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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: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 19: Work, Heat, and the First Law of ThermodynamicsProblema 30a
Capitolo 19, Problema 30a

A gas cylinder holds 0.10 mol of O₂ at 150°C and a pressure of 3.0 atm. The gas expands adiabatically until the pressure is halved. What are the final volume?

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Step 1: Understand the problem. This is an adiabatic process, meaning no heat is exchanged with the surroundings. The relationship between pressure, volume, and temperature for an adiabatic process is governed by the equation \( P V^\gamma = \text{constant} \), where \( \gamma \) (gamma) is the adiabatic index, which for diatomic gases like O₂ is approximately 1.4.
Step 2: Write down the given values. The initial pressure \( P_1 = 3.0 \ \text{atm} \), the initial temperature \( T_1 = 150^\circ \text{C} = 423 \ \text{K} \), the number of moles \( n = 0.10 \ \text{mol} \), and the final pressure \( P_2 = 1.5 \ \text{atm} \) (since the pressure is halved).
Step 3: Use the ideal gas law \( PV = nRT \) to calculate the initial volume \( V_1 \). Rearrange the equation to \( V_1 = \frac{nRT_1}{P_1} \), where \( R \) is the ideal gas constant \( 0.0821 \ \text{L·atm·mol}^{-1}\text{·K}^{-1} \). Substitute the known values to find \( V_1 \).
Step 4: Apply the adiabatic condition \( P_1 V_1^\gamma = P_2 V_2^\gamma \) to find the final volume \( V_2 \). Rearrange the equation to \( V_2 = V_1 \left( \frac{P_1}{P_2} \right)^{1/\gamma} \). Substitute \( \gamma = 1.4 \), \( P_1 \), \( P_2 \), and \( V_1 \) to calculate \( V_2 \).
Step 5: Verify the units and ensure consistency throughout the calculations. The final volume \( V_2 \) will be in liters, as the ideal gas constant \( R \) was used in \( \text{L·atm·mol}^{-1}\text{·K}^{-1} \).

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Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. This law is essential for understanding the behavior of gases under various conditions, allowing us to calculate changes in volume or pressure when other variables are altered.
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Ideal Gases and the Ideal Gas Law

Adiabatic Process

An adiabatic process is one in which no heat is exchanged with the surroundings. For an ideal gas, this means that any change in internal energy is due solely to work done on or by the gas. Understanding this concept is crucial for analyzing how the gas expands and how its temperature and pressure change during the expansion.
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Entropy & Ideal Gas Processes

First Law of Thermodynamics

The First Law of Thermodynamics states that energy cannot be created or destroyed, only transformed. In the context of an adiabatic process, this law helps us understand how the internal energy of the gas changes as it expands, affecting its temperature and pressure while conserving total energy.
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