Q.What is a spontaneous process? Derive the Gibbs-Helmholtz equation.

Gibbs Free Energy: The "Why" Behind Spontaneous Reactions

Imagine you're pushing a boulder downhill. It's going to happen naturally — you don't need to keep pushing. But if you want to roll it uphill, you have to work against gravity the whole way. Chemistry works the same way: some reactions happen on their own (spontaneous), and others need a constant push of energy.

The question is: what decides which is which? That's exactly what Gibbs Free Energy answers.

The Two Competing Forces

Two things drive every chemical change:

1. Enthalpy ($H$) — the total heat content. Nature tends to move toward lower energy. A fire releases heat; that's enthalpy driving the reaction forward. Reactions that release heat (exothermic, $\Delta H < 0$) are favoured.

2. Entropy ($S$) — the measure of disorder. Nature also tends toward more chaos. A messy room doesn't tidy itself; a gas spreads to fill its container. Reactions that increase disorder (positive $\Delta S$) are favoured.

But here's the catch: these two can pull in opposite directions. An endothermic reaction (absorbs heat, $\Delta H > 0$) might still happen if it creates enough disorder. Ice melting is a perfect example — it absorbs heat, but the liquid water is far more disordered than the crystal.

The Resolution: Gibbs Free Energy

Josiah Willard Gibbs combined these two forces into one number that tells you the net direction:

Where:

• $\Delta G$ = change in Gibbs free energy (kJ/mol)
• $\Delta H$ = change in enthalpy (kJ/mol)
• $T$ = absolute temperature (Kelvin)
• $\Delta S$ = change in entropy (J/K·mol — careful with units!)

The sign of $\Delta G$ is the final verdict:

Why Temperature Matters

Notice the $T$ in front of $\Delta S$. Temperature decides which force wins. At low temperatures, the enthalpy term ($\Delta H$) dominates. At high temperatures, the entropy term ($T\Delta S$) takes over.

This explains everyday observations:

• Ice melts spontaneously above 0°C (entropy wins at higher $T$)
• Water freezes spontaneously below 0°C (enthalpy wins at lower $T$)
• At exactly 0°C, $\Delta G = 0$ — ice and water coexist in equilibrium

The Precise Statement

Gibbs Free Energy is the maximum useful work obtainable from a closed system at constant temperature and pressure. When a reaction proceeds, the system loses free energy ($\Delta G < 0$), and that energy is available to do work — like running a muscle or powering a battery. …