Standard Gibbs energy and equilibrium
The standard molar reaction Gibbs energy change relates enthalpy and entropy. The ° superscript denotes standard states at the specified temperature. ΔH° and ΔS° are reaction changes: they must describe the same equation, stoichiometry, phases and reference temperature. The thermodynamic constant K, defined using activities, is dimensionless.
Standard Gibbs energy
ΔG∘=ΔH∘−TΔS∘ Equilibrium constant
K=exp(−RTΔG∘) With R in J/(mol·K), the equations require T in K, ΔH° in J/mol and ΔS° in J/(mol·K). Inputs can use the units available in the selectors.
Gibbs-Helmholtz and the van’t Hoff line
The identity above relates to the differential Gibbs-Helmholtz equation, which describes how ΔG°/T varies with temperature at constant pressure:
(∂T∂(ΔG∘/T))p=−T2ΔH∘ Approximating ΔH° and ΔS° as constant makes ln K a linear function of 1/T:
lnK=−RΔH∘T1+RΔS∘ The slope is −ΔH°/R and has units of K; the intercept is ΔS°/R and is dimensionless.
Four regimes and inversion temperature
The inversion temperature T_inv is the crossover where ΔG° changes sign and K passes through 1. For constant ΔH° and ΔS°, with ΔS° ≠ 0, T_inv = ΔH°/ΔS° if the ratio is positive. The four sign combinations are interpreted within this approximation:
| ΔH° / ΔS° | Sign of ΔG° | Sign example |
|---|
| − / + | Negative for T > 0. | 2 H₂O₂(l) → 2 H₂O(l) + O₂(g), near 298 K. |
| + / − | Positive for T > 0. | 2 H₂O(l) + O₂(g) → 2 H₂O₂(l), near 298 K. |
| − / − | Negative below T_inv; positive above. | Freezing: H₂O(l) → H₂O(s), near its melting point. |
| + / + | Positive below T_inv; negative above. | Melting: H₂O(s) → H₂O(l), near its melting point. |
The chemical examples illustrate signs for the phases and temperatures specified. The calculator’s numerical examples are independent of those data.
With ΔS° = 0 and ΔH° ≠ 0, there is no inversion temperature: ΔG° stays constant with T, although K varies. With ΔH° = ΔS° = 0, ΔG° = 0 and K = 1 for every T > 0 within the approximation; there is no isolated inversion.
Standard energy and reaction direction
ΔG° determines K, but K alone does not specify equilibrium concentrations. For a mixture, the reaction quotient Q is constructed from product and reactant activities, raised to their stoichiometric coefficients:
ΔG=ΔG∘+RTlnQ=RTln(KQ) At constant T and p, Q < K implies ΔG < 0 for the forward reaction; Q > K implies ΔG > 0 and favors the reverse direction. Equilibrium requires Q = K and ΔG = 0, even when ΔG° is nonzero. The sign of ΔG° describes the favored direction when Q = 1; it does not determine the reaction rate.
Ideal equilibrium composition is covered in ICE Table. In electrochemistry, ΔG = −nFE relates energy to cell potential; the Nernst equation is presented in Galvanic Cell. Kc and Kp expressions must be interpreted with their units and reference states.
Limits of the constant-parameter model
ΔCₚ is the reaction heat capacity change. When it is not negligible, ΔH° and ΔS° vary with T. The ΔCₚ ≈ 0 approximation requires an interval where that variation is small and phases remain unchanged. An inversion temperature far from the data interval requires checking its validity.
The absence of a crossover within the chart does not imply that none exists at another temperature. The T_inv result indicates whether there is a positive inversion temperature.
Reading very large or small constants
Logarithms allow comparisons of orders of magnitude: log₁₀ K = 3 means K = 10³, and log₁₀ K = −3 means K = 10⁻³. A zero value of ln K or log₁₀ K corresponds to K = 1.
The displayed digits describe result rounding; they do not represent the uncertainty of the thermodynamic data.
Constants and references
- Standard state
- Usual reference pressure: 1 bar. Temperature must be specified; 298.15 K (25 °C) is a common tabulation temperature, rather than a mandatory standard-state condition.