The Carnot cycle is an ideal model of a heat engine that sets the maximum possible efficiency for converting heat into work. It operates between a hot reservoir at temperature T_H and a cold reservoir at temperature T_C. Engineers use it as a benchmark because no real engine working between the same two temperatures can be more efficient.
The cycle also shows why temperature difference, not just heat input, controls the potential for useful work.
A Carnot engine moves through four reversible processes: isothermal expansion, adiabatic expansion, isothermal compression, and adiabatic compression. On a PV diagram, the enclosed area represents the net work output per cycle. On a TS diagram, heat transfer appears as area because Q_rev = TΔS for an isothermal reversible step.
Real engines fall short because of friction, turbulence, finite temperature differences during heat transfer, heat leaks, and other irreversible effects.
Key Facts
- Carnot efficiency: η_C = 1 - T_C/T_H, with temperatures in kelvins.
- Net work per cycle: W_net = Q_H - Q_C.
- Thermal efficiency: η = W_net/Q_H.
- For a reversible Carnot cycle: Q_C/Q_H = T_C/T_H.
- Isothermal processes occur at constant temperature, so ΔU = 0 for an ideal gas and Q = W.
- Adiabatic reversible processes have no heat transfer, so Q = 0 and PV^γ = constant for an ideal gas.
Vocabulary
- Carnot cycle
- An ideal reversible heat engine cycle made of two isothermal processes and two adiabatic processes.
- Heat reservoir
- A large thermal body that can absorb or supply heat while remaining at nearly constant temperature.
- Isothermal process
- A thermodynamic process that occurs at constant temperature.
- Adiabatic process
- A thermodynamic process in which no heat is transferred into or out of the working substance.
- Entropy
- A state variable that measures energy dispersal and determines the direction and limits of heat transfer.
Common Mistakes to Avoid
- Using Celsius temperatures in η_C = 1 - T_C/T_H is wrong because thermodynamic temperature ratios must use kelvins.
- Assuming a Carnot engine has 100 percent efficiency is wrong because efficiency reaches 1 only if T_C = 0 K, which is physically unattainable.
- Confusing the PV diagram area with total heat input is wrong because the PV loop area represents net work, not Q_H alone.
- Treating real engines as reversible is wrong because friction, rapid expansion, heat loss, and finite temperature differences create entropy and reduce efficiency.
Practice Questions
- 1 A Carnot engine operates between a hot reservoir at 600 K and a cold reservoir at 300 K. What is its maximum thermal efficiency?
- 2 A reversible engine absorbs 1200 J of heat from a 500 K reservoir and rejects heat to a 300 K reservoir. Find Q_C and W_net.
- 3 Explain why increasing the hot reservoir temperature or decreasing the cold reservoir temperature can improve the maximum possible efficiency of a heat engine.