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

Knight Calc 5th Edition

Ch 19: Work, Heat, and the First Law of Thermodynamics

Problem 30a

Chapter 19, Problem 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?

Verified step by step guidance

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 \( \text{L·atm·mol}^{-1}\text{·K}^{-1} \).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

11m

Was this helpful?

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 essential for understanding the behavior of gases under various conditions, allowing us to calculate changes in volume or pressure when other variables are altered.

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.

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.

Recommended video:

Guided course

08:04

The First Law of Thermodynamics

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?

views

Textbook Question

A container holds 1.0 g of oxygen at a pressure of 8.0 atm. How much will the temperature increase if this amount of heat energy is transferred to the gas at constant volume?

views

Textbook Question

A container holds 1.0 g of oxygen at a pressure of 8.0 atm. How much heat is required to increase the temperature by 100°C at constant pressure?

Textbook Question

A monatomic gas follows the process 1→2→3 shown in FIGURE EX19.26. How much heat is needed for (a) process 1→2 and (b) process 2→3?

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?

views