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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 32a
Chapter 19, Problem 32a

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 volume is doubled. What are the final pressure?

Verified step by step guidance
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Step 1: Understand the problem. This is an adiabatic process, meaning no heat is exchanged with the surroundings. For an adiabatic process involving an ideal gas, the relationship between pressure, volume, and temperature is governed by the adiabatic condition: \( P V^\gamma = \text{constant} \), where \( \gamma \) (gamma) is the adiabatic index, defined as \( \gamma = \frac{C_p}{C_v} \). For diatomic gases like O₂, \( \gamma \approx 1.4 \).
Step 2: Write the adiabatic condition for the initial and final states. Let the initial pressure, volume, and temperature be \( P_1 \), \( V_1 \), and \( T_1 \), and the final pressure, volume, and temperature be \( P_2 \), \( V_2 \), and \( T_2 \). Since \( V_2 = 2V_1 \), the adiabatic condition becomes: \( P_1 V_1^\gamma = P_2 V_2^\gamma \). Substitute \( V_2 = 2V_1 \) into the equation.
Step 3: Solve for the final pressure \( P_2 \). Rearrange the adiabatic condition to isolate \( P_2 \): \( P_2 = P_1 \left( \frac{V_1}{V_2} \right)^\gamma \). Substitute \( V_2 = 2V_1 \) and simplify: \( P_2 = P_1 \left( \frac{1}{2} \right)^\gamma \). Use \( \gamma = 1.4 \) for diatomic gases.
Step 4: Use the ideal gas law to relate the initial conditions. The ideal gas law is \( PV = nRT \). For the initial state, \( P_1 V_1 = nRT_1 \). Convert the given temperature from Celsius to Kelvin: \( T_1 = 150 + 273 = 423 \, \text{K} \). This relationship can help verify the initial volume if needed.
Step 5: Recognize that the final temperature \( T_2 \) can also be determined using the adiabatic relation between temperature and volume: \( T_1 V_1^{\gamma-1} = T_2 V_2^{\gamma-1} \). Rearrange to solve for \( T_2 \): \( T_2 = T_1 \left( \frac{V_1}{V_2} \right)^{\gamma-1} \). Substitute \( V_2 = 2V_1 \) and simplify: \( T_2 = T_1 \left( \frac{1}{2} \right)^{\gamma-1} \).

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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 a gas through the equation PV = nRT. This law is fundamental in understanding the behavior of gases under various conditions, allowing us to calculate changes in state variables when the gas undergoes transformations.
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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. In such processes, the relationship between pressure and volume can be described by the adiabatic equation, which is crucial for determining the final state of the gas after expansion.
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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 an adiabatic process, this law implies that the work done by the gas during expansion results in a change in internal energy, affecting its temperature and pressure. Understanding this principle is essential for analyzing energy changes in thermodynamic systems.
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Related Practice
Textbook Question

The volume of a gas is halved during an adiabatic compression that increases the pressure by a factor of 2.5. By what factor does the temperature increase?

Textbook Question

The volume of a gas is halved during an adiabatic compression that increases the pressure by a factor of 2.5. What is the specific heat ratio γ?

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Textbook Question

You are boiling pasta and absentmindedly grab a copper stirring spoon rather than your wooden spoon. The copper spoon has a 20 mm ×1.5 mm rectangular cross section, and the distance from the boiling water to your 35°C hand is 18 cm. How long does it take the spoon to transfer 25 J of energy to your hand?

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Textbook Question

A 5.0-m-diameter garden pond is 30 cm deep. Solar energy is incident on the pond at an average rate of 400 W/m2. If the water absorbs all the solar energy and does not exchange energy with its surroundings, how many hours will it take to warm from 15°C to 25°C?

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Textbook Question

The ends of a 20-cm-long, 2.0-cm-diameter rod are maintained at 0°C and 100°C by immersion in an ice-water bath and boiling water. Heat is conducted through the rod at 4.5×104 J per hour. Of what material is the rod made?

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Textbook Question

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 temperature?

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