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NEET PHYSICSMedium

The temperature inside a refrigerator is t2Ct_2^\circ\text{C} and the room temperature is t1Ct_1^\circ\text{C}. The amount of heat delivered to the room for each joule of electrical energy consumed ideally will be:

A

t1t1t2\frac{t_1}{t_1-t_2}

B

t1+273t1t2\frac{t_1+273}{t_1-t_2}

C

t2+273t1+t2\frac{t_2+273}{t_1+t_2}

D

t1+t2t1+273\frac{t_1+t_2}{t_1+273}

Step-by-Step Solution

The coefficient of performance (β\beta) of a refrigerator is given by β=Q2W=T2T1T2\beta = \frac{Q_2}{W} = \frac{T_2}{T_1 - T_2}, where Q2Q_2 is the heat extracted from the refrigerator (cold reservoir), WW is the work done (electrical energy consumed), T1T_1 is the absolute temperature of the room (hot reservoir), and T2T_2 is the absolute temperature inside the refrigerator. The total heat delivered to the room is Q1=Q2+WQ_1 = Q_2 + W. We need to find the amount of heat delivered per joule of work done, which is Q1W\frac{Q_1}{W}. Q1W=Q2+WW=Q2W+1=β+1\frac{Q_1}{W} = \frac{Q_2 + W}{W} = \frac{Q_2}{W} + 1 = \beta + 1 Substituting the expression for β\beta: Q1W=T2T1T2+1=T2+T1T2T1T2=T1T1T2\frac{Q_1}{W} = \frac{T_2}{T_1 - T_2} + 1 = \frac{T_2 + T_1 - T_2}{T_1 - T_2} = \frac{T_1}{T_1 - T_2} Converting the given Celsius temperatures to Kelvin: T1=t1+273T_1 = t_1 + 273 T2=t2+273T_2 = t_2 + 273 Now, substitute the temperatures into the ratio: Q1W=t1+273(t1+273)(t2+273)=t1+273t1t2\frac{Q_1}{W} = \frac{t_1 + 273}{(t_1 + 273) - (t_2 + 273)} = \frac{t_1 + 273}{t_1 - t_2} For each joule of electrical energy consumed (W=1 JW = 1\text{ J}), the heat delivered is t1+273t1t2\frac{t_1 + 273}{t_1 - t_2}.

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