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The Carnot Cycle and Heat Engines
Thermodynamics
The Carnot cycle describes the most efficient heat engine possible between two temperatures. No real engine can beat it, and it sets the limit every power station and car engine must live within.
What the Carnot cycle is
A heat engine takes heat from a hot reservoir, turns part of it into work, and rejects the rest to a cold reservoir. The Carnot cycle is an idealised sequence of four reversible steps: isothermal expansion, adiabatic expansion, isothermal compression and adiabatic compression. During the isothermal steps the gas exchanges heat at constant temperature, and during the adiabatic steps it changes temperature with no heat flow. The cycle returns the gas to its starting state, so the net work done equals the net heat taken in. Because every step is reversible, the cycle produces no extra entropy, which is what makes it the most efficient possible engine.
Maximum efficiency
The efficiency of a Carnot engine depends only on the absolute temperatures of its two reservoirs: efficiency equals one minus the cold temperature divided by the hot temperature. To raise efficiency, raise the hot temperature or lower the cold temperature. A power station uses very hot steam and a cold river or cooling tower for exactly this reason. The formula also shows why 100 percent efficiency is impossible: it would need a cold reservoir at absolute zero or a hot reservoir at infinite temperature. Any real engine, with friction and irreversible heat flow, is less efficient than the Carnot limit for the same temperatures.
Where you see it
Steam turbines, petrol engines and jet engines are all heat engines, and their efficiencies are judged against the Carnot limit. A typical car engine is about 25 to 30 percent efficient, while a modern combined-cycle power station can reach about 60 percent, because it uses the waste heat from one cycle to drive a second. Refrigerators and heat pumps are Carnot cycles run in reverse: work is put in to move heat from cold to hot. The same temperature ratio that limits engines also limits how efficiently a heat pump can move heat, which is why heat pumps work best when the temperature difference is small.
Worked example: efficiency of a Carnot engine
A Carnot engine works between a hot reservoir at Th = 600 K and a cold reservoir at Tc = 300 K. Efficiency eta = 1 - Tc/Th = 1 - 300/600 = 1 - 0.50 = 0.50, or 50 percent. If it takes in Qh = 1000 J of heat, the work done is W = eta x Qh = 0.50 x 1000 = 500 J. The remaining heat rejected is Qc = Qh - W = 1000 - 500 = 500 J. No engine working between 600 K and 300 K can do better than 50 percent efficiency.
Common mistakes
- Using Celsius temperatures: Th and Tc must be in kelvin, or the efficiency comes out wrong.
- Thinking a Carnot engine is a real machine: it is an ideal model with perfectly reversible steps, used as a benchmark, not a design to build.
- Forgetting the rejected heat: even a perfect engine must reject some heat to the cold reservoir; efficiency is never 100 percent.
Practice questions
- A Carnot engine works between 800 K and 400 K. What is its maximum efficiency?
Show answer
eta = 1 - 400/800 = 0.50, or 50 percent.
- If that engine takes in 2000 J, how much work does it do?
Show answer
W = 0.50 x 2000 = 1000 J.
- What happens to efficiency if the hot reservoir gets hotter?
Show answer
It increases, because Tc/Th gets smaller, so 1 - Tc/Th gets larger.
Frequently asked questions
Could a Carnot engine ever reach 100 percent efficiency?
Only if the cold reservoir were at 0 K or the hot reservoir infinitely hot, both impossible in practice.
Can a real engine be more efficient than a Carnot engine?
No. A real engine always has friction, heat loss and irreversible processes, so it is always less efficient than a Carnot engine between the same temperatures.