Entropy and Gibbs Free Energy
Entropy and Gibbs Free Energy
Entropy (S) is a measure of the disorder or number of possible arrangements (microstates) of particles in a system. The more disordered a system, the higher its entropy.
Entropy Values
Entropy has units of J K⁻¹ mol⁻¹ (note: joules, not kilojoules).
Typical entropy values increase in the order:
Solids < Liquids < Gases
This is because:
- In solids, particles are in fixed positions with very few arrangements
- In liquids, particles can move past each other — more possible arrangements
- In gases, particles are widely spaced and move randomly — vastly more possible arrangements
More complex molecules have higher entropy than simpler ones because they have more ways to vibrate, rotate, and distribute energy.
Entropy Change of a Reaction (ΔS)
The entropy change for a reaction is calculated from standard entropy values:
ΔS = ΣS(products) − ΣS(reactants)
Worked Example
For the decomposition of calcium carbonate:
CaCO₃(s) → CaO(s) + CO₂(g)
Given: S(CaCO₃) = 93, S(CaO) = 40, S(CO₂) = 214 J K⁻¹ mol⁻¹
ΔS = (40 + 214) − (93) = +161 J K⁻¹ mol⁻¹
The positive value is expected — a gas is produced from a solid, greatly increasing disorder.
Predicting the Sign of ΔS
Entropy increases (ΔS positive) when:
- A solid melts or a liquid boils (change of state to a more disordered phase)
- A solid dissolves in a solvent
- The number of gas moles increases (e.g. 1 mol gas → 2 mol gas)
- Temperature increases
Entropy decreases (ΔS negative) when:
- Gases combine to form fewer gas molecules
- A gas dissolves in a liquid
- A precipitate forms from solution
The Second Law of Thermodynamics
The second law states that for any spontaneous process, the total entropy of the universe increases:
ΔS(total) = ΔS(system) + ΔS(surroundings) > 0
The entropy change of the surroundings depends on the enthalpy change:
ΔS(surroundings) = −ΔH / T
Where ΔH is in J mol⁻¹ and T is in kelvin.
An exothermic reaction (negative ΔH) increases the entropy of the surroundings (energy disperses into them). An endothermic reaction decreases the entropy of the surroundings.
Gibbs Free Energy
Gibbs free energy (ΔG) combines enthalpy and entropy into a single criterion for spontaneity:
ΔG = ΔH − TΔS
Where:
- ΔG is in kJ mol⁻¹
- ΔH is in kJ mol⁻¹
- T is temperature in kelvin
- ΔS is in kJ K⁻¹ mol⁻¹ (convert from J K⁻¹ mol⁻¹ by dividing by 1000)
A reaction is thermodynamically feasible (spontaneous) when ΔG < 0 (negative).
Four Possible Combinations
| ΔH | ΔS | ΔG | Feasibility |
|---|---|---|---|
| − (exo) | + (increase) | Always negative | Feasible at all temperatures |
| − (exo) | − (decrease) | Negative at low T | Feasible at low temperatures only |
| + (endo) | + (increase) | Negative at high T | Feasible at high temperatures only |
| + (endo) | − (decrease) | Always positive | Never feasible |
Finding the Temperature at Which a Reaction Becomes Feasible
At the boundary, ΔG = 0:
0 = ΔH − TΔS
T = ΔH / ΔS
Worked Example
For CaCO₃(s) → CaO(s) + CO₂(g):
- ΔH = +178 kJ mol⁻¹
- ΔS = +161 J K⁻¹ mol⁻¹ = +0.161 kJ K⁻¹ mol⁻¹
T = 178 / 0.161 = 1106 K (833 °C)
Above this temperature, the reaction becomes feasible — consistent with the observation that limestone must be heated strongly to decompose.
Gibbs Free Energy and Equilibrium
ΔG is related to the equilibrium constant K:
ΔG = −RT ln K
Where R = 8.314 J K⁻¹ mol⁻¹ and T is in kelvin.
| ΔG | K | Meaning |
|---|---|---|
| Large negative | K >> 1 | Products strongly favoured |
| Zero | K = 1 | At equilibrium (products and reactants equally favoured) |
| Large positive | K << 1 | Reactants strongly favoured |
Important Caveats
- Feasible does not mean fast. A negative ΔG tells you a reaction is thermodynamically favourable but says nothing about the rate. Diamond → graphite has ΔG < 0 at room temperature, but the rate is negligible (kinetic stability).
- ΔG values calculated from standard data apply at standard conditions. Under non-standard conditions, concentrations and pressures shift the position of equilibrium.
- Always convert ΔS to kJ K⁻¹ mol⁻¹ before substituting into the Gibbs equation — mixing units is the most common calculation error.
Ellingham Diagrams (Extension)
An Ellingham diagram plots ΔG against temperature for the oxidation of metals. The line for each metal oxide slopes upward (because forming a solid from a gas decreases entropy, and −TΔS becomes more positive at higher T). A metal whose line lies below another can reduce that metal's oxide — used in extractive metallurgy to choose reducing agents.
Exam Tips
- Show unit conversions explicitly (J → kJ) in calculations — examiners award marks for this
- When explaining feasibility, state that ΔG must be negative AND note that this does not guarantee the reaction will occur at an observable rate
- Link entropy to the number of moles of gas as a quick way to predict sign