The Carnot Cycle and Heat Engine Efficiency
Carnot cycle
Definition
Carnot cycle
The Carnot cycle represents an idealized model of a heat engine that achieves the maximum possible efficiency for given reservoir temperatures.
The Carnot cycle consists of four reversible processes:
- Isothermal Expansion (Process 1-2)
- Gas expands at constant temperature $T_h$
- Absorbs heat $Q_h$ from hot reservoir
- Work is done by the gas
- Adiabatic Expansion (Process 2-3)
- No heat exchange with surroundings
- Temperature drops from $T_h$ to $T_c$
- Work is done by the gas
- Isothermal Compression (Process 3-4)
- Gas compressed at constant temperature $T_c$
- Releases heat $Q_c$ to cold reservoir
- Work is done on the gas
- Adiabatic Compression (Process 4-1)
- No heat exchange with surroundings
- Temperature increases from $T_c$ to $T_h$
- Work is done on the gas
The Adiabatic Equation for a Monatomic Ideal Gas
- During an adiabatic process, the relationship between pressure and volume for a monatomic ideal gas is given by: $$PV^{\frac{5}{3}} = \text{constant}$$
- This equation arises from combining the ideal gas law with the first law of thermodynamics under adiabatic conditions.
- Suppose a gas undergoes adiabatic expansion from an initial state $P_1$, $V_1$ to a final state $P_2$, $V_2$.
- The adiabatic equation ensures: $$P_1V_1^{\frac{5}{3}} = P_2V_2^{\frac{5}{3}}$$
