Overview
This lecture explains the Carnot cycle, its steps, how it sets the theoretical efficiency limit for heat engines, and its relationship with the second law of thermodynamics.
The Carnot Cycle and Its Steps
- The Carnot cycle is a theoretical model for the most efficient heat engine cycle possible between two heat reservoirs.
- Step 1: Isothermal expansion—gas absorbs heat (Q_h) from hot reservoir at (T_h), does work with no change in internal energy.
- Step 2: Adiabatic expansion—gas expands further without heat exchange, temperature drops from (T_h) to (T_c).
- Step 3: Isothermal compression—gas is compressed at (T_c), releases heat (Q_c) to cold reservoir.
- Step 4: Adiabatic compression—gas is compressed without heat exchange, temperature rises from (T_c) back to (T_h).
- Total work done by the cycle equals the area inside the cycle on a (pV) diagram.
- Over one cycle, the system's internal energy returns to its original value ((\Delta E_{int} = 0))._
Formulas and Efficiency of the Carnot Engine
- Efficiency ((e)) of a Carnot engine: (e = 1 - \frac{T_c}{T_h}), where temperatures are in kelvin.
- Ratio of heats exchanged: (\frac{Q_c}{Q_h} = \frac{T_c}{T_h}).
- All reversible engines between the same two reservoirs have equal efficiency.
- Real engines are always less efficient than a Carnot engine due to irreversibilities.
Carnot Refrigerator and Heat Pump
- Carnot refrigerator: cycle runs in reverse, extracting heat (Q_c) from cold reservoir while work is done on the system.
- Coefficient of performance (refrigerator): (K_R = \frac{T_c}{T_h - T_c}).
- Coefficient of performance (heat pump): (K_P = \frac{T_h}{T_h - T_c}).
- Refrigerators are less effective as the outside temperature drops far below room temperature.
Carnot's Principle and the Second Law
- Carnot's Principle: No engine between two reservoirs is more efficient than a reversible engine.
- This principle is equivalent to the Kelvin and Clausius statements of the second law of thermodynamics.
- The Carnot engine sets a universal upper limit on efficiency, independent of working substance.
Key Terms & Definitions
- Carnot Cycle — the sequence of processes (two isothermal and two adiabatic) forming the most efficient possible heat engine cycle.
- Isothermal Process — thermodynamic process at constant temperature.
- Adiabatic Process — process with no heat exchange between system and surroundings.
- Carnot Efficiency — maximum possible efficiency: (e = 1 - \frac{T_c}{T_h}).
- Coefficient of Performance ((K_R, K_P)) — ratio measuring efficiency of refrigerators and heat pumps.
Action Items / Next Steps
- Practice problems: efficiency and coefficient of performance calculations for Carnot engines, refrigerators, and heat pumps.
- Review Check Your Understanding 4.3 and 4.4 to test your grasp of Carnot engine and refrigerator concepts.