A Carnot engine has an efficiency of 0.700, and the...

A Carnot engine has an efficiency of $0.700$, and the temperature of its cold reservoir is $378 \mathrm{~K}$. (a) Determine the temperature of its hot reservoir. (b) If $5230 \mathrm{~J}$ of heat is rejected to the cold reservoir, what amount of heat is put into the engine?

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Key Concepts

Thermal Reservoirs

Thermal reservoirs are large systems that maintain a constant temperature while exchanging heat with a heat engine. One acts as the hot source that supplies heat and the other as the cold sink where excess heat is rejected. These reservoirs are central to the operation of idealized engines like the Carnot engine, as they define the fundamental temperature limits for the conversion of heat into work.

Energy Conservation in Heat Engines

The principle of energy conservation in heat engines, also known as the first law of thermodynamics, states that the total energy input as heat is partitioned between work output and the waste heat expelled to the cold reservoir. This balance, expressed as the input heat equaling the sum of work done and rejected heat, is critical for analyzing and calculating energy flows within heat cycles.

Carnot Efficiency

Carnot efficiency represents the maximum possible efficiency that any heat engine operating between two fixed temperatures can achieve. It is defined by the formula efficiency = 1 - (T_cold/T_hot), where the temperatures are measured on an absolute scale. This concept sets an upper bound on engine performance and highlights the limitations imposed by the second law of thermodynamics.

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