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Chapter 15: Problem 78

In a heat engine, 3.00 mol of a monatomic ideal gas, initially at 4.00 atm of pressure, undergoes an isothermal expansion, increasing its volume by a factor of 9.50 at a constant temperature of \(650.0 \mathrm{K} .\) The gas is then compressed at a constant pressure to its original volume. Finally, the pressure is increased at constant volume back to the original pressure. (a) Draw a \(P V\) diagram of this three-step heat engine. (b) For each step of this process, calculate the work done on the gas, the change in internal energy, and the heat transferred into the gas. (c) What is the efficiency of this engine?

Short Answer

Expert verified
Question: Calculate the efficiency of the heat engine in the given three-step process. Answer: The efficiency of the heat engine can be calculated using the formula: \(\eta = 1 - \frac{Q_c}{Q_h}\), where \(Q_c\) is the heat released during constant pressure compression (step 2, \(Q_{BC}\)) and \(Q_h\) is the heat absorbed during the isothermal expansion (step 1, \(Q_{AB}\)). To find the efficiency, substitute the values of \(Q_{BC}\) and \(Q_{AB}\) obtained from the previous steps in the formula and calculate the result.

Step by step solution

01

Draw PV-diagram

Sketch a PV-diagram with the three steps of the process: isothermal expansion, constant pressure compression, and constant volume pressure increase. In the diagram, label each point as follows: - Point A: Initial state (pressure: \(P_A = 4.00\ \mathrm{atm}\), volume: \(V_A = \frac{nRT}{P_A}\)) - Point B: After isothermal expansion (pressure: \(P_B = \frac{P_A}{9.50}\), volume: \(V_B = 9.50V_A\)) - Point C: After constant pressure compression (pressure: \(P_C = P_B\), volume: \(V_C = V_A\)) - Point D: After constant volume pressure increase (pressure: \(P_D = P_A\), volume: \(V_D = V_A\))
02

Calculate work done during each step

Now, we'll calculate the work done on the gas during each step of the process: 1. Isothermal expansion (A to B): \(W_{AB} = -nRT_A\ln{\frac{V_B}{V_A}}\) 2. Constant pressure compression (B to C): \(W_{BC} = -P_B\Delta V_{BC} = -P_B(V_C - V_B)\) 3. Constant volume pressure increase (C to D): \(W_{CD} = 0\)
03

Calculate the change in internal energy for each step

Next, we'll calculate the change in internal energy for each step of the process: 1. Isothermal expansion (A to B): \(\Delta U_{AB} = \frac{3}{2}nR\Delta T_{AB} = 0\) (since \(\Delta T_{AB} = 0\) due to isothermal process) 2. Constant pressure compression (B to C): \(\Delta U_{BC} = \frac{3}{2}nR\Delta T_{BC}\) 3. Constant volume pressure increase (C to D): \(\Delta U_{CD} = \frac{3}{2}nR\Delta T_{CD}\)
04

Calculate the heat transferred for each step

Now, we'll calculate the heat transferred into the gas during each step of the process: 1. Isothermal expansion (A to B): \(Q_{AB} = \Delta U_{AB} + W_{AB}\) 2. Constant pressure compression (B to C): \(Q_{BC} = \Delta U_{BC} + W_{BC}\) 3. Constant volume pressure increase (C to D): \(Q_{CD} = \Delta U_{CD} + W_{CD}\)
05

Calculate the efficiency of the heat engine

Finally, we'll calculate the efficiency of the engine using the formula: \(\eta = 1 - \frac{Q_c}{Q_h}\) Here, \(Q_c\) is the heat released during constant pressure compression (step 2, \(Q_{BC}\)) and \(Q_h\) is the heat absorbed during the isothermal expansion (step 1, \(Q_{AB}\)). So, the efficiency is: \(\eta = 1 - \frac{Q_{BC}}{Q_{AB}}\) Calculate the efficiency using the values obtained in previous steps.

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Most popular questions from this chapter

An engine operates between temperatures of \(650 \mathrm{K}\) and $350 \mathrm{K}\( at \)65.0 \%$ of its maximum possible efficiency. (a) What is the efficiency of this engine? (b) If $6.3 \times 10^{3} \mathrm{J}$ is exhausted to the low temperature reservoir, how much work does the engine do?
For a more realistic estimate of the maximum coefficient of performance of a heat pump, assume that a heat pump takes in heat from outdoors at $10^{\circ} \mathrm{C}$ below the ambient outdoor temperature, to account for the temperature difference across its heat exchanger. Similarly, assume that the output must be \(10^{\circ} \mathrm{C}\) hotter than the house (which itself might be kept at \(20^{\circ} \mathrm{C}\) ) to make the heat flow into the house. Make a graph of the coefficient of performance of a reversible heat pump under these conditions as a function of outdoor temperature (from $\left.-15^{\circ} \mathrm{C} \text { to }+15^{\circ} \mathrm{C} \text { in } 5^{\circ} \mathrm{C} \text { increments }\right)$
An electric power station generates steam at \(500.0^{\circ} \mathrm{C}\) and condenses it with river water at \(27^{\circ} \mathrm{C} .\) By how much would its theoretical maximum efficiency decrease if it had to switch to cooling towers that condense the steam at \(47^{\circ} \mathrm{C} ?\)
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 $3.5 \mathrm{J} /\left(\mathrm{g} \cdot^{\circ} \mathrm{C}\right)$.
How much heat does a heat pump with a coefficient of performance of 3.0 deliver when supplied with \(1.00 \mathrm{kJ}\) of electricity?
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