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Chapter 15: Problem 38

A heat engine uses the warm air at the ground as the hot reservoir and the cooler air at an altitude of several thousand meters as the cold reservoir. If the warm air is at \(37^{\circ} \mathrm{C}\) and the cold air is at $25^{\circ} \mathrm{C},$ what is the maximum possible efficiency for the engine?

Short Answer

Expert verified
Answer: The maximum possible efficiency for the engine is approximately 3.87%.

Step by step solution

01

Convert the given temperatures to Kelvin

To convert temperatures from degrees Celsius to Kelvin, we simply add 273.15. So, \(T_h = 37^{\circ} \mathrm{C} + 273.15 = 310.15\,K\) \(T_c = 25^{\circ} \mathrm{C} + 273.15 = 298.15\,K\)
02

Calculate the maximum possible efficiency

We can now use the Carnot efficiency formula with the converted temperature values: Efficiency = \(1 - \frac{298.15\,K}{310.15\,K}\) Calculate the efficiency: Efficiency = \(1 - \frac{298.15}{310.15} = 1 - 0.9613 \approx 0.0387\)
03

Express the efficiency as a percentage

To express the efficiency as a percentage, we multiply the result by 100: Efficiency = \(0.0387 \times 100 = 3.87\%\) So, the maximum possible efficiency for the engine is approximately \(3.87\%\).

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Most popular questions from this chapter

A town is considering using its lake as a source of power. The average temperature difference from the top to the bottom is \(15^{\circ} \mathrm{C},\) and the average surface temperature is \(22^{\circ} \mathrm{C} .\) (a) Assuming that the town can set up a reversible engine using the surface and bottom of the lake as heat reservoirs, what would be its efficiency? (b) If the town needs about \(1.0 \times 10^{8} \mathrm{W}\) of power to be supplied by the lake, how many \(\mathrm{m}^{3}\) of water does the heat engine use per second? (c) The surface area of the lake is $8.0 \times 10^{7} \mathrm{m}^{2}$ and the average incident intensity (over \(24 \mathrm{h})\) of the sunlight is \(200 \mathrm{W} / \mathrm{m}^{2} .\) Can the lake supply enough heat to meet the town's energy needs with this method?
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?
A monatomic ideal gas at \(27^{\circ} \mathrm{C}\) undergoes a constant volume process from \(A\) to \(B\) and a constant pressure process from \(B\) to \(C\) Find the total work done during these two processes.
A fish at a pressure of 1.1 atm has its swim bladder inflated to an initial volume of \(8.16 \mathrm{mL}\). If the fish starts swimming horizontally, its temperature increases from \(20.0^{\circ} \mathrm{C}\) to $22.0^{\circ} \mathrm{C}$ as a result of the exertion. (a) since the fish is still at the same pressure, how much work is done by the air in the swim bladder? [Hint: First find the new volume from the temperature change. \(]\) (b) How much heat is gained by the air in the swim bladder? Assume air to be a diatomic ideal gas. (c) If this quantity of heat is lost by the fish, by how much will its temperature decrease? The fish has a mass of \(5.00 \mathrm{g}\) and its specific heat is about $3.5 \mathrm{J} /\left(\mathrm{g} \cdot^{\circ} \mathrm{C}\right)$.
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.
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