Entropy: Measuring Disorder

Entropy (S) is a thermodynamic quantity that measures the degree of disorder or randomness in a system. The second law of thermodynamics states that the total entropy of the universe always increases for a spontaneous process.

States of Matter and Entropy

Solid

Low entropy -- ordered

→

Liquid

Medium entropy

→

Gas

High entropy -- disordered

Factors that increase entropy (ΔS > 0):

Phase changes from solid → liquid → gas
Increase in the number of moles of gas
Dissolving a solute in a solvent
Increasing temperature

Units: Entropy is measured in J K⁻¹ mol⁻¹ (joules per kelvin per mole). Note: joules, not kilojoules -- a common source of error in calculations!

Gibbs Free Energy and Spontaneity

Gibbs free energy (G) combines enthalpy and entropy into a single value that predicts whether a reaction is spontaneous. The key equation is:

ΔG = ΔH − TΔS

ΔG = Gibbs free energy change (kJ/mol)

ΔH = enthalpy change (kJ/mol)

T = absolute temperature (K)

ΔS = entropy change (convert to kJ K⁻¹ mol⁻¹ by dividing J value by 1000)

< 0

ΔG < 0

Spontaneous -- the reaction proceeds in the forward direction without external input.

= 0

ΔG = 0

The system is at equilibrium -- no net change in either direction.

> 0

ΔG > 0

Non-spontaneous -- the reaction requires continuous energy input to proceed.

Important: "Spontaneous" does not mean "fast." A spontaneous reaction may still be extremely slow if the activation energy is very high (e.g., diamond converting to graphite).

Temperature Dependence of Spontaneity

Since ΔG = ΔH − TΔS, the sign of ΔG can change with temperature. The table below shows the four possible combinations:

ΔH	ΔS	ΔG	Spontaneity
− (exo)	+ (increase)	Always −	Spontaneous at all T
+ (endo)	− (decrease)	Always +	Non-spontaneous at all T
− (exo)	− (decrease)	Depends on T	Spontaneous at low T
+ (endo)	+ (increase)	Depends on T	Spontaneous at high T

Finding the crossover temperature: When ΔG = 0, the reaction is at equilibrium. Setting ΔH − TΔS = 0 gives T = ΔH / ΔS. Above or below this temperature, spontaneity changes.

Key Vocabulary

Entropy (S)

A measure of the degree of disorder or number of possible microstates in a system. Units: J K⁻¹ mol⁻¹.

Gibbs Free Energy (G)

A thermodynamic potential that determines whether a process is spontaneous at constant temperature and pressure.

Spontaneous

A process that can occur without continuous external energy input (ΔG < 0). Does not imply anything about reaction rate.

Equilibrium

The state where the rate of forward and reverse reactions are equal and ΔG = 0. No net change occurs.

Worked Examples

1

Calculate ΔG at 298 K for a reaction with ΔH = −120 kJ/mol and ΔS = +45 J K⁻¹ mol⁻¹. Is it spontaneous?

Step 1: Convert ΔS to kJ: ΔS = +45 / 1000 = +0.045 kJ K⁻¹ mol⁻¹.

Step 2: Apply ΔG = ΔH − TΔS = (−120) − (298 × 0.045) = −120 − 13.41 = −133.41 kJ/mol.

Answer: ΔG = −133.4 kJ/mol. Since ΔG < 0, the reaction is spontaneous at 298 K.

2

A reaction has ΔH = +180 kJ/mol and ΔS = +250 J K⁻¹ mol⁻¹. At what temperature does it become spontaneous?

Step 1: At the crossover, ΔG = 0, so T = ΔH / ΔS.

Step 2: Convert ΔS: 250 J K⁻¹ mol⁻¹ = 0.250 kJ K⁻¹ mol⁻¹.

Step 3: T = 180 / 0.250 = 720 K.

Answer: The reaction becomes spontaneous above 720 K (approximately 447 °C).

3

Predict the sign of ΔS for: CaCO₃(s) → CaO(s) + CO₂(g).

Step 1: Count moles of gas: Reactants have 0 moles of gas; products have 1 mole of gas (CO₂).

Step 2: An increase in moles of gas means greater disorder.

Answer: ΔS is positive (ΔS > 0) because the number of moles of gas increases and a solid decomposes into a solid plus a gas, increasing overall disorder.

Knowledge Check

Select the correct answer for each question. Click "Check Answer" to see if you are right.

Question 1

Which of the following processes would have the greatest increase in entropy?

Water freezing to ice Water evaporating to steam Compressing a gas into a smaller volume A gas condensing to a liquid

Question 2

A reaction has ΔH = −50 kJ/mol and ΔS = +100 J K⁻¹ mol⁻¹. At 300 K, ΔG equals:

+20 kJ/mol −20 kJ/mol −80 kJ/mol +80 kJ/mol

Question 3

If ΔG for a reaction is positive, the reaction is:

Spontaneous in the forward direction At equilibrium Exothermic Non-spontaneous in the forward direction

Question 4

A reaction with ΔH > 0 and ΔS > 0 becomes spontaneous:

Only at high temperatures Only at low temperatures At all temperatures Never

Question 5

At what temperature (K) does a reaction with ΔH = +100 kJ/mol and ΔS = +200 J K⁻¹ mol⁻¹ reach equilibrium?

200 K 500 K 1000 K 2000 K

Entropy and Gibbs Free Energy

Entropy: Measuring Disorder

States of Matter and Entropy

Gibbs Free Energy and Spontaneity

ΔG < 0

ΔG = 0

ΔG > 0

Temperature Dependence of Spontaneity

Key Vocabulary

Entropy (S)

Gibbs Free Energy (G)

Spontaneous

Equilibrium

Worked Examples

Calculate ΔG at 298 K for a reaction with ΔH = −120 kJ/mol and ΔS = +45 J K⁻¹ mol⁻¹. Is it spontaneous?

A reaction has ΔH = +180 kJ/mol and ΔS = +250 J K⁻¹ mol⁻¹. At what temperature does it become spontaneous?

Predict the sign of ΔS for: CaCO₃(s) → CaO(s) + CO₂(g).

Knowledge Check

Key Concepts Summary

Entropy and Gibbs Free Energy

Entropy: Measuring Disorder

States of Matter and Entropy

Gibbs Free Energy and Spontaneity

ΔG < 0

ΔG = 0

ΔG > 0

Temperature Dependence of Spontaneity

Key Vocabulary

Entropy (S)

Gibbs Free Energy (G)

Spontaneous

Equilibrium

Worked Examples

Calculate ΔG at 298 K for a reaction with ΔH = −120 kJ/mol and ΔS = +45 J K−1 mol−1. Is it spontaneous?

A reaction has ΔH = +180 kJ/mol and ΔS = +250 J K−1 mol−1. At what temperature does it become spontaneous?

Predict the sign of ΔS for: CaCO3(s) → CaO(s) + CO2(g).

Knowledge Check

Key Concepts Summary

Calculate ΔG at 298 K for a reaction with ΔH = −120 kJ/mol and ΔS = +45 J K⁻¹ mol⁻¹. Is it spontaneous?

A reaction has ΔH = +180 kJ/mol and ΔS = +250 J K⁻¹ mol⁻¹. At what temperature does it become spontaneous?

Predict the sign of ΔS for: CaCO₃(s) → CaO(s) + CO₂(g).