A Carnot engine is operated between two heat reservoirs at temperatures of 520 K and 300 K. (c) What is the thermal efficiency of the engine?

Verified step by step guidance

1. Identify the temperatures of the hot and cold reservoirs. In this case, the hot reservoir temperature (T_h) is 520 K, and the cold reservoir temperature (T_c) is 300 K.
2. Recall the formula for the thermal efficiency (\(\eta\)) of a Carnot engine, which is given by \(\eta = 1 - \frac{T_c}{T_h}\), where \(T_c\) and \(T_h\) are the absolute temperatures of the cold and hot reservoirs, respectively.
3. Substitute the given temperatures into the formula. Replace \(T_h\) with 520 K and \(T_c\) with 300 K.
4. Calculate the value inside the parentheses, \(\frac{T_c}{T_h}\), which represents the fraction of heat rejected to the cold reservoir.
5. Subtract the result from 1 to find the thermal efficiency of the Carnot engine.

Key Concepts

Carnot Engine

A Carnot engine is an idealized heat engine that operates on the reversible Carnot cycle. It is designed to achieve maximum efficiency by operating between two thermal reservoirs at different temperatures. The efficiency of a Carnot engine depends solely on the temperatures of the hot and cold reservoirs, making it a benchmark for real engines.

Thermal Efficiency

Thermal efficiency is a measure of how well an engine converts heat energy from a fuel into work. It is defined as the ratio of the work output of the engine to the heat input from the hot reservoir. For a Carnot engine, the thermal efficiency can be calculated using the formula: η = 1 - (T_c / T_h), where T_c is the absolute temperature of the cold reservoir and T_h is the absolute temperature of the hot reservoir.

Absolute Temperature

Absolute temperature is a temperature measurement on the Kelvin scale, where 0 K represents absolute zero, the point at which molecular motion ceases. In thermodynamics, using absolute temperatures is crucial because it ensures that calculations involving thermal energy and efficiency are accurate. The Kelvin scale is used in the Carnot efficiency formula to maintain consistency in temperature measurements.