A heat engine absorbs 52.4 kJ of heat and exhausts 36.2 kJ of heat each cycle. Calculate (a) the efficiency and (b) the work done by the engine per cycle.

To tackle the problem of calculating the efficiency and work done by a heat engine, we can use some fundamental principles from thermodynamics. Let's break it down step by step.

Understanding the Heat Engine

A heat engine operates by absorbing heat from a high-temperature source, converting part of that heat into work, and then expelling the remaining heat to a low-temperature sink. The efficiency of a heat engine is a measure of how well it converts the absorbed heat into useful work.

Given Data

• Heat absorbed (Q_in) = 52.4 kJ
• Heat exhausted (Q_out) = 36.2 kJ

Calculating Work Done

The work done (W) by the engine in each cycle can be calculated using the first law of thermodynamics, which states that the work done by the engine is equal to the heat absorbed minus the heat exhausted:

W = Q_in - Q_out

Substituting the values:

W = 52.4 kJ - 36.2 kJ

W = 16.2 kJ

Determining Efficiency

The efficiency (η) of a heat engine is defined as the ratio of the work done by the engine to the heat absorbed from the source. It can be expressed mathematically as:

η = W / Q_in

Now, substituting the values we have:

η = 16.2 kJ / 52.4 kJ

Calculating this gives:

η = 0.3094

To express efficiency as a percentage, we multiply by 100:

η = 0.3094 × 100 = 30.94%

Summary of Results

In summary, for the heat engine in question:

• The work done by the engine per cycle is 16.2 kJ.
• The efficiency of the engine is approximately 30.94%.

This means that about 30.94% of the heat energy absorbed is converted into useful work, while the rest is expelled as waste heat. Understanding these concepts is crucial for analyzing the performance of heat engines in various applications, from power plants to automobiles.