04 // Interactive Formula Compendium // CHEMISTRY-THERMOη_max = 1 − (T_C / T_H) = W_out / Q_H
Carnot Maximum Thermodynamic Efficiency Formula
Carnot's theorem proves no heat engine operating between absolute temperatures T_H and T_C can exceed thermal efficiency η_max = 1 − T_C/T_H.
Formula Reference
Main Formula
η_max = 1 − (T_C / T_H) = W_out / Q_H
Rearranged Forms
COP_heating = T_H / (T_H − T_C)
COP_cooling = T_C / (T_H − T_C)
Variables & Units
| Symbol | Name | Unit |
|---|---|---|
| eta | Maximum Carnot Efficiency | ratio (0–1) |
| TH | Hot Reservoir Temperature T_H | K |
| TC | Cold Sink Temperature T_C | K |
Inputs & Parameters
[K]
[K]
Calculated Result (Maximum Carnot Efficiency)
eta =0.647059ratio (0–1)
Formula Notes & Background
- A reversible cycle of two isotherms and two adiabats satisfies ΔS_cycle = Q_H/T_H − Q_C/T_C = 0 ⇒ Q_C/Q_H = T_C/T_H.
- Since W = Q_H − Q_C, efficiency η = W/Q_H = 1 − Q_C/Q_H = 1 − T_C/T_H.
Step-by-Step Examples
#1: Steam turbine operating between superheated steam at T_H = 850 K and condenser cooling water at T_C = 300 K.
T_H = 850 KT_C = 300 K
η_max = 1 − (300 / 850) = 1 − 0.3529
→ η_max = 0.6471 (64.71% theoretical ceiling)
#2: Ideal heat pump delivering indoor heat at T_H = 295 K from outdoor air at T_C = 270 K.
T_H = 295 KT_C = 270 K
COP_max = 295 / (295 − 270) = 295 / 25
→ COP_max = 11.8 (real heat pumps achieve ~3.5–4.5)
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Frequently Asked Questions
Can Carnot efficiency ever reach 100%?+
Only if T_C = 0 K (impossible by the Third Law of Thermodynamics) or T_H → ∞.
Why must T_H and T_C be in Kelvin?+
Thermodynamic entropy scales with absolute Kelvin temperature dS = δQ_rev/T; Celsius ratios produce physically meaningless numbers.
Why can a heat pump COP exceed 1.0 without violating energy conservation?+
Because electrical work moves existing ambient thermal energy Q_C from outdoors rather than generating all heat resistively: Q_H = W + Q_C.