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Chapter 20: Problem 43

A proposal is submitted for a novel engine that will operate between \(400 . \mathrm{K}\) and \(300 . \mathrm{K}\) a) What is the theoretical maximum efficiency of the engine? b) What is the total entropy change per cycle if the engine operates at maximum efficiency?

Short Answer

Expert verified
Answer: The theoretical maximum efficiency of the engine is 25%, and the total entropy change per cycle if the engine operates at maximum efficiency is 0.

Step by step solution

01

Calculate the theoretical maximum efficiency of the engine

For an engine operating between two temperatures, the theoretical maximum efficiency is given by the Carnot efficiency formula: Efficiency = \(1 - \frac{T_{cold}}{T_{hot}}\) Where \(T_{cold}\) and \(T_{hot}\) are the cold and hot reservoir temperatures, respectively. In this case, we have: \(T_{cold} = 300 \,\mathrm{K}\) and \(T_{hot} = 400 \,\mathrm{K}\) Now we will insert these values into the Carnot efficiency formula: Efficiency = \(1 - \frac{300}{400}\)
02

Simplify the expression

We will simplify the expression to find the efficiency: Efficiency = \(1 - \frac{300}{400} = 1 - \frac{3}{4} = \frac{1}{4}\) So, the theoretical maximum efficiency of the engine is \(\frac{1}{4}\) or 25%.
03

Calculate the total entropy change per cycle

When the engine operates at its maximum efficiency (Carnot efficiency), we know that the total entropy change per cycle is zero: Total entropy change per cycle = 0
04

Write down the answers

a) The theoretical maximum efficiency of the engine is 25%. b) The total entropy change per cycle if the engine operates at maximum efficiency is 0.

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Most popular questions from this chapter

Suppose an atom of volume \(V_{\mathrm{A}}\) is inside a container of volume \(V\). The atom can occupy any position within this volume. For this simple model, the number of states available to the atom is given by \(V / V_{A}\). Now suppose the same atom is inside a container of volume \(2 V .\) What will be the change in entropy?

A certain refrigerator is rated as being \(32.0 \%\) as ef ficient as a Carnot refrigerator. To remove \(100 .\) J of heat from the interior at \(0^{\circ} \mathrm{C}\) and eject it to the outside at \(22^{\circ} \mathrm{C}\), how much work must the refrigerator motor do?

An ideal gas undergoes an isothermal expansion. What will happen to its entropy? a) It will increase. c) It's impossible to determine. b) It will decrease. d) It will remain unchanged.

A refrigerator with a coefficient of performance of 3.80 is used to \(\operatorname{cool} 2.00 \mathrm{~L}\) of mineral water from room temperature \(\left(25.0^{\circ} \mathrm{C}\right)\) to \(4.00^{\circ} \mathrm{C} .\) If the refrigerator uses \(480 . \mathrm{W}\) how long will it take the water to reach \(4.00^{\circ} \mathrm{C}\) ? Recall that the heat capacity of water is \(4.19 \mathrm{~kJ} /(\mathrm{kg} \mathrm{K}),\) and the density of water is \(1.00 \mathrm{~g} / \mathrm{cm}^{3}\). Assume the other contents of the refrig. erator are already at \(4.00^{\circ} \mathrm{C}\).

Suppose a person metabolizes \(2000 .\) kcal/day. a) With a core body temperature of \(37.0^{\circ} \mathrm{C}\) and an ambient temperature of \(20.0^{\circ} \mathrm{C}\), what is the maximum (Carnot) efficiency with which the person can perform work? b) If the person could work with that efficiency, at what rate, in watts, would they have to shed waste heat to the surroundings? c) With a skin area of \(1.50 \mathrm{~m}^{2}\), a skin temperature of \(27.0^{\circ} \mathrm{C}\) and an effective emissivity of \(e=0.600,\) at what net rate does this person radiate heat to the \(20.0^{\circ} \mathrm{C}\) surroundings? d) The rest of the waste heat must be removed by evaporating water, either as perspiration or from the lungs At body temperature, the latent heat of vaporization of water is \(575 \mathrm{cal} / \mathrm{g}\). At what rate, in grams per hour, does this person lose water? e) Estimate the rate at which the person gains entropy. Assume that all the required evaporation of water takes place in the lungs, at the core body temperature of \(37.0^{\circ} \mathrm{C}\).
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