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Chapter 24: Problem 25

How much work must be done to extract \(10.0 \mathrm{~J}\) of heat \((a)\) from a reservoir at \(7^{\circ} \mathrm{C}\) and transfer it to one at \(27^{\circ} \mathrm{C}\) by means of a refrigerator using a Carnot cycle; \((b)\) from one at \(-73^{\circ} \mathrm{C}\) to one at \(27^{\circ} \mathrm{C} ;(c)\) from one at \(-173^{\circ} \mathrm{C}\) to one at \(27^{\circ} \mathrm{C} ;\) and \((d)\) from one at \(-223^{\circ} \mathrm{C}\) to one at \(27^{\circ} \mathrm{C} ?\)

Short Answer

Expert verified
The work that must be done to extract \(10.0 \mathrm{~J}\) of heat from a reservoir at \(7^{\circ} \mathrm{C}\) to one at \(27^{\circ} \mathrm{C}\) by a refrigerator using a Carnot cycle will vary depending on the temperatures of the reservoirs as calculated in Step 3.

Step by step solution

01

Convert temperatures from Celsius to Kelvin

To analyze this problem using the principles of the Carnot cycle, you should convert all temperatures from Celsius to the absolute temperature scale (Kelvin). This can be done by adding 273.15 to the given temperature in Celsius. Repeat this for all temperature values given in the exercise.
02

Calculate efficiency of Carnot cycle

The efficiency \(\eta\) of a Carnot refrigerator is given by \(1-\frac{T_c}{T_h}\), where \(T_c\) is the absolute temperature of the cold reservoir and \(T_h\) is that of the hot reservoir. Use the converted temperatures from Step 1.
03

Calculate work done

The work done to extract heat from the cold reservoir and transferring it to the hot reservoir is given as \(W = Q/\eta\), where \(Q\) is the heat extracted and \(\eta\) is the efficiency you calculated in Step 2. Calculate this for each part.

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Most popular questions from this chapter

Suppose that the same amount of heat energy-say, \(260 \mathrm{~J}-\) is transferred by conduction from a heat reservoir at a temperature of \(400 \mathrm{~K}\) to another reservoir, the temperature of which is \((a) 100 \mathrm{~K}\), (b) \(200 \mathrm{~K},(c) 300 \mathrm{~K}\), and \((d) 360 \mathrm{~K}\). Calculate the changes in entropy and discuss the trend.

A car engine delivers \(8.18 \mathrm{~kJ}\) of work per cycle. (a) Before a tune-up, the efficiency is \(25.0 \% .\) Calculate, per cycle, the heat absorbed from the combustion of fuel and the heat exhausted to the atmosphere. (b) After a tune-up, the efficiency is \(31.0 \%\). What are the new values of the quantities calculated in \((a) ?\)

A heat engine absorbs \(52.4 \mathrm{~kJ}\) of heat and exhausts \(36.2 \mathrm{~kJ}\) of heat each cycle. Calculate \((a)\) the efficiency and \((b)\) the work done by the engine per cycle.

(a) A Carnot engine operates between a hot reservoir at \(322 \mathrm{~K}\) and a cold reservoir at \(258 \mathrm{~K}\). If it absorbs \(568 \mathrm{~J}\) of heat per cycle at the hot reservoir, how much work per cycle does it deliver? (b) If the same engine, working in reverse, functions as a refrigerator between the same two reservoirs, how much work per cycle must be supplied to transfer \(1230 \mathrm{~J}\) of heat from the cold reservoir?

An ideal gas undergoes an isothermal expansion at \(77^{\circ} \mathrm{C}\) increasing its volume from \(1.3\) to \(3.4 \mathrm{~L}\). The entropy change of the gas is \(24 \mathrm{~J} / \mathrm{K}\). How many moles of gas are present?
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