Question

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

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

Answer 1

The Correct Option is C

To solve the problem, we need to use the formula for the Coefficient of Performance (COP) of a refrigerator, which is defined as:

\[\text{COP} = \frac{T_{\text{C}}}{T_{\text{H}} - T_{\text{C}}}\]

where:

• T_{\text{C}} is the absolute temperature inside the freezer (cold reservoir).
• T_{\text{H}} is the absolute temperature of the surroundings (hot reservoir).

Given:

• COP = 5
• Temperature inside freezer T_{\text{C}} = -20^{\circ}C

First, convert the temperature inside the freezer from Celsius to Kelvin:

T_{\text{C}} = -20^{\circ}C + 273 = 253 \, \text{K}

Substitute the values into the formula for COP:

5 = \frac{253}{T_{\text{H}} - 253}

Solve for T_{\text{H}}:

5(T_{\text{H}} - 253) = 253

5T_{\text{H}} - 1265 = 253

5T_{\text{H}} = 1518

T_{\text{H}} = \frac{1518}{5} = 303.6 \, \text{K}

Convert the temperature of the surroundings from Kelvin back to Celsius:

T_{\text{H}} = 303.6 \, \text{K} - 273 = 30.6^{\circ}C \approx 31^{\circ}C

Therefore, the temperature of the surroundings to which the refrigerator rejects heat is approximately 31^{\circ}C. This matches with the correct answer: 31^{\circ}C.