/*! 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 38 Some incandescent light bulbs ar... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 38

Some incandescent light bulbs are filled with argon gas. What is \(v_{\mathrm{rms}}\) for argon atoms near the filament, assuming their temperature is \(2500 \mathrm{K} ?\)

Short Answer

Expert verified
The root mean square velocity (\(v_{rms}\)) of argon atoms near the filament in an incandescent light bulb at a temperature of 2500 K is approximately 620 m/s.

Step by step solution

01

Set up the root mean square velocity formula

The formula for the root mean square velocity (\(v_{rms}\)) of a gas is given by: \(v_{rms} = \sqrt{\frac {3kT} {m}}\), where \(k\) is the Boltzmann constant (\(1.38 \times 10^{-23} JK^{-1}\)), \(T\) is the temperature of the gas, and \(m\) is the mass of one particle of the gas. In our case, the temperature \(T = 2500 K\), and we need to find the mass of one argon atom.
02

Find the mass of one argon atom

Argon has a molar mass of \(39.95 \thinspace g/mol\). In order to find the mass of one argon atom, we first need to obtain the mass in kilograms and then divide it by the Avogadro's number (\(6.022 \times 10^{23} \thinspace mol^{-1}\)). The mass of one argon atom (m) in kilograms is: \[m = \frac{39.95\thinspace g/mol}{1000\thinspace g/kg} \times \frac{1\thinspace mol}{6.022\times 10^{23}} = 6.63 \times 10^{-26} kg\]
03

Calculate the root mean square velocity

Now that we have the temperature and the mass of one argon atom, we can plug these values into the root mean square velocity formula: \[v_{rms} = \sqrt{\frac {3kT} {m}}\] Plugging in the values, we get: \[v_{rms} = \sqrt{\frac {3(1.38 \times 10^{-23} \thinspace JK^{-1})(2500 \thinspace K)} {6.63 \times 10^{-26} \thinspace kg}}\]
04

Solve for the root mean square velocity

Now, we need to solve for \(v_{rms}\): \[v_{rms} = \sqrt{\frac {3(1.38 \times 10^{-23} \thinspace JK^{-1})(2500 \thinspace K)} {6.63 \times 10^{-26} \thinspace kg}} = 620 \thinspace m/s\] The root mean square velocity of argon atoms near the filament in an incandescent light bulb at a temperature of 2500 K is approximately 620 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.

Kinetic Theory of Gases
The Kinetic Theory of Gases offers a scientific framework to comprehend the motion and behavior of gas molecules. This theory posits that gas consists of a multitude of small particles, often atoms or molecules, that move randomly and are in constant motion.
  • These particles collide with each other and the walls of their container.
  • Such collisions are responsible for the pressure exerted by the gas.
  • The theory assumes that the volume of the particles themselves is negligible compared to the volume of the gas as a whole.
  • It also considers that there are no intermolecular forces acting between particles except during collisions.
In essence, the kinetic theory helps to illustrate how temperature, volume, and pressure interrelate with the molecular motion of gases.
Temperature and Molecular Motion
Temperature is fundamentally a measure of the average kinetic energy of the molecules in a substance. Simply put, when a substance is heated, its molecules move faster.
  • Each increase in temperature leads to an increase in the average speed of molecular motion.
  • In gases, like argon, this motion is random and spans in all directions.
  • The kinetic energy heavily influences properties like pressure and volume, as they stem from molecular collisions.
Thus, by understanding temperature as a facilitator for increased molecular motion, we understand how it impacts other properties, such as the root mean square velocity (RMS velocity), which is a measure of these molecular speeds within a gas.
Argon Gas Properties
Argon is a noble gas, characterized by properties that make it particularly interesting for various applications. It is colorless, odorless, and chemically inert due to its complete outer electron shell.
  • Argon's atomic mass is 39.95 amu, a detail crucial for scientific calculations involving gases.
  • As a monatomic gas, Argon exists as individual atoms, unlike diatomic molecules like Nitrogen or Oxygen.
  • Given its non-reactive nature, Argon is used in environments where materials need to be kept from easily oxidizing.
These properties make Argon ideal for use in incandescent light bulbs, where it protects sensitive filament materials from reacting with oxygen, leading to a longer bulb life.
RMS Velocity Calculation
The Root Mean Square (RMS) Velocity provides a measure of the speed of particles in a gas and is specifically useful in understanding energy and temperature relationships. The formula for RMS velocity is:\[v_{rms} = \sqrt{\frac{3kT}{m}}\]Here:
  • \( k \) is the Boltzmann constant \( (1.38 \times 10^{-23} \thinspace J/K) \).
  • \( T \) represents the temperature in Kelvin.
  • \( m \) is the mass of a gas particle, often determined through the molar mass and Avogadro's number.
In the context of Argon in a light bulb at \( 2500 \thinspace K \), calculating the \( v_{rms} \) helps predict how fast argon atoms are moving near the filament. By substituting the known values into the formula, one determines the RMS velocity as approximately \( 620 \thinspace m/s \). This outcome provides insight into the kinetic energy and speed these atoms attain at such high temperatures.

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

Dry air consists of approximately \(78 \%\) nitrogen, \(21 \%\) oxygen, and \(1 \%\) argon by mole, with trace amounts of other gases. A tank of compressed dry air has a volume of 1.76 cubic feet at a gauge pressure of 2200 pounds per square inch and a temperature of \(293 \mathrm{K}\). How much oxygen does it contain in moles?

An airtight dispenser for drinking water is sign artight dispenser for drinking water is \(25 \mathrm{cm} \times 10 \mathrm{cm}\) in horizontal dimensions and \(20 \mathrm{cm}\) tall. It has a tap of negligible volume that opens at the level of the bottom of the dispenser. Initially, it contains water to a level \(3.0 \mathrm{cm}\) from the top and air at the ambient pressure, \(1.00 \mathrm{atm},\) from there to the top. When the tap is opened, water will flow out until the gauge pressure at the bottom of the dispenser, and thus at the opening of the tap, is 0 . What volume of water flows out? Assume the temperature is constant, the dispenser is perfectly rigid, and the water has a constant density of \(1000 \mathrm{kg} / \mathrm{m}^{3}\)

If one kind of molecule has double the radius of another and eight times the mass, how do their mean free paths under the same conditions compare? How do their mean free times compare?

Under what circumstances would you expect a gas to behave significantly differently than predicted by the ideal gas law?

A constant-volume gas thermometer contains a fixed amount of gas. What property of the gas is measured to indicate its temperature?
See all solutions

Recommended explanations on Physics Textbooks

Linear Momentum

Read Explanation

Medical Physics

Read Explanation

Scientific Method Physics

Read Explanation

Physics of Motion

Read Explanation

Atoms and Radioactivity

Read Explanation

Rotational Dynamics

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.