The Carnot Cycle and Maximum Efficiency

The Carnot cycle is the ideal reversible heat engine cycle, consisting of two isothermal and two adiabatic processes between a hot reservoir at Th and a cold reservoir at Tc, and it has the maximum possible efficiency of any engine operating between these temperatures, ηCarnot = 1 − Tc ÷ Th (temperatures in kelvins). In brief: no real engine can exceed the Carnot efficiency, and all reversible engines between the same two reservoirs have the same efficiency. To raise the efficiency, raise Th or lower Tc. The maximum coefficients of performance for refrigerators and heat pumps are COPR = Tc ÷ (Th − Tc) and COPHP = Th ÷ (Th − Tc).

Carnot’s analysis gives the theoretical limit for the efficiency of every engine, from steam turbines to jet engines, and sets the scale for the temperature that Kelvin used to define the absolute scale. This guide describes the four steps of the cycle, derives the efficiency, discusses why real engines fall short, compares ideal and real engines in a table, works through numerical examples and corrects common mistakes.

What are the four steps?

An ideal gas in a cylinder with a piston goes through the following steps (A → B → C → D → A) on a PV diagram:

  1. Isothermal expansion (A → B) at Th: the gas absorbs heat Qh = nRTh ln(VB/VA) from the hot reservoir and does work. Since the gas and the reservoir are at the same temperature, the heat transfer is reversible.
  2. Adiabatic expansion (B → C): the gas continues to expand without heat exchange, doing work and cooling from Th to Tc.
  3. Isothermal compression (C → D) at Tc: work is done on the gas, and it rejects heat Qc = nRTc ln(VC/VD) to the cold reservoir.
  4. Adiabatic compression (D → A): the gas is compressed without heat exchange, and warms from Tc back to Th.

The net work is the area enclosed by the cycle: W = Qh − Qc. The cycle is reversible: run backward, it is a refrigerator.

Pressure-volume diagram of a Carnot cycle between 500 K and 300 K Volume (arbitrary units) Pressure (arbitrary units) A B C D Th = 500 K Tc = 300 K W
A Carnot cycle for a monatomic ideal gas between 500 K and 300 K, in arbitrary units. A to B and C to D are isothermal (solid) and B to C and D to A are adiabatic (dashed). The cycle runs clockwise, and the shaded enclosed area is the net work W = Qh − Qc, with efficiency 1 − 300 ÷ 500 = 40 percent.

What is the maximum efficiency?

For the adiabatic steps, TVγ−1 is constant, which gives VB/VA = VC/VD. The logarithms in Qh and Qc are therefore equal, and Qc ÷ Qh = Tc ÷ Th. The efficiency is

ηCarnot = 1 − Qc ÷ Qh = 1 − Tc ÷ Th.

Carnot’s theorem: (1) no engine operating between two given reservoirs can be more efficient than a reversible engine operating between the same reservoirs; (2) all reversible engines between the same two reservoirs have the same efficiency, independent of the working substance. The proof is by contradiction: a more efficient engine could be used to drive a Carnot refrigerator and transfer heat from cold to hot without work, violating the second law. The Carnot efficiency is 100 percent only if Tc = 0 K, which is unattainable; it follows that no engine can be 100 percent efficient. A Carnot engine would have to operate infinitely slowly to be reversible, so it would deliver zero power: real engines are designed to give a useful power with a lower efficiency. For maximum power from an engine, the efficiency is η = 1 − √(Tc/Th) (Curzon-Ahlborn), which is closer to the observed efficiencies of power plants.

What about refrigerators?

The reversed Carnot cycle absorbs heat Qc at Tc and rejects Qh at Th, using work W = Qh − Qc. The maximum coefficients of performance are COPR,max = Tc ÷ (Th − Tc) and COPHP,max = Th ÷ (Th − Tc). They are large when the temperature difference is small: a heat pump works better in mild weather than in severe cold, and a refrigerator uses more energy when the room is warmer. Absolute zero cannot be reached since COP would tend to zero (the third law).

Ideal and real efficiencies compared

Carnot limits and typical real efficiencies
Engine Th (K) Tc (K) Carnot limit Typical real
Steam power plant80030062.5%about 35 to 40%
Combined-cycle gas plant1,50030080%up to about 60%
Car engineabout 2,000 (combustion)30085%about 25 to 30%
Geothermal plant45030033%about 10 to 15%

Worked example: Carnot limits

  1. Maximum efficiency. An engine operates between 600 K and 300 K: η = 1 − 300 ÷ 600 = 0.50 = 50 percent. If the hot reservoir were raised to 900 K: η = 1 − 300 ÷ 900 = 67 percent.
  2. Steam plant. Steam at 527 °C (800 K) and a condenser at 27 °C (300 K): ηmax = 1 − 300 ÷ 800 = 62.5 percent. The real efficiency of 38 percent is 61 percent of the Carnot limit.
  3. An impossible claim. An inventor claims an engine with an efficiency of 40 percent operating between 400 K (steam at 127 °C) and 300 K. Carnot limit: 1 − 300 ÷ 400 = 25 percent. The claim of 40 percent violates the second law.
  4. Heat and work per cycle. A Carnot engine between 500 K and 300 K takes 1,000 J from the hot reservoir. η = 1 − 300 ÷ 500 = 40 percent, so W = 400 J and Qc = 600 J (= 1,000 × 300 ÷ 500 ✓). The entropy: −1,000 ÷ 500 + 600 ÷ 300 = −2 + 2 = 0, so the process is reversible.
  5. Refrigerator. A kitchen refrigerator keeps the inside at 270 K (−3 °C) in a room at 300 K. COPmax = 270 ÷ (300 − 270) = 9.0. A real refrigerator has COP of about 2 to 4. To extract 900 J of heat, the minimum work is 900 ÷ 9.0 = 100 J.
  6. Heat pump. Heating a house at 293 K (20 °C) from outside air at 273 K (0 °C): COPHP,max = 293 ÷ 20 = 14.7. At −10 °C (263 K): 293 ÷ 30 = 9.8. Real heat pumps reach COP of about 3 to 4, and decrease in cold weather.
500 K 300 K 1,000 J 400 J 600 J 600 0.5 900 0.667 300 K

Who was Carnot?

Nicolas Léonard Sadi Carnot (1796 to 1832) published Reflections on the Motive Power of Fire in 1824, at the age of 28, analyzing the limit of the efficiency of steam engines, before the first law was known: he used the caloric theory, in which heat is a conserved fluid, but his main conclusion survived. He died of cholera, and his work was ignored until Émile Clapeyron presented it graphically in 1834 and William Thomson and Rudolf Clausius used it in the 1840s and 1850s to develop the second law and the absolute temperature scale.

Which mistakes are common?

Mistake 1: Using Celsius temperatures in the Carnot efficiency

Absolute temperatures are required. Using °C gives a wrong value.

Mistake 2: Believing that a real engine can reach the Carnot efficiency

Friction, heat losses and irreversibility reduce the efficiency below the limit.

Mistake 3: Thinking that the efficiency depends on the working substance

For a reversible engine, it depends only on the two temperatures.

Mistake 4: Applying the Carnot formula to refrigerators as an efficiency

Use the coefficient of performance: Qc ÷ W, which can exceed 1.