Entropy

Learning Objectives (OCR Spec 5.2.2)

By the end of this lesson you should be able to:

• Explain entropy as a measure of the disorder of a system
• Recall the units of entropy as J K^-1 mol^-1 (note: joules, not kilojoules)
• Predict the sign of an entropy change (ΔS) for a reaction
• Calculate ΔS° from standard entropy values
• Relate entropy to the number of accessible microstates of a system

What is Entropy?

Thermodynamics is more than just enthalpy. Many reactions that are endothermic nevertheless happen spontaneously - a piece of ammonium nitrate dissolved in water chills the beaker, yet the process proceeds without any input of energy. Enthalpy alone cannot predict which reactions are thermodynamically favourable. We need another variable: entropy.

Entropy (S) is a measure of the disorder or spread of energy in a system. More rigorously, it is proportional to the number of ways the particles and their energies can be arranged while still giving the same overall macroscopic state. The more arrangements (microstates) available, the higher the entropy.

A pack of playing cards, neatly ordered from Ace of Spades to King of Hearts, has only one "sorted" arrangement. Throw the cards in the air and the number of disordered arrangements is vast. Tossed cards have high entropy; the sorted deck has low entropy.

At the particulate level:

• A crystalline solid has particles locked in place - very few ways to arrange them - low entropy.
• A gas has particles moving randomly throughout a large volume - vastly many ways to arrange them - high entropy.

States of Matter and Entropy

The clearest guide to entropy is the physical state of a substance.

State	Particle arrangement	Entropy
Solid	regular, fixed	lowest
Liquid	disordered, free to move	intermediate
Gas	random, high KE, large volume	highest

graph LR
  A[Solid<br/>low S] -->|melting| B[Liquid<br/>medium S]
  B -->|vaporising| C[Gas<br/>high S]
  C -->|much higher than liquid| D[Gases dominate<br/>the entropy term]

Vaporisation gives a particularly large entropy increase because the volume available to each particle jumps from a few cubic nanometres (liquid) to a few cubic centimetres (gas at 1 bar, 298 K) - an increase of about 1000-fold in accessible space.

Predicting the Sign of ΔS

Rules of thumb for predicting the sign of an entropy change:

1. Increase in moles of gas -> positive ΔS (largest contribution)
2. Solid -> liquid -> gas -> positive ΔS
3. Dissolving a solid usually gives positive ΔS (except for some hydrations)
4. Reactions where the number of particles increases give positive ΔS
5. Mixing two substances gives positive ΔS
6. Higher temperature -> higher entropy (more thermal motion)

Counter-examples exist. Hydrating a highly charged ion can DECREASE the entropy of water because the orderly water shell around the ion is more structured than bulk liquid water.

Worked Examples: Predicting ΔS

(a) 2H2(g) + O2(g) -> 2H2O(l)

Moles of gas: 3 -> 0. Entropy decreases sharply. ΔS° is very negative (about -327 J K^-1 mol^-1).

(b) CaCO3(s) -> CaO(s) + CO2(g)

Moles of gas: 0 -> 1. Entropy increases. ΔS° is positive (about +161 J K^-1 mol^-1).

(c) NH4NO3(s) -> NH4+(aq) + NO3-(aq)

Solid dissolving to give aqueous ions. Entropy increases. ΔS° is positive (about +108 J K^-1 mol^-1), and this is why the process is spontaneous even though it is endothermic.

(d) N2(g) + 3H2(g) -> 2NH3(g)

Moles of gas: 4 -> 2. Entropy decreases. ΔS° is negative (about -198 J K^-1 mol^-1). Despite this, the Haber reaction can proceed because it is strongly exothermic (see lesson 7 for the full treatment).

Calculating ΔS°

The standard entropy change of a reaction is calculated from the standard entropies of the reactants and products:

ΔS°_reaction = ΣS°_products - ΣS°_reactants

Just as with enthalpy, you take "products minus reactants". The difference is that standard entropies are tabulated as absolute values (referenced to S = 0 at 0 K), not relative values. A pure substance has a well-defined standard entropy.

Units: J K^-1 mol^-1 (joules, not kilojoules)