The coefficient of performance of a refrigerator is $5.$ If the temperature inside freezer is $-20^o C,$ the temperature of the surroundings to which it rejects heat is ........ $^oC$
AIPMT 2015, Medium
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The coefficient of performance of a refrigerator is
$\alpha = \frac{{{T_2}}}{{{T_1} - {T_2}}}$
where $T_1$ and $T_2$ are the temperatures of hot and cold reservoirs $(in\,kelvin)$ respectively.
$Here,\alpha = 5,{T_2} = - {20^ \circ }C = - 20 + 273\,K = 253\,K$
${T_1} = ?$
$\therefore \,\,5 = \frac{{253K}}{{{T_1} - 253\,K}}$
$5{T_1} - 5\left( {253\,K} \right) = 253\,K$
$5{T_1} = 253\,K + 5\left( {253\,K} \right) = 6\left( {253\,K} \right)$
${T_1} = \frac{6}{5}\left( {253K} \right) = 303.6\,K = 303.6 - 273$
$ = {30.6^ \circ }C \approx {31^ \circ }C$
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