Chapter 21: Q. 47 (page 596)

A Carnot heat engine and an ordinary refrigerator with coefficient of performance $2.00$ operate between reservoirs at $350 K$ and $250 K$ . The work done by the Carnot heat engine drives the refrigerator. If the heat engine extracts $10.0 J$ of energy from the hot reservoir, how much energy does the refrigerator exhaust to the hot reservoir?

Short Answer

Expert verified

The refrigerator rejects $8.57 J$ thermal energy to the hot reservoir.

Step by step solution

Step : 1 Given Information

Temperatures of the hot and cold reservoirs between which Carnot heat engine operates are: $T_{H} = 350 K$ and $T_{C} = 250 K$ respectively. The amount of heat extracted from the hot reservoir (source) is $Q_{1} = 10 J$ . Let the amount of heat rejected to the cold reservoir (sink) is $Q_{2}$ and the amount of work done by the Carnot heat engine is $W_{o u t}$ . This heat engine drives a refrigerator whose coefficient of performance is $β = 2$ . Let the amount of heat extracted from the cold reservoir (source) is role="math" localid="1648621324594" $Q'_{2}$ , and the amount of heat rejected to the hot reservoir (sink) is $Q'_{1}$ .

Step : 2 Calculation

Formula used:

Efficiency of a Carnot heat engine, $= \frac{W_{o u t}}{Q_{1}} = \frac{Q_{1} - Q_{2}}{Q_{1}} = \frac{T_{H} - T_{C}}{T_{H}}$ ;Coefficient of performance of Carnot refrigerator, $β = \frac{Q'_{2}}{W_{i n}} = \frac{Q'_{2}}{Q'_{1} - Q'_{2}}$

Calculation:

Efficiency of the Carnot heat engine : $η = \frac{Q_{1} - Q_{2}}{Q_{1}} = \frac{T_{H} - T_{C}}{T_{H}}$

$\Rightarrow \frac{Q_{1} - Q_{2}}{Q_{1}} = \frac{350 - 250}{350} = \frac{100}{350} = \frac{2}{7}$

or, $\frac{10 - Q_{2}}{10} = \frac{2}{7} \Rightarrow Q_{2} = \frac{50}{7} J$

Step : 3 Calculation of Q'1

Thus, $W_{o u t} = Q_{1} - Q_{2} = 10 - \frac{50}{7} = \frac{20}{7} J .$ While amount of work input to the refrigerator $W_{i n} = {Q^{'}}_{1} - {Q^{'}}_{2} = W_{o u t} = \frac{20}{7} J \Rightarrow {Q^{'}}_{1} = \frac{20}{7} + {Q^{'}}_{2} .$ Using this relation between ${Q^{'}}_{1}$ and ${Q^{'}}_{2}$ in coefficient of performance, $β = \frac{{Q^{'}}_{2}}{W_{i n}} = \frac{{Q^{'}}_{2}}{{Q^{'}}_{1} - {Q^{'}}_{2}} = 2$