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Ch 19: Work, Heat, and the First Law of Thermodynamics
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 19: Work, Heat, and the First Law of ThermodynamicsProblem 73
Chapter 19, Problem 73

One cylinder in the diesel engine of a truck has an initial volume of 600 cm3. Air is admitted to the cylinder at 30°C and a pressure of 1.0 atm. The piston rod then does 400 J of work to rapidly compress the air. What are its final temperature and volume?

Verified step by step guidance
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Start by identifying the given values: initial volume \( V_1 = 600 \; \text{cm}^3 \), initial temperature \( T_1 = 30^\circ \text{C} = 303 \; \text{K} \), initial pressure \( P_1 = 1.0 \; \text{atm} \), and work done on the gas \( W = 400 \; \text{J} \). Note that the process is rapid, so it can be approximated as adiabatic (no heat exchange).
Use the first law of thermodynamics for an adiabatic process: \( \Delta U = W \), where \( \Delta U \) is the change in internal energy and \( W \) is the work done on the gas. For an ideal gas, \( \Delta U = n C_V \Delta T \), where \( n \) is the number of moles, \( C_V \) is the molar heat capacity at constant volume, and \( \Delta T \) is the temperature change.
Relate the initial and final states using the adiabatic condition: \( P_1 V_1^\gamma = P_2 V_2^\gamma \), where \( \gamma = \frac{C_P}{C_V} \) is the adiabatic index (for air, \( \gamma \approx 1.4 \)). Rearrange this equation to express the final volume \( V_2 \) in terms of \( P_1, V_1, P_2, \) and \( \gamma \).
To find the final pressure \( P_2 \), use the ideal gas law \( P V = nRT \). Since \( n \) and \( R \) are constants, the ratio \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \) holds. Combine this with the adiabatic condition to solve for \( P_2 \) and \( T_2 \).
Finally, calculate the final temperature \( T_2 \) using the relationship \( T_2 = T_1 \left( \frac{V_1}{V_2} \right)^{\gamma - 1} \). Substitute the known values and solve step by step to find \( T_2 \) and \( V_2 \).

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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 how gases behave under varying conditions of pressure and temperature, particularly in processes like compression where the volume and temperature change.
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First Law of Thermodynamics

The First Law of Thermodynamics states that energy cannot be created or destroyed, only transformed. In the context of the problem, the work done on the gas increases its internal energy, which affects its temperature. This principle is crucial for analyzing energy transfers in thermodynamic processes.
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Work Done on a Gas

Work done on a gas during compression is defined as the force applied over a distance, which in this case is quantified as 400 J. This work increases the internal energy of the gas, leading to a rise in temperature. Understanding how work affects gas properties is vital for solving the problem.
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Textbook Question

In Problems 75, 76, and 77 you are given the equation used to solve a problem. For each of these, you are to write a realistic problem for which this is the correct equation.

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Textbook Question

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