Lesson 2 · 16 min
Heat and engines
The first law is bookkeeping. The second law is why you cannot bookkeep your way to 100%.
Two ways in, one ledger
A system’s internal energy U changes when heat Q enters or when work W is done. Engineers write ΔU = Q − W, with W the work done by the system (pistons that go out, turbines that spin). Heat capacity: Q = mcΔT for a temperature change with no phase change. Phase changes eat (or release) heat at constant T: Q = mL.
ΔU = Q − W η = W_net / Q_in η_Carnot = 1 − T_C / T_H
Temperatures in kelvin. Carnot is the ceiling, not a design. Real heat engines live well below it.
Worked example
A small heat engine
- Given: Q_in = 400 J per cycle
- Given: Q_out = 280 J
- Given: T_H = 600 K
- Given: T_C = 300 K
Find: Actual η vs Carnot
- W_net = Q_in − Q_out = 120 J
- η = 120/400 = 0.30
- η_C = 1 − 300/600 = 0.50
30% actual, 50% Carnot ceiling. The gap is friction, finite-rate heat transfer, incomplete expansion.
Check
Carnot efficiency between 25 °C and 100 °C is about