Gibbs Energy Calculator (ΔG = ΔH − TΔS)

About the Gibbs Energy Calculator (ΔG = ΔH − TΔS)

Four calculations around the Gibbs energy, G = H − TS. ΔG = ΔH − TΔS: type any three of ΔG, ΔH, ΔS and T and get the fourth, with the units sorted out for you (kJ against J is the classic slip), whether the reaction is spontaneous, the crossover temperature T* = ΔH/ΔS where it changes, a table and a chart of ΔG at other temperatures, and the four-case map of the signs of ΔH and ΔS. From a reaction: type the equation and ΔrH°, ΔrS° and ΔrG° come from a table of 414 standard formation values (elements, compounds, ions in water, organic compounds), each value editable, with K and ΔrG at another temperature. ΔG° and K: ΔG° = −RT ln K both ways, with K far beyond the range of a calculator (10³⁰⁰ and more). With Q: ΔG = ΔG° + RT ln Q tells which way a mixture that is not at equilibrium will go.

Without a pass, the free preview shows the Gibbs energy ΔG, or the value you solve for in full, with the first rows of each table, the first step and a watermarked chart; other figures are hidden; with a Pro pass, or after unlocking this result, you can copy the full answer or download it as CSV.

Common uses

• Thermodynamics homework: ΔG, ΔH, ΔS or T from the other three, with the units converted and every step shown.
• Finding the temperature where a reaction becomes spontaneous, such as the decomposition of limestone.
• Turning ΔG° into an equilibrium constant (or a Ksp or Ka into ΔG°) and back.
• Checking which way a reaction mixture will move from its reaction quotient.

The four cases

• ΔH < 0 and ΔS > 0: ΔG is negative at every temperature, so the reaction is always spontaneous.
• ΔH < 0 and ΔS < 0: ΔG is negative when cold: spontaneous below T* = ΔH/ΔS (enthalpy-driven).
• ΔH > 0 and ΔS > 0: ΔG is negative when hot: spontaneous above T* = ΔH/ΔS (entropy-driven).
• ΔH > 0 and ΔS < 0: ΔG is positive at every temperature: never spontaneous, and the reverse reaction always is.

At T* itself ΔG = 0: the reaction is at equilibrium with every substance in its standard state. Use the same energy unit for both terms: ΔS in J/(mol·K) is divided by 1000 before TΔS is subtracted from ΔH in kJ/mol.

ΔG°, K and Q

• ΔG° = −RT ln K. The standard Gibbs energy of reaction fixes the equilibrium constant at that temperature: ΔG° < 0 gives K > 1 (products favoured), ΔG° > 0 gives K < 1. Every 5.7 kJ/mol at 25 °C is a factor of 10 in K.
• ΔG = ΔG° + RT ln Q = RT ln(Q/K). Q has the form of K but uses the present pressures (in bar) or concentrations (in mol/L). Q < K: ΔG < 0, the reaction runs forward; Q > K: it runs backwards; Q = K: equilibrium, ΔG = 0.
• R = 8.314 462 618 J/(mol·K), exact since the SI was redefined (it is the Avogadro constant times the Boltzmann constant).
• K here is the thermodynamic constant: dimensionless, with pressures relative to 1 bar and concentrations relative to 1 mol/L.

About the data

• 414 entries at 298.15 K and 1 bar: ΔfH° and ΔfG° in kJ/mol and S° in J/(mol·K), each with its source — the same table as the reaction enthalpy calculator.
• CODATA Key Values for Thermodynamics (ΔfG° worked out from ΔfH° and S°), the NIST-JANAF tables, the NBS tables (every entry with all three values checked against ΔfG° = ΔfH° − TΔfS°) and the NIST Chemistry WebBook.
• An element in its reference state has ΔfH° = ΔfG° = 0 by definition, but its entropy S° is not zero. Ions in water use the convention that H⁺(aq) has zero for all three.
• Any two of the reaction’s sums are enough: when an S° is missing, ΔrS° comes from (ΔrH° − ΔrG°)/T.

Sources

• IUPAC, Quantities, Units and Symbols in Physical Chemistry (the Green Book), 3rd edition: G = H − TS, ΔrG° = −RT ln K°
• NIST, molar gas constant R (CODATA recommended values)
• CODATA, Key Values for Thermodynamics
• NIST, NIST-JANAF Thermochemical Tables
• Wagman et al., The NBS Tables of Chemical Thermodynamic Properties, J. Phys. Chem. Ref. Data 11, Supplement 2
• NIST, Chemistry WebBook

Limitations

• ΔH and ΔS are taken as constant with temperature, so ΔG far from 298 K is an estimate; heat capacities would refine it.
• Spontaneous means thermodynamically favourable, not fast: the rate is a question of kinetics.
• Table values are for the standard state (1 bar, 1 mol/L, infinite dilution for ions); real solutions and high pressures need activities.
• Substances that are not in the table need their ΔfH°, ΔfG° or S° typed in.

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