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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

A \(0.50-\mathrm{kg}\) block of iron $[c=0.44 \mathrm{kJ} /(\mathrm{kg} \cdot \mathrm{K})]\( at \)20.0^{\circ} \mathrm{C}$ is in contact with a \(0.50-\mathrm{kg}\) block of aluminum \([c=\) $0.900 \mathrm{kJ} /(\mathrm{kg} \cdot \mathrm{K})]$ at a temperature of \(20.0^{\circ} \mathrm{C} .\) The system is completely isolated from the rest of the universe. Suppose heat flows from the iron into the aluminum until the temperature of the aluminum is \(22.0^{\circ} \mathrm{C}\) (a) From the first law, calculate the final temperature of the iron. (b) Estimate the entropy change of the system. (c) Explain how the result of part (b) shows that this process is impossible. [Hint: since the system is isolated, $\left.\Delta S_{\text {System }}=\Delta S_{\text {Universe }} .\right]$
A balloon contains \(160 \mathrm{L}\) of nitrogen gas at \(25^{\circ} \mathrm{C}\) and 1.0 atm. How much energy must be added to raise the temperature of the nitrogen to \(45^{\circ} \mathrm{C}\) while allowing the balloon to expand at atmospheric pressure?
Suppose 1.00 mol of oxygen is heated at constant pressure of 1.00 atm from \(10.0^{\circ} \mathrm{C}\) to \(25.0^{\circ} \mathrm{C} .\) (a) How much heat is absorbed by the gas? (b) Using the ideal gas law, calculate the change of volume of the gas in this process. (c) What is the work done by the gas during this expansion? (d) From the first law, calculate the change of internal energy of the gas in this process.
(a) What is the entropy change of 1.00 mol of \(\mathrm{H}_{2} \mathrm{O}\) when it changes from ice to water at \(0.0^{\circ} \mathrm{C} ?\) (b) If the ice is in contact with an environment at a temperature of \(10.0^{\circ} \mathrm{C}\) what is the entropy change of the universe when the ice melts?
(a) Calculate the efficiency of a reversible engine that operates between the temperatures \(600.0^{\circ} \mathrm{C}\) and \(300.0^{\circ} \mathrm{C} .\) (b) If the engine absorbs \(420.0 \mathrm{kJ}\) of heat from the hot reservoir, how much does it exhaust to the cold reservoir?
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