Carnot Engine & Efficiency

A heat engine converts thermal energy into useful mechanical work by exploiting a temperature difference between a hot reservoir and a cold reservoir. Every real engine — steam turbines, car engines, jet engines — does exactly this. But how efficient can such an engine ever be? The Carnot engine answers that question by describing the most efficient heat engine theoretically possible operating between any two fixed temperatures. Sadi Carnot proved in 1824 that no engine can be more efficient than a reversible engine operating between the same two temperatures, and that all reversible engines operating between the same two temperatures have identical efficiency. This is not an engineering limitation but a fundamental law of nature rooted in the second law of thermodynamics. The Carnot efficiency sets an absolute ceiling that no engine, no matter how cleverly designed, can exceed. Understanding Carnot efficiency teaches us two powerful lessons: first, it tells us exactly how much work we can extract from a given temperature difference; second, it reveals that even a 'perfect' engine wastes heat — not because of friction or poor design, but because thermodynamics demands it. Real engines always fall short of the Carnot limit due to irreversibilities like friction, turbulence, and heat conduction across finite temperature gaps.

Key takeaways
  • The Four Carnot Processes
  • Defining Thermal Efficiency
  • The Carnot Efficiency Formula
  • Why Real Engines Fall Short

The Four Carnot Processes

The Carnot cycle consists of four reversible steps performed on an ideal gas. (1) Isothermal expansion at the hot temperature T_H: the gas absorbs heat Q_H from the hot reservoir while expanding slowly enough that temperature stays constant. (2) Adiabatic expansion: the gas expands with no heat exchange, so it cools from T_H down to T_C while doing work. (3) Isothermal compression at the cold temperature T_C: the gas releases heat Q_C to the cold reservoir at constant temperature. (4) Adiabatic compression: the gas is compressed with no heat exchange, heating back up from T_C to T_H, completing the cycle. Every step is perfectly reversible — run the cycle backwards and you have a perfect refrigerator.

Defining Thermal Efficiency

Efficiency η (eta) measures how well the engine converts input heat into useful work. It is defined as the net work output W divided by the heat absorbed from the hot source Q_H. Since energy is conserved, W = Q_H − Q_C, where Q_C is the heat dumped into the cold reservoir. So η = W/Q_H = (Q_H − Q_C)/Q_H = 1 − Q_C/Q_H. This ratio is always less than 1, because some heat must always be rejected — you cannot convert heat entirely into work without a cold reservoir.

The Carnot Efficiency Formula

For the Carnot cycle specifically, one can show using the properties of ideal-gas isothermal and adiabatic processes that the ratio of heats equals the ratio of absolute temperatures: Q_C/Q_H = T_C/T_H, where temperatures are measured in Kelvin. Substituting into the efficiency formula gives the celebrated Carnot efficiency: η_Carnot = 1 − T_C/T_H. This depends only on the two reservoir temperatures — not on the working substance or any design detail. To maximise efficiency, you want T_H as large as possible and T_C as small as possible.

Why Real Engines Fall Short

Real engines are irreversible: friction generates extra heat, gas flows through valves with turbulence, and heat leaks across temperature gradients. Every such irreversibility increases the entropy of the universe and forces the engine to reject more heat than the Carnot minimum. A modern coal power station typically achieves about 35–45% efficiency, while the Carnot efficiency for its steam temperature (~600 °C) versus a cold reservoir (~30 °C) would be about 66%. The gap reflects real irreversibilities. Car engines reach roughly 25–35% efficiency in practice.

The Key Equation

ηCarnot=1−TCTH\eta_{\text{Carnot}} = 1 - \frac{T_C}{T_H}
  • \eta_{\text{Carnot}} — Carnot efficiency (dimensionless (0 to 1)): The maximum fraction of input heat that can be converted to useful work; multiply by 100 for percentage.
  • T_C — Cold reservoir temperature (K (kelvin)): The absolute temperature of the cold sink into which waste heat is dumped. Must be in kelvin, not Celsius.
  • T_H — Hot reservoir temperature (K (kelvin)): The absolute temperature of the hot source supplying heat to the engine. Must be in kelvin.

Worked Example

Problem. A steam power plant operates with steam entering the turbine at 500 °C and exhausting heat into a river at 20 °C. (a) Calculate the maximum (Carnot) efficiency of this plant. (b) If the plant absorbs 800 MJ of heat from the steam per second, what is the maximum power output? (c) How much heat per second must be discharged to the river at this maximum efficiency?

η=62.1%,W=497 MW,QC=303 MJ s−1\eta = 62.1\%, \quad W = 497\text{ MW}, \quad Q_C = 303\text{ MJ s}^{-1}

Answer. Carnot efficiency ≈ 62.1%; maximum power output ≈ 497 MW; heat discharged to river ≈ 303 MJ per second.

Figure
Diagram
HOT RESERVOIRTₕ (high temp)COLD RESERVOIRT_C (low temp)HEATENGINEQₕQ_CWWORKOUTPUTη = 1 − T_C / Tₕ
Origins · 1 minNicolas Léonard Sadi Carnot and how this was discovered
NL
Nicolas Léonard Sadi Carnot
1796–1832 · French

Carnot published 'Reflections on the Motive Power of Fire' in 1824 at just 28 years old, establishing the theoretical maximum efficiency of heat engines — now called the Carnot efficiency. His work laid the foundation for the second law of thermodynamics, decades before entropy was formally defined by Clausius. Carnot developed his revolutionary theory before the law of conservation of energy was established, and he initially used the (incorrect) caloric theory of heat. Despite this flawed starting point, his conclusions about maximum efficiency were entirely correct — a testament to the power of careful logical reasoning.

  • Second Law of Thermodynamics — The Carnot efficiency limit is a direct consequence of the second law; no engine can be more efficient than a reversible (Carnot) engine because that would violate the second law.
  • Entropy — In a Carnot cycle, entropy exchanged with the hot reservoir equals entropy returned to the cold reservoir (ΔS_total = 0 for the reversible cycle), explaining why Q_C/Q_H = T_C/T_H.
  • First Law of Thermodynamics — Energy conservation (W = Q_H − Q_C) is used in every efficiency calculation and underpins the Carnot cycle analysis.
  • Refrigerators and Heat Pumps — A Carnot engine run in reverse is a Carnot refrigerator — the most efficient possible refrigerator — making both concepts two sides of the same thermodynamic coin.