Entropy and spontaneity

Entropy

• Entropy, $S$, is a measure of how spread out the energy and the particles of a system are.

• A system with its particles and energy more spread out and more disordered has a higher entropy.

• $\Delta S$ is positive when entropy increases and negative when it decreases.

• You are never asked to calculate an entropy value at Level 3 — only to work out the sign and use it.

When entropy increases

• Solid → liquid → gas. Each step gives the particles more freedom, so entropy increases.

• A solid dissolving. The ordered lattice breaks up into ions or molecules moving freely through the solution, so entropy increases.

• More moles of gas produced than consumed. Entropy increases.

• Fewer moles of gas produced than consumed. Entropy decreases.

• Count the moles of gas on each side first. Gases have far higher entropy than liquids or solids, so a change in the number of moles of gas swamps everything else and decides the sign in almost every exam question.

• For $\text{CaCO}_3(s) \rightarrow \text{CaO}(s) + \text{CO}_2(g)$: 0 mol of gas becomes 1 mol of gas, so $\Delta S_{sys}$ is positive.

• For $\text{N}_2(g) + 3\text{H}_2(g) \rightarrow 2\text{NH}_3(g)$: 4 mol of gas becomes 2 mol, so $\Delta S_{sys}$ is negative.

Two entropy changes, not one

This is where most marks are won and lost. There are two systems to think about.

• $\Delta S_{sys}$ — the entropy change of the reaction itself.

  • Found by counting moles of gas and looking at states.

• $\Delta S_{surr}$ — the entropy change of everything else.

  • An exothermic reaction releases energy into the surroundings, spreading energy out there, so $\Delta S_{surr}$ is positive.
  • An endothermic reaction absorbs energy from the surroundings, so $\Delta S_{surr}$ is negative.

• Note the link: the sign of $\Delta S_{surr}$ is fixed entirely by the sign of $\Delta_rH$, and it is the opposite sign.

Spontaneity

$$\Delta S_{total} = \Delta S_{sys} + \Delta S_{surr}$$

• A reaction is spontaneous when $\Delta S_{total}$ is positive.

• A reaction is not spontaneous when $\Delta S_{total}$ is negative.

• The four cases:

$\Delta S_{sys}$	$\Delta S_{surr}$ (from $\Delta_rH$)	Spontaneous?
+	+ (exothermic)	Always
−	− (endothermic)	Never
+	− (endothermic)	Only if $\Delta S_{sys}$ is larger
−	+ (exothermic)	Only if $\Delta S_{surr}$ is larger

• The bottom two rows are where the interesting questions live. Say which term wins and why.

Temperature

• The same amount of energy released into the surroundings makes a bigger difference when the surroundings are cold than when they are hot — spreading energy out matters more where there is less of it already.

• So raising the temperature makes $\Delta S_{surr}$ smaller in magnitude, which weakens whichever way the enthalpy term was pushing.

• Consequently:

  • A reaction with $\Delta S_{sys}$ positive but endothermic becomes spontaneous at high temperature, because the unfavourable $\Delta S_{surr}$ shrinks and $\Delta S_{sys}$ takes over.
  • A reaction with $\Delta S_{sys}$ negative but exothermic becomes non-spontaneous at high temperature, because the favourable $\Delta S_{surr}$ shrinks away.

• This is a qualitative argument only. No calculation is required or expected.

Spontaneous does not mean fast

• Spontaneous means the reaction can happen without a continuous supply of energy. It says nothing about how quickly.
• A mixture of petrol and air is thermodynamically spontaneous and completely stable at room temperature, because its activation energy is high.
• Spontaneity is thermodynamics; speed is kinetics. They are separate questions with separate answers.