Thermodynamics — Energy & Cycles

Heat-engine efficiency — and Carnot's hard ceiling

continues from lesson 3 — values defined earlier in the course stay live here

ELI5: an engine takes in heat Q_hot, dumps some as waste Q_cold, and keeps the rest as work. Efficiency η = W / Q_hot. But Carnot proved no engine beats η = 1 − T_cold / T_hot — a limit set purely by the temperatures, not by clever engineering.

Q_{hot} := 1000 J

= 1000 J heat in from the hot source

Q_{cold} := 600 J

= 600 J heat dumped to the cold sink

\eta_{real} = \frac{Q_{hot} - Q_{cold}}{Q_{hot}}

0.4000

real efficiency = 0.40

T_{hot} := 600 K

= 600 K hot reservoir

T_{cold} := 300 K

= 300 K cold reservoir

\eta_{carnot} = 1 - \frac{T_{cold}}{T_{hot}}

0.5000

the best possible = 0.50

✓ pass

\eta_{real} \le \eta_{carnot}

no real engine beats Carnot's ceiling

1 - 300 / x

Carnot efficiency vs hot-side temperature x (K), cold side fixed at 300 K — hotter is always better

Real-world hook: this ceiling is why power plants run their boilers as hot as metallurgy allows, and why a car engine throws most of its fuel energy out as heat.