3.3
Carnot Cycle and Efficiency
The second law of thermodynamics states that no engine can be 100% efficient.
Sadi Carnot explored the maximum theoretical efficiency allowed by this law by constructing the Carnot cycle of an ideal heat engine.
The cycle begins with an ideal gas absorbing heat from a hot reservoir at temperature Th and undergoing isothermal expansion while doing work.
The gas is then isolated from the hot reservoir and undergoes reversible adiabatic expansion. As the gas does work, its temperature drops to Tc.
When the gas contacts the cold reservoir, it undergoes isothermal compression. Heat is released to the cold reservoir as work is done on the gas. Finally, removing the cold reservoir causes adiabatic compression. As work continues on the gas, it returns to its initial state.
Because the system returns to its initial state, its internal energy remains constant, and the net work produced equals the net heat transferred during the cycle.
The efficiency of the engine is defined as the ratio of work performed to heat absorbed from the hot reservoir. Efficiency can also be expressed solely in terms of the temperatures of the cold and hot reservoirs.
The Second Law of Thermodynamics asserts that it's impossible for any heat engine to achieve 100% efficiency. While contemplating the maximum possible efficiency, Nicolas Sadi Carnot conceptualized an ideal heat engine. This engine gets its energy from a high-temperature reservoir. It then performs some work and releases the remaining energy into a low-temperature reservoir.
The Carnot cycle, named after Sadi Carnot, is fully reversible. The cycle consists of four distinct stages. In the first stage, the gas undergoes isothermal expansion. During this process, the entropy of the system increases by qh/Th.
In the second stage, the gas undergoes adiabatic expansion. Because there is no heat transfer, the entropy remains constant during this step.
The third stage is an isothermal compression. In this stage, its entropy decreases by |qc|/Tc.
In the final stage, the gas undergoes adiabatic compression. At the end of this step, the system returns to its original state, completing the cycle. The net work produced by the engine equals the area enclosed by the cycle on a pressure–volume diagram.
Because the Carnot cycle is reversible, the total change in entropy over one complete cycle is zero. This leads to the relation qh /Th − |qc| /Tc = 0. Although entropy changes during individual steps, these changes cancel out over the full cycle.
Using energy conservation, efficiency can be written in terms of the heat rejected to the cold reservoir. η = 1 − |qc|\qh
Since the magnitude of rejected heat can never exceed the absorbed heat, efficiency always lies between zero and one.
The second law of thermodynamics states that no engine can be 100% efficient.
Sadi Carnot explored the maximum theoretical efficiency allowed by this law by constructing the Carnot cycle of an ideal heat engine.
The cycle begins with an ideal gas absorbing heat from a hot reservoir at temperature Th and undergoing isothermal expansion while doing work.
The gas is then isolated from the hot reservoir and undergoes reversible adiabatic expansion. As the gas does work, its temperature drops to Tc.
When the gas contacts the cold reservoir, it undergoes isothermal compression. Heat is released to the cold reservoir as work is done on the gas. Finally, removing the cold reservoir causes adiabatic compression. As work continues on the gas, it returns to its initial state.
Because the system returns to its initial state, its internal energy remains constant, and the net work produced equals the net heat transferred during the cycle.
The efficiency of the engine is defined as the ratio of work performed to heat absorbed from the hot reservoir. Efficiency can also be expressed solely in terms of the temperatures of the cold and hot reservoirs.
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