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Ch 21: Heat Engines and Refrigerators
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 21: Heat Engines and RefrigeratorsProblem 64
Chapter 21, Problem 64

A heat engine with 0.20 mol of a monatomic ideal gas initially fills a 2000 cm³ cylinder at 600 K. The gas goes through the following closed cycle: Isothermal expansion to 4000 cm³. Isochoric cooling to 300 K. Isothermal compression to 2000 cm³. Isochoric heating to 600 K. How much work does this engine do per cycle and what is its thermal efficiency?

Verified step by step guidance
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Step 1: Understand the problem and identify the key processes in the cycle. The heat engine undergoes four distinct processes: (1) Isothermal expansion, (2) Isochoric cooling, (3) Isothermal compression, and (4) Isochoric heating. Each process involves specific thermodynamic principles that we will analyze step by step.
Step 2: Calculate the work done during the isothermal processes. For an isothermal process, the work done by the gas is given by the formula: W=nRT⁢ln⁢(VfVi), where n is the number of moles, R is the gas constant, T is the temperature, and Vi and Vf are the initial and final volumes, respectively. Apply this formula to both the isothermal expansion and compression steps.
Step 3: Analyze the isochoric processes. During an isochoric process, the volume remains constant, so no work is done (W=0). However, heat is exchanged, which can be calculated using the formula: Q=nCv⁢⁢⁢(Tf-Ti), where Cv is the molar heat capacity at constant volume, and Ti and Tf are the initial and final temperatures.
Step 4: Determine the net work done per cycle. The net work done by the engine is the sum of the work done during the isothermal expansion and compression processes. Since the isochoric processes do not involve work, they do not contribute to the net work. Use the values calculated in Step 2 to find the total work done.
Step 5: Calculate the thermal efficiency of the engine. The thermal efficiency is defined as the ratio of the net work done by the engine to the total heat input: η=WQin. Identify the heat input during the isothermal expansion and isochoric heating processes, and use the net work calculated in Step 4 to determine the efficiency.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of an ideal gas through the equation PV = nRT. This law is essential for understanding the behavior of gases in various thermodynamic processes, such as isothermal and isochoric changes, which are key to analyzing the heat engine's cycle.
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Ideal Gases and the Ideal Gas Law

Thermodynamic Processes

Thermodynamic processes describe how a system changes from one state to another. In this question, the engine undergoes isothermal (constant temperature) and isochoric (constant volume) processes, which affect the work done and heat transfer. Understanding these processes is crucial for calculating work and thermal efficiency.
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Properties of Cyclic Thermodynamic Processes

Thermal Efficiency

Thermal efficiency is a measure of how well an engine converts heat energy into work, defined as the ratio of work output to heat input. For a heat engine, it is calculated using the formula η = W/Q_in, where W is the work done by the engine and Q_in is the heat absorbed during the cycle. This concept is vital for evaluating the performance of the engine described in the question.
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Related Practice
Textbook Question

The heat engine shown in FIGURE P21.63 uses 0.020 mol of a diatomic gas as the working substance. Make a table that shows ∆Eth, Ws, and Q for each of the three processes.

Textbook Question

100 mL of water at 15℃ is placed in the freezer compartment of a refrigerator with a coefficient of performance of 4.0. How much heat energy is exhausted into the room as the water is changed to ice at -15℃?

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Textbook Question

The heat engine shown in FIGURE P21.62 uses 2.0 mol of a monatomic gas as the working substance. Determine T1, T2 and T3.

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Textbook Question

The gasoline engine in your car can be modeled as the Otto cycle shown in FIGURE CP21.73. A fuel-air mixture is sprayed into the cylinder at point 1, where the piston is at its farthest distance from the spark plug. This mixture is compressed as the piston moves toward the spark plug during the adiabatic compression stroke. The spark plug fires at point 2, releasing heat energy that had been stored in the gasoline. The fuel burns so quickly that the piston doesn't have time to move, so the heating is an isochoric process. The hot, high-pressure gas then pushes the piston outward during the power stroke. Finally, an exhaust value opens to allow the gas temperature and pressure to drop back to their initial values before starting the cycle over again. Analyze the Otto cycle and show that the work done per cycle is Wout=nR1−γ(T2−T1+T4−T3)W_{\(\text{out}\)} = \(\frac{nR}{1-\gamma}\)(T_2 - T_1 + T_4 - T_3)

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Textbook Question

FIGURE CP21.70 shows two insulated compartments separated by a thin wall. The left side contains 0.060 mol of helium at an initial temperature of 600 K and the right side contains 0.030 mol of helium at an initial temperature of 300 K. The compartment on the right is attached to a vertical cylinder, above which the air pressure is 1.0 atm. A 10-cm-diameter, 2.0 kg piston can slide without friction up and down the cylinder. Neither the cylinder diameter nor the volumes of the compartments are known. How much heat is transferred from the left side to the right side?

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Textbook Question

The heat engine shown in FIGURE P21.62 uses 2.0 mol of a monatomic gas as the working substance. Make a table that shows ∆Eth, Ws, and Q for each of the three processes.

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