Let's break down the question from the image: **Question:** A refrigerator operating between temperature limits of 0°C and 30°C, removing 50 kJ of energy from lower temperature and rejecting 65 kJ of heat to the higher temperature. Calculate the energy efficiency of the cycle. --- ### Step-by-Step Solution #### 1. **Energy Efficiency of Refrigerator (COP)** The coefficient of performance (COP) for a refrigerator is given by: \[ \text{COP}_{\text{ref}} = \frac{Q_L}{W} \] where: - \( Q_L \) = heat removed from the cold reservoir (0°C), here \( Q_L = 50 \) kJ - \( W \) = work input #### 2. **Work Input Calculation** The work input \( W \) is the difference between the heat rejected to the hot reservoir (\( Q_H \)) and the heat absorbed from the cold reservoir (\( Q_L \)): \[ W = Q_H - Q_L = 65\,\text{kJ} - 50\,\text{kJ} = 15\,\text{kJ} \] #### 3. **Calculate COP** \[ \text{COP}_{\text{ref}} = \frac{Q_L}{W} = \frac{50}{15} = 3.33 \] #### 4. **Energy Efficiency of Cycle** Energy efficiency (also called thermal efficiency) is: \[ \text{Efficiency} = \frac{W}{Q_H} = \frac{15}{65} \approx 0.231 \] #### 5. **Check the Options** But the options provided are: - 0.678 - 0.459 - 0.549 - 0.647 Let's check if the question wants the **ratio of Q_L to Q_H**: \[ \text{Ratio} = \frac{Q_L}{Q_H} = \frac{50}{65} \approx 0.769 \] But none of the options match this. Let's check if they want the ratio of \( Q_L \) to the total input: \[ \frac{Q_L}{Q_L + W} = \frac{50}{50+15} = \frac{50}{65} = 0.769 \] But the closest match to the numbers is if the question is asking for the **energy efficiency** as \( \frac{W}{Q_H} \): \[ \frac{W}{Q_H} = \frac{15}{65} \approx 0.231 \] None of the answers match exactly. Let's double-check **efficiency as per the options**: **If asking for energy efficiency (for a refrigerator, sometimes defined as the ratio of energy removed to energy supplied):** \[ \text{Efficiency} = \frac{Q_L}{Q_H} = \frac{50}{65} \approx 0.769 \] Still not matching. **If asking for the work ratio:** \[ \frac{W}{Q_L} = \frac{15}{50} = 0.3 \] No match. --- ## Most Likely Answer Based on the numbers, the **closest calculation to the provided options** is not matching any standard formula. Please verify if there is a typo or mismatch in the question or the options. **For a refrigerator, standard energy efficiency is not typically calculated in the format of the provided options.** --- ### **Summary Table** | Quantity | Formula | Value | |-------------------------|------------------------|----------| | Work \( W \) | \( Q_H - Q_L \) | 15 kJ | | COP | \( Q_L/W \) | 3.33 | | Energy Efficiency | \( W/Q_H \) | 0.231 | | Energy Removed/Heat Out | \( Q_L/Q_H \) | 0.769 | --- If you can clarify whether the question is about **COP, energy efficiency**, or **ratio** (and which ratio), I can help further. Based on standard definitions, your answer is **0.231** (which is not listed), so there may be an issue with the options provided. If you want to select the closest answer, **0.459** is the nearest to \( \frac{Q_L}{Q_H+Q_L} \), but that's not a standard efficiency definition. **Let me know if you want more details or another approach!**