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Chapter 15: Problem 27

In a certain steam engine, the boiler temperature is \(127^{\circ} \mathrm{C}\) and the cold reservoir temperature is \(27^{\circ} \mathrm{C} .\) While this engine does \(8.34 \mathrm{kJ}\) of work, what minimum amount of heat must be discharged into the cold reservoir?

Short Answer

Expert verified
Answer: The minimum amount of heat that must be discharged into the cold reservoir is \(25.02\,\mathrm{kJ}\).

Step by step solution

01

Converting temperatures to Kelvin

We need to convert the given temperatures from Celsius to Kelvin, as we will be working with the absolute temperature in our calculations. To do this, we add 273.15 to the Celsius temperature: $$ T_{h} = 127^{\circ} \mathrm{C} + 273.15 = 400.15\,\mathrm{K} $$ $$ T_{c} = 27^{\circ} \mathrm{C} + 273.15 = 300.15\,\mathrm{K} $$
02

Calculating Carnot efficiency

Now we will calculate the Carnot efficiency using the formula \(1-\frac{T_{c}}{T_{h}}\): $$ \eta_{Carnot} = 1-\frac{T_{c}}{T_{h}} = 1-\frac{300.15}{400.15} = 1-0.75 = 0.25 $$
03

Finding the heat absorbed from the hot reservoir

Next, we will find the heat absorbed from the hot reservoir using the relationship between work done (W), heat absorbed (Q) and efficiency: $$ \eta_{Carnot} = \frac{W}{Q} \implies Q = \frac{W}{\eta_{Carnot}} $$ Plugging in the values, $$ Q = \frac{8.34\,\mathrm{kJ}}{0.25} = 33.36\,\mathrm{kJ} $$
04

Calculating the heat discharged in the cold reservoir

Finally, we can find the heat discharged in the cold reservoir using the conservation of energy principle. The heat absorbed from the hot reservoir minus the work done equals the heat discharged to the cold reservoir: $$ Q_{c} = Q - W $$ Plugging in the values, $$ Q_{c} = 33.36\,\mathrm{kJ} - 8.34\,\mathrm{kJ} = 25.02\,\mathrm{kJ} $$ Hence, the minimum amount of heat that must be discharged into the cold reservoir is \(25.02\,\mathrm{kJ}\).

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Most popular questions from this chapter

An ideal gas is heated at a constant pressure of $2.0 \times 10^{5} \mathrm{Pa}\( from a temperature of \)-73^{\circ} \mathrm{C}$ to a temperature of \(+27^{\circ} \mathrm{C}\). The initial volume of the gas is $0.10 \mathrm{m}^{3} .\( The heat energy supplied to the gas in this process is \)25 \mathrm{kJ} .$ What is the increase in internal energy of the gas?
The internal energy of a system increases by 400 J while \(500 \mathrm{J}\) of work are performed on it. What was the heat flow into or out of the system?
Consider a heat engine that is not reversible. The engine uses $1.000 \mathrm{mol}\( of a diatomic ideal gas. In the first step \)(\mathrm{A})$ there is a constant temperature expansion while in contact with a warm reservoir at \(373 \mathrm{K}\) from \(P_{1}=1.55 \times 10^{5} \mathrm{Pa}\) and $V_{1}=2.00 \times 10^{-2} \mathrm{m}^{3}$ to \(P_{2}=1.24 \times 10^{5} \mathrm{Pa}\) and $V_{2}=2.50 \times 10^{-2} \mathrm{m}^{3} .$ Then (B) a heat reservoir at the cooler temperature of \(273 \mathrm{K}\) is used to cool the gas at constant volume to \(273 \mathrm{K}\) from \(P_{2}\) to $P_{3}=0.91 \times 10^{5} \mathrm{Pa} .$ This is followed by (C) a constant temperature compression while still in contact with the cold reservoir at \(273 \mathrm{K}\) from \(P_{3}, V_{2}\) to \(P_{4}=1.01 \times 10^{5} \mathrm{Pa}, V_{1} .\) The final step (D) is heating the gas at constant volume from \(273 \mathrm{K}\) to \(373 \mathrm{K}\) by being in contact with the warm reservoir again, to return from \(P_{4}, V_{1}\) to \(P_{1}, V_{1} .\) Find the change in entropy of the cold reservoir in step \(\mathrm{B}\). Remember that the gas is always in contact with the cold reservoir. (b) What is the change in entropy of the hot reservoir in step D? (c) Using this information, find the change in entropy of the total system of gas plus reservoirs during the whole cycle.
An ideal gas is in contact with a heat reservoir so that it remains at a constant temperature of \(300.0 \mathrm{K}\). The gas is compressed from a volume of \(24.0 \mathrm{L}\) to a volume of 14.0 L. During the process, the mechanical device pushing the piston to compress the gas is found to expend \(5.00 \mathrm{kJ}\) of energy. How much heat flows between the heat reservoir and the gas and in what direction does the heat flow occur?
A reversible engine with an efficiency of \(30.0 \%\) has $T_{\mathrm{C}}=310.0 \mathrm{K} .\( (a) What is \)T_{\mathrm{H}} ?$ (b) How much heat is exhausted for every \(0.100 \mathrm{kJ}\) of work done?
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