Question

A refrigerator has a coefficient of performance of 5. If the temperature inside the freezer is $-20^{\circ}C$, the temperature of the surroundings to which it rejects heat is:

• A. $31^{\circ}C$
• B. $41^{\circ}C$
• C. $21^{\circ}C$
• D. $11^{\circ}C$

Hint:

Always convert Celsius to Kelvin before using thermodynamic formulas!

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
A refrigerator's efficiency is measured by its Coefficient of Performance (COP). It is the ratio of heat removed from the cold body to the work input required.
Step 2: Key Formula or Approach:
COP (\(\beta\)) \(= \frac{T_L}{T_H - T_L}\)
Where:
\(T_L\) = Temperature of cold reservoir (Kelvin).
\(T_H\) = Temperature of hot reservoir (Kelvin).
Step 3: Detailed Explanation:
Given:
COP = 5
\(T_L = -20^\circ\text{C} = 273 - 20 = 253\text{ K}\).

Substitute these into the COP formula:
\[ 5 = \frac{253}{T_H - 253} \]
Rearranging to solve for \(T_H\):
\[ 5(T_H - 253) = 253 \]
\[ 5T_H - 1265 = 253 \]
\[ 5T_H = 1518 \]
\[ T_H = \frac{1518}{5} = 303.6\text{ K} \]

Convert back to Celsius:
\[ \text{Temp in } ^\circ\text{C} = 303.6 - 273 = 30.6^\circ\text{C} \approx 31^\circ\text{C} \]
Step 4: Final Answer:
The temperature of the surroundings is \(31^\circ\text{C}\).