Question

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

Short answer

The efficiency of the heat engine is approximately \(0.3085\) or \(30.85 \%\), and the work done per cycle is \(16.2 \, kJ\).

Step-by-step solution

Calculate the Efficiency

First, calculate the efficiency of the heat engine. This is done using the formula \(η = 1 - \frac{Q_{out}}{Q_{in}}\), where \(Q_{in} = 52.4 \, kJ\) is the heat absorbed and \(Q_{out} = 36.2 \, kJ\) is the heat exhausted. Substitute the given values to obtain the efficiency.

Calculate the Work Done

Next, calculate the work done by the heat engine per cycle. The formula is \(Work \, done = Q_{in} - Q_{out}\), where \(Q_{in} = 52.4 \, kJ\) and \(Q_{out} = 36.2 \, kJ\). Substitute the given values into the equation and calculate the work done.

Key concepts

Efficiency of Heat Engines

Understanding the efficiency of heat engines is essential in thermodynamics. Efficiency is a measure of how well a heat engine converts absorbed heat into useful work. The formula for efficiency \( \eta \) is given by:\[\eta = 1 - \frac{Q_{out}}{Q_{in}}\]- \(Q_{in}\) represents the heat absorbed by the engine, in this case, 52.4 kJ.
- \(Q_{out}\) stands for the heat that is not converted into work and is exhausted, here it is 36.2 kJ.
Plugging these values into the formula allows us to find the efficiency. This efficiency value is expressed as a percentage, showing what portion of the absorbed heat is turned into work. Remember, no heat engine can be 100% efficient due to inevitable losses to friction and other factors.

Thermodynamics

Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. Heat engines are a practical application of thermodynamics, converting heat into mechanical work.
Thermodynamics operates under four fundamental laws, but the first and second laws are particularly relevant to understanding heat engines.

• The First Law of Thermodynamics focuses on energy conservation, stating that energy cannot be created or destroyed, only converted.
  
• The Second Law introduces the concept of entropy, explaining why energy conversion processes are never perfectly efficient.
  

Understanding these principles helps explain why not all absorbed heat is converted into work in heat engines, aligning with the calculated efficiency.

Work Done by Heat Engines

The work done by a heat engine is the amount of energy converted from heat into mechanical work in each cycle. To find the work done, we use the relation:\[Work \ done = Q_{in} - Q_{out}\]Here, the absorbed heat \(Q_{in}\) is 52.4 kJ, and the exhausted heat \(Q_{out}\) is 36.2 kJ. By substituting these values into the equation, we can calculate the work done, which provides valuable insight into the engine's performance.
This equation highlights a fundamental aspect of heat engines: not all absorbed heat is usable; the difference between \(Q_{in}\) and \(Q_{out}\) gives us the exact amount of energy performing useful work. This understanding is crucial in designs and improvements of energy-efficient engines.