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Step-by-Step Solution
Step 1: Identify given information
• Coefficient of performance (COP) of the refrigerator is 5.
• The inside freezer temperature, denoted T2, is −20°C.
For thermodynamic calculations, we first convert this into the Kelvin scale:
$
T_2 = 273 - 20 = 253 \,\text{K}.
$
Step 2: Write down the formula for the coefficient of performance
The COP for a refrigerator can be expressed as:
$
\text{COP} = \frac{T_2}{T_1 - T_2}
$
where:
• T1 represents the higher temperature (the surroundings in Kelvin).
• T2 represents the lower temperature (inside the freezer in Kelvin).
Step 3: Substitute known values
Given COP = 5 and T2 = 253 K, we have:
$
5 = \frac{253}{T_1 - 253}.
$
Step 4: Solve for T1 (in Kelvin)
Rearrange the equation:
$
5 (T_1 - 253) = 253
$
$
5\,T_1 - 5 \times 253 = 253
$
$
5\,T_1 = 253 + 5 \times 253
$
$
5\,T_1 = 253 + 1265 = 1518
$
$
T_1 = \frac{1518}{5} = 303.6\text{ K}.
$
Step 5: Convert T1 to the Celsius scale
To convert from Kelvin to Celsius:
$
T_{1(\text{°C})} = T_1 - 273 = 303.6 - 273 \approx 30.6\,^\circ\text{C}.
$
This is approximately 31°C.
Step 6: State the final answer
Therefore, the temperature of the surroundings to which the refrigerator rejects heat is about 31°C.
