Gibbs Free Energy: Why It Is the Criterion for Spontaneity

The problem Gibbs energy solves

The second law states that a spontaneous process increases the entropy of the universe:

That is correct but inconvenient, because it demands information about the surroundings. Gibbs energy removes that requirement.

The derivation

At constant temperature and pressure, heat flows to the surroundings equal to minus the enthalpy change of the system, so:

Substituting into the second law:

Multiplying through by −T, which reverses the inequality:

Defining G = H − TS gives ΔG < 0 for a spontaneous process at constant T and P. Everything about the surroundings has been absorbed into quantities measurable in the system.

The enthalpy–entropy competition

The last two rows are where questions concentrate, because they have a crossover temperature at which ΔG changes sign. Setting ΔG = 0 gives T = ΔH/ΔS, and that temperature is exactly where the two terms balance.

The link to equilibrium

Under non-standard conditions the Gibbs energy change is

At equilibrium ΔG = 0 and Q = K, giving the relation that connects thermodynamics to equilibrium:

Temperature dependence

From dG = −S dT + V dP, the temperature derivative at constant pressure is

Since entropy is always positive, Gibbs energy always falls as temperature rises — and it falls faster for a system with higher entropy. This is why gases become more favoured at high temperature, and it is the thermodynamic basis of many industrial decisions.

Frequently asked questions

Does a negative ΔG mean the reaction will be fast?

No. Thermodynamics says whether a process can happen, kinetics says how quickly. Many strongly favourable reactions are immeasurably slow without a catalyst, and confusing the two is a standard trap.

What does the word "free" mean?

It is the portion of the energy change available to do non-expansion work. At constant T and P, −ΔG is the maximum useful work obtainable — which is why it appears in electrochemistry as ΔG = −nFE.

Can an endothermic reaction be spontaneous?

Yes, provided ΔS is positive and the temperature is high enough for TΔS to exceed ΔH. Dissolution processes that cool the solution are everyday examples.

Why is ΔG zero at equilibrium?

Because at equilibrium there is no net drive in either direction. It does not mean nothing is happening — forward and reverse rates are equal, so the composition is unchanging.