Gibbs Free Energy and Reaction Spontaneity

The Gibbs free energy change, ΔG = ΔH − TΔS, combines enthalpy and entropy into a single quantity that predicts whether a process is spontaneous at constant temperature and pressure: ΔG < 0 means spontaneous, ΔG > 0 means non-spontaneous and ΔG = 0 means equilibrium. In brief: for the formation of ammonia, ΔH = −92.2 kJ and ΔS = −198.7 J/K, so at 298 K ΔG = −92.2 − (298)(−0.1987) = −33.0 kJ and the reaction is spontaneous.

Enthalpy alone does not decide whether a reaction occurs: some endothermic processes are spontaneous, and some exothermic ones are not. The Gibbs free energy resolves this by accounting for the entropy of the system and of the surroundings at once. This article derives the equation, explains the four possible cases, gives data on standard free energies of formation and works through calculations.

What is Gibbs free energy?

The second law requires that ΔS_universe > 0 for a spontaneous process. At constant temperature and pressure, ΔS_surroundings = −ΔH_system / T, so ΔS_universe = ΔS − ΔH/T. Multiplying by −T gives −TΔS_universe = ΔH − TΔS, and this quantity is defined as the change in Gibbs free energy:

ΔG = ΔH − TΔS

A positive ΔS_universe corresponds to a negative ΔG. The advantage is that ΔG refers only to the system, so it can be calculated from the properties of the reactants and products. Under standard conditions, ΔG° can be calculated from the standard free energies of formation: ΔG°_rxn = ΣnΔG_f°(products) − ΣmΔG_f°(reactants).

When is a reaction spontaneous?

Sign of ΔG for the four combinations of ΔH and ΔS
ΔH	ΔS	ΔG = ΔH − TΔS	Spontaneity	Example
−	+	Always negative	Spontaneous at all T	2H₂O₂ → 2H₂O + O₂
+	−	Always positive	Non-spontaneous at all T	3O₂ → 2O₃
−	−	Negative at low T	Spontaneous at low T	Freezing of water, ammonia synthesis
+	+	Negative at high T	Spontaneous at high T	CaCO₃ → CaO + CO₂, melting

“Spontaneous” means that the process can occur on its own, without continuous input of energy. It does not say anything about speed: the conversion of diamond to graphite is spontaneous, but is so slow that it is not observed. Also, a reaction at equilibrium has ΔG = 0.

Schematic plot of ΔG = ΔH − TΔS against temperature. Each label gives (sign of ΔH, sign of ΔS). The (−, +) line is always below zero and the (+, −) line is always above it. The (−, −) line starts negative and crosses zero upward, and the (+, +) line starts positive and crosses zero downward, so those two reactions change spontaneity at a switching temperature.

Standard free energies of formation

Standard Gibbs free energies of formation at 298.15 K (kJ/mol)
Substance	ΔG_f°	Substance	ΔG_f°
H₂O(l)	−237.1	NH₃(g)	−16.4
CO₂(g)	−394.4	NO(g)	+86.6
CO(g)	−137.2	NaCl(s)	−384.1
CH₄(g)	−50.5	C₂H₅OH(l)	−174.8

As with enthalpies of formation, the free energy of formation of an element in its standard state is zero.

Worked examples

Example 1: ΔG from ΔH and ΔS

For N₂ + 3H₂ → 2NH₃, ΔH° = −92.2 kJ and ΔS° = −198.7 J/K = −0.1987 kJ/K. At 298.15 K:

ΔG° = −92.2 − (298.15)(−0.1987) = −92.2 + 59.2 = −33.0 kJ.

Note that ΔS must be converted from J/K to kJ/K before it is combined with ΔH in kJ. The result is negative: the reaction is spontaneous at 298 K.

Example 2: ΔG from free energies of formation

CH₄(g) + 2O₂(g) → CO₂(g) + 2H₂O(l)

ΔG° = [−394.4 + 2(−237.1)] − [−50.5 + 0] = −868.6 + 50.5 = −818.1 kJ. The highly negative value shows that methane in air is thermodynamically unstable, though the mixture needs a spark to react.

Example 3: melting of ice at two temperatures

For H₂O(s) → H₂O(l), ΔH = +6.01 kJ and ΔS = +22.0 J/K = +0.0220 kJ/K.

At 263.15 K (−10 °C): ΔG = 6.01 − (263.15)(0.0220) = 6.01 − 5.79 = +0.22 kJ. Ice does not melt.
At 283.15 K (+10 °C): ΔG = 6.01 − (283.15)(0.0220) = 6.01 − 6.23 = −0.22 kJ. Ice melts.
At 273.15 K: ΔG = 6.01 − 6.01 = 0. Ice and water are in equilibrium.

Example 4: a coupled reaction

In living cells, reactions that are not spontaneous are driven by coupling with a reaction that is strongly spontaneous, such as the hydrolysis of ATP, for which ΔG°′ is about −30.5 kJ/mol (at pH 7). If a process needs +17 kJ/mol, then coupling it with ATP hydrolysis gives a total of about −13.5 kJ/mol, so the combined process can take place.

What does “free” mean?

The change in Gibbs free energy equals the maximum amount of useful non-expansion work that can be obtained from a process at constant temperature and pressure: w_max = ΔG. The word “free” refers to energy that is free to do useful work, as in the electrical work of a battery. For a reaction in an electrochemical cell, ΔG = −nFE, which links thermodynamics to electrochemistry. The rest of the energy, TΔS, is lost as heat in a reversible process.

Who introduced free energy?

Josiah Willard Gibbs, an American physicist at Yale, developed the theory in a series of papers from 1873 to 1878, including the long paper “On the Equilibrium of Heterogeneous Substances”. He called the function the “available energy”. Hermann von Helmholtz had earlier introduced a similar function for constant volume. The notation ΔG was chosen in honor of Gibbs, and the IUPAC name is the Gibbs energy.

Which mistakes are common?

Mixing units: convert ΔS from J/K to kJ/K, or ΔH to J.
Using Celsius: T must be in kelvins.
Equating spontaneous with fast: ΔG says nothing about the rate.
Using ΔG° as if it were ΔG: under non-standard conditions, ΔG = ΔG° + RT ln Q.
Forgetting that ΔG° of formation of elements is zero: but not their S° values.