Entropy
Entropy measures energy dispersal and irreversibility. Learn ΔS = ∫δQ/T, entropy change of solids, liquids and ideal gases, entropy generation, isentropic processes and isentropic efficiency.
Why does a hot coffee cool down but never spontaneously heat up? Why is no real turbine perfect? Entropy gives a number to the direction of natural processes: in an isolated system it always increases, and every increase represents lost opportunity to do work.
Engineers use entropy to define ideal (isentropic) processes and to measure how far real machines fall short.
An analogy
A tidy deck of cards shuffled becomes disordered; shuffling more never brings it back to order. Entropy is the measure of that 'spread-outness' of energy.
The increase-of-entropy principle
For any real process, S_gen > 0; for an ideal reversible process S_gen = 0; S_gen < 0 is impossible. Friction, mixing, unrestrained expansion and heat transfer across finite temperature differences all generate entropy.
Calculating entropy change
Entropy is a property, so ΔS depends only on end states. Use m c ln(T₂/T₁) for incompressible substances, the ideal-gas relation for gases, and tables (s_f, s_fg, s_g) for steam and refrigerants.
Isentropic processes and efficiency
An adiabatic, reversible process keeps s constant — the ideal for turbines, compressors, pumps and nozzles. Real devices are compared with it: good steam turbines reach η ≈ 85–92 %, compressors ≈ 80–88 %.
T–s diagrams and lost work
On a T–s diagram, area under a reversible process curve equals heat transferred, and the enclosed cycle area equals net work. Lost work is T₀·S_gen — the Gouy–Stodola theorem — allowing a cost to be attached to each irreversibility.
Formulas
- Definition (reversible):
dS = δQ_rev / T - Solid / liquid:
ΔS = m c ln(T₂/T₁) - Ideal gas:
Δs = c_p ln(T₂/T₁) − R ln(p₂/p₁) - Isentropic ideal gas:
T₂/T₁ = (p₂/p₁)^((γ−1)/γ) - Turbine efficiency:
η_T = (h₁ − h₂)/(h₁ − h₂s)
Worked example: Heat flow between reservoirs
Given: 1000 kJ from a 500 K reservoir to a 300 K reservoir
Find: Entropy generated.
- ΔS_hot = −1000/500 = −2.0 kJ/K
- ΔS_cold = +1000/300 = +3.33 kJ/K
- S_gen = 1.33 kJ/K > 0 ✓
Answer: 1.33 kJ/K (lost work = 300 × 1.33 = 400 kJ)
Worked example: Heating water
Given: 2 kg water, 20 → 80 °C; c = 4.18 kJ/kg·K
Find: ΔS of the water.
- ΔS = 2 × 4.18 × ln(353.15/293.15) = 1.56 kJ/K
Answer: +1.56 kJ/K
Common mistakes
- Wrong: Using Q/T with °C. — Right: Always kelvin in entropy calculations.
- Wrong: Assuming adiabatic means isentropic. — Right: Only adiabatic and reversible is isentropic; friction adds entropy.
Applications
- Turbomachinery rating: Isentropic efficiency is the key performance number for compressors and turbines.
- Plant optimisation: Exergy/entropy-generation maps show where to invest to improve efficiency.
- Refrigeration: T–s and p–h diagrams track the refrigerant through each component.
Key takeaways
- ΔS = ∫δQ_rev/T; S_gen ≥ 0.
- Solids: mc ln(T₂/T₁); gases: include pressure term.
- Isentropic = ideal adiabatic reference.
- Lost work = T₀·S_gen.
Going further
- Exergy efficiency: Second-law efficiency = useful exergy out / exergy in — more meaningful than first-law efficiency for heating systems.
- Entropy in heat exchangers: Minimising temperature differences reduces S_gen but increases area — a classic cost trade-off.
- Information entropy: Shannon's entropy has the same mathematical form; Landauer's principle links erasing a bit to kT ln 2 of heat.