/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 27 (a) What is the total translatio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 18: Problem 27

(a) What is the total translational kinetic energy of the air in an empty room that has dimensions 8.00 m \(\times\) 12.00 m \(\times\) 4.00 m if the air is treated as an ideal gas at 1.00 atm? (b) What is the speed of a 2000-kg automobile if its kinetic energy equals the translational kinetic energy calculated in part (a)?

Short Answer

Expert verified
(a) The translational kinetic energy is approximately \(5.83 \times 10^6\) Joules. (b) The automobile's speed is \(76.27\) m/s.

Step by step solution

01

Calculate Volume of the Room

First, find the volume of the room. The room is a rectangular box, so the volume \( V \) is calculated by multiplying its length, width, and height: \[V = 8.00 \times 12.00 \times 4.00 = 384 \text{ m}^3.\]
02

Determine Number of Moles

Use the ideal gas law to determine the number of moles of air in the room. The ideal gas law is given by \( PV = nRT \). Rearrange it to solve for \( n \), the number of moles: \[n = \frac{PV}{RT} \]where \( P = 1.00 \text{ atm} = 101325 \text{ Pa} \) (since 1 atm = 101325 Pa), \( R = 8.31 \text{ J/(mol路K)} \), and \( T = 273.15 + 20 = 293.15 \text{ K} \) (assuming room temperature of about 20掳C). Calculate \( n \): \[n = \frac{101325 \times 384}{8.31 \times 293.15} \approx 15973 \text{ mol}.\]
03

Calculate Translational Kinetic Energy

The translational kinetic energy of an ideal gas is given by the formula: \[E_k = \frac{3}{2} nRT.\]Substitute the known values:\[E_k = \frac{3}{2} \times 15973 \times 8.31 \times 293.15 \approx 5.83 \times 10^6 \text{ Joules}.\]
04

Calculate Speed of Automobile

Use the formula for kinetic energy to find the speed of a 2000-kg automobile, where kinetic energy \( E_k = \frac{1}{2} mv^2 \). Set \( E_k \) equal to the translational kinetic energy calculated in step 3 and solve for \( v \): \[5.83 \times 10^6 = \frac{1}{2} \times 2000 \times v^2.\]Rearrange and solve for \( v \):\[v^2 = \frac{5.83 \times 10^6 \times 2}{2000} \v = \sqrt{5815} \approx 76.27 \text{ m/s}.\]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ideal Gas Law
Understanding the ideal gas law is essential when dealing with gases under certain conditions. This law is expressed with the formula \( PV = nRT \), where:
  • \( P \) is the pressure of the gas.
  • \( V \) is the volume occupied by the gas.
  • \( n \) is the number of moles of gas.
  • \( R \) is the ideal gas constant, approximately \( 8.31 \, \text{J/(mol路K)} \).
  • \( T \) is the absolute temperature in Kelvin.
This equation shows that if you know the pressure, volume, and temperature of a gas, you can determine the number of moles of that gas.
The ideal gas law assumes that the gas behaves ideally, meaning it follows the assumptions of kinetic molecular theory without any interactions between its molecules.
Kinetic Energy Calculation
When calculating the translational kinetic energy of a system like an ideal gas, we use the formula \( E_k = \frac{3}{2} nRT \). This equation derives from the behavior of gas molecules moving randomly in the space.
  • \( E_k \) represents the translational kinetic energy.
  • \( n \) is the number of moles of the gas.
  • \( R \) is the ideal gas constant.
  • \( T \) is the temperature in Kelvin.
This formula shows that kinetic energy is proportional to the temperature of the gas. If the temperature increases, the molecules have more energy and move faster. The kinetic energy depends also on how much gas is present, reflected by the number of moles \( n \).
Moles of Gas
Calculating the number of moles in a gas sample is needed to figure out how much air is in a space, like a room. In our scenario, using the formula \( n = \frac{PV}{RT} \), we can rearrange the ideal gas law to solve for the number of moles:
  • \( P \) is set as \( 101325 \, \text{Pa} \), the pressure for \( 1 \, \text{atm} \).
  • \( V \) is the volume of the space, in this case, a room volume calculated from its dimensions \( V = 8.00 \, \times \ 12.00 \, \times \ 4.00 \).
  • \( R \) remains consistent at \( 8.31 \, \text{J/mol \, K} \).
  • \( T \), the temperature, converts to Kelvin, usually room temperature, \( 293.15 \, K \).
This method enables you to determine the quantity of gas in terms of moles, which subsequently allows for deeper calculations like kinetic energy.
Volume of a Room
Knowing the volume of a room is crucial in many physics calculations involving gases, such as applying the ideal gas law. To compute the room's volume in cubic meters, apply the formula for the volume of a rectangular prism, \( V = \text{length} \times \text{width} \times \text{height} \).
  • Begin with measuring the dimensions: 8.00 m (length), 12.00 m (width), and 4.00 m (height).
  • Multiply these three measurements together to get the volume \( V = 8.00 \, \times \ 12.00 \, \times \ 4.00 = 384 \, \text{m}^3 \).
This calculation gives us the space available for the air molecules to occupy. It's a fundamental step when working with the ideal gas law to find out how much gas can fit in the room.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A physics lecture room at 1.00 atm and 27.0\(^\circ\)C has a volume of 216 m\(^3\). (a) Use the ideal-gas law to estimate the number of air molecules in the room. Assume that all of the air is N\(_2\). Calculate (b) the particle density-that is, the number of N\(_2 \) molecules per cubic centimeter-and (c) the mass of the air in the room.

A welder using a tank of volume 0.0750 m\(^3\) fills it with oxygen (molar mass 32.0 g/mol) at a gauge pressure of 3.00 \(\times\) 10\({^5}\) Pa and temperature of 37.0\(^\circ\)C. The tank has a small leak, and in time some of the oxygen leaks out. On a day when the temperature is 22.0\(^\circ\)C, the gauge pressure of the oxygen in the tank is 1.80 \(\times\) 10\({^5}\) Pa. Find (a) the initial mass of oxygen and (b) the mass of oxygen that has leaked out.

The \(vapor\) \(pressure\) is the pressure of the vapor phase of a substance when it is in equilibrium with the solid or liquid phase of the substance. The \(relative\) \(humidity\) is the partial pressure of water vapor in the air divided by the vapor pressure of water at that same temperature, expressed as a percentage. The air is saturated when the humidity is 100%. (a) The vapor pressure of water at 20.0\(^\circ\)C is 2.34 \(\times\) 103 Pa. If the air temperature is 20.0\(^\circ\)C and the relative humidity is 60%, what is the partial pressure of water vapor in the atmosphere (that is, the pressure due to water vapor alone)? (b) Under the conditions of part (a), what is the mass of water in 1.00 m\(^3\) of air? (The molar mass of water is 18.0 g/mol. Assume that water vapor can be treated as an ideal gas.)

A flask contains a mixture of neon (Ne), krypton (Kr), and radon (Rn) gases. Compare (a) the average kinetic energies of the three types of atoms and (b) the root-mean-square speeds. (Hint: Appendix D shows the molar mass (in g/mol) of each element under the chemical symbol for that element.)

Three moles of an ideal gas are in a rigid cubical box with sides of length 0.300 m. (a) What is the force that the gas exerts on each of the six sides of the box when the gas temperature is 20.0\(^\circ\)C? (b) What is the force when the temperature of the gas is increased to 100.0\(^\circ\)C?
See all solutions

Recommended explanations on Physics Textbooks

Modern Physics

Read Explanation

Rotational Dynamics

Read Explanation

Force

Read Explanation

Thermodynamics

Read Explanation

Solid State Physics

Read Explanation

Electric Charge Field and Potential

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.