IMAT Chemistry · Equilibrium, Kinetics and Energetics

Entropy, Gibbs Free Energy and Spontaneous Change

Whether a reaction can happen by itself depends on both its enthalpy change and its entropy change, combined in the Gibbs free energy ΔG = ΔH − TΔS.

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Why this matters

No past IMAT question has been set on this page yet, but entropy and free energy are in the official syllabus. Expect a qualitative question: the sign of an entropy change, or whether a reaction is spontaneous at high or low temperature.

Concept 1 of 2: Entropy: a measure of the spreading of matter and energy

A drop of ink spreads through water and never gathers back into a drop by itself. There are vastly more ways for the particles to be spread out than gathered up, so spreading is overwhelmingly likely. Entropy measures how many ways the particles and their energy can be arranged: the more spread out, the higher the entropy.

Definition

Entropy $S$ measures the dispersal (often called the disorder) of particles and energy. Units: J/(K mol).
Entropy increases from solid to liquid to gas, with a big jump on forming a gas.
A reaction that increases the number of moles of gas usually has $\Delta S > 0$ . Dissolving, mixing and heating also increase entropy.
The second law of thermodynamics: in a spontaneous change, the total entropy of the system plus surroundings increases. The entropy of the system alone can fall, if enough heat is released to the surroundings.

Change	Entropy of the system	Why
Ice melting	Increases	Particles become free to move
Steam condensing	Decreases	Gas becomes liquid
$\mathrm{CaCO_3(s) \rightarrow CaO(s) + CO_2(g)}$	Increases	A gas is produced from a solid
$\mathrm{N_2(g) + 3H_2(g) \rightarrow 2NH_3(g)}$	Decreases	4 mol of gas become 2 mol
Salt dissolving in water	Usually increases	Ions spread through the solution

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The entropy of the system can fall in a spontaneous change

Water freezes spontaneously below 0 °C although its entropy falls. The heat released raises the entropy of the surroundings by more. The second law is about the total entropy, not the system alone.

Concept 2 of 2: Gibbs free energy and whether a reaction is spontaneous

Two tendencies drive chemical change: to release energy (negative ΔH) and to spread out (positive ΔS). The Gibbs free energy weighs them against each other, with temperature deciding how much the entropy term counts. When ΔG is negative the reaction can go by itself.

Definition

$\Delta G = \Delta H - T\Delta S$ , with $T$ in kelvin. Convert $\Delta S$ from J to kJ before combining it with $\Delta H$ in kJ.
$\Delta G < 0$ : the reaction is spontaneous (feasible). $\Delta G > 0$ : not spontaneous; the reverse reaction is. $\Delta G = 0$ : at equilibrium.
$\Delta H < 0$ and $\Delta S > 0$ : spontaneous at all temperatures.
$\Delta H > 0$ and $\Delta S < 0$ : never spontaneous.
$\Delta H < 0$ and $\Delta S < 0$ : spontaneous only at low temperature. $\Delta H > 0$ and $\Delta S > 0$ : only at high temperature. The switch happens at $T = \Delta H / \Delta S$ .
Spontaneous says nothing about speed: diamond turning into graphite has $\Delta G < 0$ but is far too slow to see.

Gibbs free energy

\Delta G = \Delta H - T\Delta S

$\Delta G$ free energy change, in kJ/mol
$\Delta H$ enthalpy change, in kJ/mol
$T$ absolute temperature, in K
$\Delta S$ entropy change, in kJ/(K mol) when combined with ΔH in kJ

Worked example

For

\mathrm{CaCO_3(s) \rightarrow CaO(s) + CO_2(g)}

\Delta H = +178\ \text{kJ/mol}

and

\Delta S = +161\ \text{J K}^{-1}\text{mol}^{-1}

. Is it spontaneous at 298 K? Above what temperature does it become spontaneous?

Practice this conceptself-check · 3 quick reps

Endothermic reactions can be spontaneous

Ice melting above 0 °C and ammonium nitrate dissolving are endothermic yet happen by themselves, because their entropy increase is large enough. A positive ΔH alone does not rule out a spontaneous change.

Spontaneous does not mean fast

ΔG tells you whether a reaction can go, not how quickly. A mixture of hydrogen and oxygen has a very negative ΔG for forming water, yet it can sit unchanged for years until a spark supplies the activation energy.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Gibbs free energy and whether a reaction is spontaneous
Gibbs free energy
$\Delta G = \Delta H - T\Delta S$

Reference tables (1)

Entropy: a measure of the spreading of matter and energy5 rows

Change	Entropy of the system	Why
Ice melting	Increases	Particles become free to move
Steam condensing	Decreases	Gas becomes liquid
$\mathrm{CaCO_3(s) \rightarrow CaO(s) + CO_2(g)}$	Increases	A gas is produced from a solid
$\mathrm{N_2(g) + 3H_2(g) \rightarrow 2NH_3(g)}$	Decreases	4 mol of gas become 2 mol
Salt dissolving in water	Usually increases	Ions spread through the solution

Watch out for (3)

The entropy of the system can fall in a spontaneous change
→ Entropy: a measure of the spreading of matter and energy
Endothermic reactions can be spontaneous
→ Gibbs free energy and whether a reaction is spontaneous
Spontaneous does not mean fast
→ Gibbs free energy and whether a reaction is spontaneous

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