Figure 20-32 represents a Carnot engine that works between...

Figure $20-32$ represents a Carnot engine that works between temperatures $T_{1}=400 \mathrm{~K}$ and $T_{2}=150 \mathrm{~K}$ and drives a Carnot refrigerator that works between temperatures $\quad T_{3}=325 \mathrm{~K} \quad$ and $T_{4}=225 \mathrm{~K}$. What is the ratio $Q_{3} / Q_{1} ?$

Key Concepts

Carnot Cycle

The Carnot cycle is an idealized thermodynamic cycle that sets the upper limit for efficiency in any heat engine or refrigerator operating between two temperatures. It consists of two isothermal processes and two adiabatic processes, providing a framework where the relationship between heat exchanges and reservoir temperatures is exact and reversible.

Reversible Processes

Reversible processes are key to the Carnot cycle, as they occur without entropy production. This property allows for a strict proportionality between the amount of heat transferred and the temperature at which the transfer occurs, captured in the relation Q/T being constant for the cycle. This characteristic is essential for comparing different parts of the cycle, such as in heat engines and refrigerators.

Heat Transfer Relations in Carnot Systems

In reversible Carnot systems, the heat transferred from the hot reservoir and to the cold reservoir is directly proportional to their absolute temperatures. This principle (often expressed as Q_hot/T_hot = Q_cold/T_cold) is critical in determining efficiencies and performance metrics, as it connects the thermal energy flows with the operating temperatures.

Carnot Engine Efficiency

The efficiency of a Carnot engine is determined solely by the temperatures of the hot and cold reservoirs via the equation ? = 1 - (T_cold/T_hot). This efficiency represents the theoretical maximum conversion of heat into work and is foundational in understanding the limits of real heat engine performance.

Carnot Refrigerator and Coefficient of Performance (COP)

A Carnot refrigerator is essentially a reversed Carnot engine that uses work to remove heat from a cold space and expel it to a warmer space. Its performance is measured by the coefficient of performance (COP), which is the ratio of the heat extracted from the cold reservoir to the work input. This relationship helps in comparing the effectiveness of refrigeration cycles operating between different temperatures.

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