# Solution: Thermal Efficiency of a Heat Engine ## Step 1: Write Given Data - Heat supplied from the source (\( Q_1 \)): **900 J** - Heat rejected to the sink (\( Q_2 \)): **630 J** ## Step 2: Recall the Formula for Thermal Efficiency The thermal efficiency (\( \eta \)) of a heat engine is given by: \[ \eta = \frac{\text{Net work output}}{\text{Heat supplied}} = \frac{Q_1 - Q_2}{Q_1} \] ## Step 3: Substitute the Values \[ \eta = \frac{900\,\text{J} - 630\,\text{J}}{900\,\text{J}} \] \[ \eta = \frac{270\,\text{J}}{900\,\text{J}} \] ## Step 4: Calculate the Efficiency \[ \eta = .3 \] To express as a percentage: \[ \text{Thermal Efficiency} = .3 \times 100\% = 30\% \] ## Step 5: Final Answer **The thermal efficiency of the engine is \(\boxed{30\%}\).**

# Explanation of Thermal Efficiency Calculation ## Concept Used The concept applied in this problem is the **thermal efficiency** of a heat engine, which measures how effectively the engine converts heat energy into work. ### Key Definitions - **Heat Input (\( Q_1 \))**: The total heat energy absorbed by the engine from the heat source. - **Heat Output (\( Q_2 \))**: The heat energy expelled or rejected by the engine to the heat sink. - **Net Work Output**: The difference between the heat absorbed and the heat rejected, which is the work output of the engine. ## Formula for Thermal Efficiency The thermal efficiency (\( \eta \)) is defined as: \[ \eta = \frac{\text{Net work output}}{\text{Heat supplied}} = \frac{Q_1 - Q_2}{Q_1} \] This formula represents the ratio of useful work produced by the engine to the heat energy provided to it. ### Steps in the Calculation 1. **Identify Given Values**: Recognize the values for heat input and heat output. 2. **Apply the Efficiency Formula**: Substitute the values into the formula to find the net work output. 3. **Calculate the Efficiency**: Perform the arithmetic to determine the efficiency and convert it to a percentage. ## Importance of the Concept Understanding thermal efficiency is crucial for evaluating the performance of engines and optimizing their design for better energy conversion.