Red = isothermal at Th |
Orange = adiabatic expansion |
Blue = isothermal at Tc |
Purple = adiabatic compression
The Carnot cycle is the most efficient possible thermodynamic cycle for converting heat into work between two fixed-temperature reservoirs. It consists of four reversible stages shown on the P-V diagram:
The green shaded area inside the loop equals the net work output W per cycle. The ratio W / Qh is the thermal efficiency.
Carnot's theorem (1824) states: No heat engine operating between two heat reservoirs can be more efficient than a Carnot engine operating between the same reservoirs. This is a direct consequence of the Second Law of Thermodynamics.
Why can't any engine beat the Carnot limit? Because to do so would require transferring heat spontaneously from a cold body to a hot body — which violates the Clausius statement of the Second Law. The Carnot cycle is reversible (zero entropy production), making it the theoretical ceiling for real engines that always involve irreversible losses such as friction, heat leaks, and finite temperature differences.
A modern coal power plant operates at roughly Th ≈ 800 K and Tc ≈ 300 K, giving a Carnot ceiling of η ≈ 62.5 %. Actual plant efficiency is ~40 % due to friction, incomplete combustion, and heat losses. Gas turbines (Th ≈ 1500 K) can approach ~60 % in combined-cycle configurations. Refrigerators run the Carnot cycle in reverse — consuming work to pump heat from cold to hot.
Increasing Th or decreasing Tc always improves the Carnot efficiency — drag the sliders to explore this directly.
The Carnot cycle is the theoretical gold standard of heat engine efficiency, devised by French engineer Sadi Carnot in 1824. It consists of four reversible processes operating between two reservoirs at temperatures T_H (hot) and T_C (cold): isothermal expansion (the gas absorbs heat Q_H at constant temperature T_H), adiabatic expansion (the gas expands without heat exchange, cooling to T_C), isothermal compression (the gas rejects heat Q_C at T_C), and adiabatic compression (the gas is compressed back to its initial state). The remarkable result is that no real engine operating between the same two temperatures can exceed the Carnot efficiency: η = 1 − T_C/T_H.
This simulation animates the Carnot cycle on a pressure-volume (P-V) diagram, showing the four stages as curves. You can adjust the hot and cold reservoir temperatures to see how efficiency changes, observe the work done as the enclosed area on the P-V diagram, and compare Carnot efficiency with a hypothetical less-efficient real engine cycle.
What is Carnot efficiency and why is it the maximum possible?
Carnot efficiency η = 1 − T_C/T_H (where temperatures are in kelvin) is the maximum efficiency any heat engine can achieve operating between two temperature reservoirs. This limit follows from the second law of thermodynamics: any real engine includes irreversibilities (friction, heat leaks, finite temperature differences), which generate entropy and reduce efficiency below the Carnot value. The Carnot engine is reversible — it generates no entropy — making it the theoretical ceiling.
What are the four stages of the Carnot cycle?
1) Isothermal expansion: the working gas absorbs heat Q_H from the hot reservoir at constant temperature T_H, expanding and doing work. 2) Adiabatic expansion: the gas expands further with no heat exchange, cooling from T_H to T_C. 3) Isothermal compression: the gas is compressed at constant temperature T_C, rejecting heat Q_C to the cold reservoir. 4) Adiabatic compression: the gas is compressed from T_C back to T_H with no heat exchange, completing the cycle.
What is an adiabatic process?
An adiabatic process occurs with no heat transfer between the system and its surroundings (Q = 0). For an ideal gas undergoing adiabatic change, the relationship PV^γ = constant holds, where γ = C_p/C_v is the heat capacity ratio (≈1.4 for diatomic gases like air). Adiabatic processes are rapid (no time for heat exchange) or well-insulated; they cause temperature changes because all the work done comes from the internal energy of the gas.
Real engines (Otto cycle in petrol engines, Diesel cycle in diesel engines, Rankine cycle in steam turbines) deviate from Carnot for several reasons: combustion is irreversible; heat transfer occurs across finite temperature differences; friction dissipates energy; and the working fluid is not an ideal gas. A modern petrol engine achieves about 35–40% efficiency; a combined-cycle gas turbine power station reaches about 60%; the Carnot limit for a 1200 K combustion temperature and 300 K ambient is 75%.
Efficiency increases as T_H rises or T_C falls. For a steam power plant with T_H = 600°C (873 K) and T_C = 30°C (303 K), the Carnot limit is 1 − 303/873 ≈ 65%. Increasing T_H to 700°C improves the limit to 70%. In practice, materials constraints (turbine blade strength, steam pipeline pressure) limit T_H, while cooling water or ambient air sets T_C. Geothermal power plants, with T_H ≈ 150°C, have Carnot limits of only ~25%.
Rudolf Clausius proved that for any cyclic process, ∮ dQ/T ≤ 0, with equality for reversible (Carnot) cycles. This led to the definition of entropy: dS = dQ_rev/T. For an irreversible process, dS > dQ/T, meaning entropy always increases in the universe. The Carnot cycle is the unique cycle where ΔS_universe = 0 for each complete cycle, which is why it achieves the maximum efficiency.
Yes — the reversed Carnot cycle is the ideal refrigerator or heat pump. Running in reverse, it uses work to transfer heat from T_C to T_H. The coefficient of performance (COP) as a refrigerator is COP_R = T_C/(T_H − T_C), and as a heat pump is COP_HP = T_H/(T_H − T_C). A heat pump heating a building (T_H = 20°C, T_C = 0°C) has a theoretical COP_HP = 293/20 ≈ 14.7, meaning it delivers nearly 15 units of heat per unit of electrical work — far better than direct electric heating.
The Rankine cycle is the practical steam power cycle used in thermal power stations. It uses a condensable working fluid (water/steam) and consists of: pump (adiabatic pressurisation of liquid), boiler (isobaric heat addition producing steam), turbine (adiabatic expansion generating work), and condenser (isobaric heat rejection). Modern supercritical Rankine cycles operate above the critical point of water (374°C, 22 MPa), achieving thermal efficiencies of 45–48%.
Entropy S (in J/K) quantifies the number of microscopic states (microstates) Ω available to a system: S = k_B ln Ω (Boltzmann's formula). In thermodynamics, entropy measures the "spread" of energy across microstates; high entropy means energy is distributed among many states, making it unavailable to do work. The second law (entropy of an isolated system never decreases) is a statistical statement: systems evolve towards the most probable macrostate, which has the most microstates.
The Otto cycle models the ideal petrol engine with four stages on a P-V diagram: adiabatic compression (ratio r = V_max/V_min), constant-volume heat addition (fuel combustion), adiabatic expansion (power stroke), and constant-volume heat rejection (exhaust). The Otto efficiency is η = 1 − r^(1−γ). For a compression ratio of 10:1 and γ = 1.4, η ≈ 60% — but real engines achieve only 35% because of heat losses, incomplete combustion, and friction. Higher compression ratios improve efficiency but risk knocking (premature ignition).