21.8
Efficiency of The Carnot Cycle
The hypothetical Carnot cycle consists of an ideal gas subjected to two isothermal and two adiabatic processes. Since the internal energy of an ideal…
In a Carnot cycle, the known quantities are the temperature of the hot reservoir, T-h, and the temperature of the cold reservoir, T-c.
The ideal gas absorbs heat Q-h during its isothermal expansion at T-h and rejects heat Q-c during its isothermal compression at T-c.
Since the internal energy of an ideal gas depends only on its temperature, it remains constant during these two steps. Using the first law of thermodynamics, the absorbed heat can be calculated. While Q-h depends on the gas’s volumes before and after the expansion, Q-c depends on the gas’s volumes before and after the contraction.
For the adiabatic processes in the Carnot cycle, the relationship between the temperature and volume of gases can be used. The four equations can be combined to relate the ratio of heat exchanged and the temperatures of the reservoirs.
When combined with the expression of the efficiency of a heat engine, it implies that the efficiency of a Carnot cycle depends only on the temperatures T-c and T-h.
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Q1: What are the four processes that make up the Carnot cycle?
The Carnot cycle consists of two isothermal and two adiabatic processes. During isothermal expansion at the hot reservoir temperature, the ideal gas absorbs heat. During isothermal compression at the cold reservoir temperature, it rejects heat. The two adiabatic processes connect these isothermal steps, with temperature and volume relationships governing the gas behavior throughout.
Q2: Why does the internal energy of an ideal gas remain constant during isothermal processes in the Carnot cycle?
Internal energy of an ideal gas depends only on temperature. Since isothermal processes occur at constant temperature, the internal energy remains unchanged during these steps. This allows the first law of thermodynamics to relate the absorbed or rejected heat directly to the work done by or on the gas.
Q3: How is the efficiency of the Carnot cycle calculated?
Carnot cycle efficiency is determined by combining equations relating heat exchanged, volumes, and temperatures across all four processes. The analysis reveals that efficiency depends only on the temperatures of the hot and cold reservoirs, expressed as a function of their temperature difference. This makes the Carnot cycle the most efficient theoretical heat engine operating between two fixed temperatures.
Q4: What is the relationship between Carnot cycle efficiency and temperature difference?
Carnot cycle efficiency increases as the temperature difference between the hot and cold reservoirs increases. The efficiency can approach unity only if the cold reservoir temperature approaches absolute zero, which is physically impossible. This demonstrates a fundamental limitation imposed by nature on heat engine performance.
Q5: How does the p-V diagram represent work done in the Carnot cycle?
The total work done by the Carnot cycle equals the area enclosed by the cycle on the pressure-volume diagram. This graphical representation shows the net work output as the difference between work done during expansion and work done during compression. The enclosed area directly quantifies the mechanical energy produced per cycle.
Q6: What role do adiabatic processes play in connecting the isothermal steps of the Carnot cycle?
Adiabatic processes connect the two isothermal steps by changing the gas temperature without heat exchange. The relationship between temperature and volume during adiabatic compression and expansion allows the gas to transition between the hot and cold reservoir temperatures. These processes are essential for completing the reversible cycle and achieving maximum theoretical efficiency.
Q7: Why is the total heat exchanged equal to the total work done in the Carnot cycle?
Since internal energy returns to its initial value after one complete Carnot cycle, the first law of thermodynamics requires that total heat absorbed minus total heat rejected equals total work done. The cycle's reversible nature ensures no energy is lost to irreversible processes, making this energy balance exact and fundamental to calculating cycle efficiency.