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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 73a
Chapter 21, Problem 73a

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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Step 1: Understand the Otto cycle. The Otto cycle consists of four processes: (1) adiabatic compression, (2) isochoric heating, (3) adiabatic expansion, and (4) isochoric cooling. The work done per cycle is the net work done during the adiabatic processes, which can be calculated using the first law of thermodynamics.
Step 2: Write the expression for the work done in the cycle. The work done per cycle is the difference between the work done during the adiabatic expansion (process 3) and the work done during the adiabatic compression (process 1). Mathematically, \( W_{cycle} = W_{expansion} - W_{compression} \).
Step 3: Use the thermodynamic relationships for adiabatic processes. For an adiabatic process, the work done can be expressed as \( W = \frac{P_2 V_2 - P_1 V_1}{\gamma - 1} \), where \( \gamma \) is the adiabatic index (ratio of specific heats), \( P \) is pressure, and \( V \) is volume. Apply this formula to both the compression and expansion processes.
Step 4: Relate the heat added during the isochoric heating process to the change in internal energy. For an isochoric process, the heat added is \( Q = n C_v \Delta T \), where \( n \) is the number of moles, \( C_v \) is the specific heat at constant volume, and \( \Delta T \) is the temperature change. This heat contributes to the energy available for the adiabatic expansion.
Step 5: Combine the results to express the net work done per cycle. Substitute the expressions for \( W_{expansion} \), \( W_{compression} \), and the heat added during the isochoric process into the equation for \( W_{cycle} \). Simplify the terms to show the relationship between the work done per cycle and the thermodynamic properties of the Otto cycle.

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Key Concepts

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

Otto Cycle

The Otto cycle is a thermodynamic cycle that describes the functioning of a gasoline engine. It consists of four main processes: adiabatic compression, isochoric heat addition, adiabatic expansion, and isochoric heat rejection. This cycle illustrates how the engine converts fuel into mechanical work by compressing a fuel-air mixture, igniting it, and allowing the resulting gas expansion to drive the piston.
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Adiabatic Process

An adiabatic process is one in which no heat is exchanged with the surroundings. In the context of the Otto cycle, both the compression and expansion strokes are adiabatic, meaning that the gas temperature changes due to work done on or by the gas, rather than heat transfer. This principle is crucial for understanding how the engine efficiently converts thermal energy into mechanical work.
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Isochoric Process

An isochoric process occurs at constant volume, meaning that the gas does not expand or contract during the process. In the Otto cycle, the heat addition from the spark plug happens during the isochoric phase, where the volume of the gas remains fixed while its temperature and pressure increase. This process is essential for understanding how energy is stored in the gas before it expands and does work on the piston.
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Related Practice
Textbook Question

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?

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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

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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