A Carnot engine is operated between two heat reservoirs at temperatures of 520 K and 300 K. (b) How much mechanical work is performed by the engine during each cycle?

Verified step by step guidance

1. Identify the temperatures of the hot and cold reservoirs. The hot reservoir temperature (Th) is 520 K, and the cold reservoir temperature (Tc) is 300 K.
2. Calculate the efficiency of the Carnot engine using the formula: \( \eta = 1 - \frac{T_c}{T_h} \), where \( \eta \) is the efficiency, \( T_c \) is the temperature of the cold reservoir, and \( T_h \) is the temperature of the hot reservoir.
3. Determine the amount of heat absorbed from the hot reservoir during each cycle. This value is typically given or can be calculated based on other provided data such as the heat rejected or the energy input.
4. Calculate the work done by the engine using the efficiency and the heat absorbed. Use the formula: \( W = \eta \times Q_h \), where \( W \) is the work done, \( \eta \) is the efficiency, and \( Q_h \) is the heat absorbed from the hot reservoir.
5. Interpret the result as the mechanical work performed by the engine during each cycle, which is the product of the efficiency and the heat absorbed from the hot reservoir.

Key Concepts

Carnot Engine

A Carnot engine is an idealized heat engine that operates between two thermal reservoirs, achieving maximum efficiency as dictated by the second law of thermodynamics. It is defined by its two temperatures: the hot reservoir (T_H) and the cold reservoir (T_C). The efficiency of a Carnot engine is given by the formula η = 1 - (T_C/T_H), where temperatures are in Kelvin.

Thermal Efficiency

Thermal efficiency is a measure of how well an engine converts heat energy from a fuel into work. For a Carnot engine, this efficiency is determined solely by the temperatures of the heat reservoirs. The higher the temperature difference between the hot and cold reservoirs, the greater the efficiency, which is crucial for calculating the work output of the engine.

Work Output

The work output of a heat engine is the useful energy produced during a cycle, calculated by multiplying the heat absorbed from the hot reservoir by the engine's efficiency. In the case of a Carnot engine, the work done can be expressed as W = Q_H * η, where Q_H is the heat absorbed from the hot reservoir. This relationship is essential for determining the mechanical work performed by the engine.