows why fuel cells beat heat engines.
Temperature Crossover and Reaction Direction
Calcium carbonate decomposition CaCO₃→CaO+CO₂, ΔH=+178 kJ/mol, ΔS=+161 J/(mol·K). T_eq=178000/161=1106 K (833°C).
Below 833°C ΔG>0 (limestone stable); above 833°C ΔG<0 (decomposes to lime + CO₂). This is why cement kilns run >1400°C.
Conversely, water formation ΔS<0 is spontaneous at all T<474K; above that (impossible physically) reverse. Temperature drives the direction only when ΔH and ΔS have the same sign.
FAQ
What is the relationship between Gibbs free energy and spontaneity?
At constant T,P: ΔG<0 forward reaction spontaneous; ΔG=0 equilibrium; ΔG>0 non-spontaneous (needs external work). ΔG combines enthalpy (ΔH, energy release) and entropy (ΔS, disorder) effects: ΔG=ΔH−TΔS. Exothermic (ΔH<0) and entropy-increasing (ΔS>0) both favor spontaneity. Only ΔG, not ΔH alone, determines direction — endothermic but entropy-driven reactions (e.g. salt dissolving) can be spontaneous.
Why is maximum work −ΔG?
At constant T,P the maximum useful (non-PV) work a system can do equals the decrease in Gibbs free energy: W_max=−ΔG. Proof: from the first law dU=δQ−δW and dS≥δQ/T, at constant T,P the reversible work δW_max=−dG. In a fuel cell ΔG is converted to electrical work; in a battery to electrical work. Expansion (PV) work is excluded (it pushes the atmosphere). For example H₂ fuel cell W_max=237 kJ/mol at 298K.
What is the difference between ΔG and ΔG°?
ΔG° is the standard Gibbs free energy change at 1 bar, all species in standard states (298K reference). Actual ΔG=ΔG°+RT ln Q (Q is the reaction quotient). At equilibrium Q=K and ΔG=0, so ΔG°=−RT ln K. ΔG° tells the intrinsic tendency; actual ΔG also depends on concentrations/pressures. A reaction with ΔG°>0 can still proceed if Q is small enough (reactants in excess).
How is it used in electrochemistry?
Cell EMF E=−ΔG/(nF), where n is the number of electrons transferred, F=96485 C/mol. ΔG<0 means E>0 (galvanic, spontaneous). Example: Daniell cell Zn|Zn²⁺||Cu²⁺|Cu, ΔG=−212 kJ/mol, n=2 → E=212000/(2×96485)=1.10 V. This links thermodynamics to voltage. Battery state-of-charge and open-circuit voltage are both ΔG-based. Charging reverses: external voltage must exceed E to drive ΔG>0.
How does temperature affect spontaneity?
ΔG=ΔH−TΔS. Two same-sign cases have a crossover temperature T_eq=ΔH/ΔS: (1) ΔH>0, ΔS>0 (melting, vaporization) — spontaneous only above T_eq; (2) ΔH<0, ΔS<0 (some syntheses) — spontaneous only below T_eq. Opposite-sign cases: ΔH<0,ΔS>0 always spontaneous; ΔH>0,ΔS<0 never spontaneous. Temperature shifts the balance between energy and entropy terms. Ice melting at 0°C is the classic T_eq example.
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References
Content reviewed by the Calculatorism editorial team. Results are for reference only; please refer to the relevant authorities for the official figures.