Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

For diatomic carbon dioxide gas (CO2, molar mass 44.0 g/mol) at T = 300 K, calculate (a) the most probable speed υmp; (b) the average speed υav; (c) the root-mean-square speed υrms.

Short Answer

Expert verified
For CO2 at 300 K, calculate: υmp395 m/s, υav420 m/s, υrms430 m/s.

Step by step solution

01

Define the Problem

We need to calculate three types of speeds (most probable, average, and root-mean-square) for CO2 gas at temperature T=300 K. These speeds are related to the kinetic theory of gases and depend on the temperature and molar mass of the gas.
02

Calculate the Most Probable Speed

The most probable speed, υmp, can be calculated using the formula: υmp=2kTmwhere k is the Boltzmann constant (1.38×1023 J/K), T=300 K is the temperature, and m is the mass of one molecule of CO2. The molar mass of CO2 is 44.0 g/mol, so m=44.01000×6.022×10237.30×1026 kg.Using these values, calculate υmp.
03

Calculate the Average Speed

The average speed, υav, is given by:υav=8kTπmSubstitute k=1.38×1023 J/K, T=300 K, and m7.30×1026 kg into the formula and compute υav.
04

Calculate the Root-Mean-Square Speed

The root-mean-square speed, υrms, can be calculated using:υrms=3kTmUsing k=1.38×1023 J/K, T=300 K, and m7.30×1026 kg, find υrms.
05

Perform Calculations

Compute the values for υmp, υav, and υrms using the formulas and substitutions performed in the previous steps. Ensure correct unit conversion and arithmetic operations.

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 Vaia!

Key Concepts

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

Molecular Speeds
In the kinetic theory of gases, we explore different ways to measure how fast gas molecules move. This concept of molecular speeds helps us understand how individual gas particles behave. Molecular speeds are essential because they influence how gases interact with each other. For example, faster molecules will collide more and may escape a container faster.
  • Most Probable Speed: The speed at which most gas molecules are moving.
  • Average Speed: The mean speed of all the molecules, which balances out slower and faster molecules.
  • Root-Mean-Square Speed: A measure that squares the speed of each molecule, finds the average, and then takes the square root.
Each of these speeds provides different insights into the behavior of gas molecules at a given temperature.
Boltzmann Constant
The Boltzmann constant is a fundamental constant in physics, denoted by the symbol, k, and has a value of 1.38×1023 J/K. It plays a crucial role in the kinetic theory of gases, linking the macroscopic and microscopic worlds. It helps us translate temperature into energy when studying gases.Whenever we calculate molecular speeds such as the most probable, average, or root-mean-square speed, the Boltzmann constant is part of the formula. By doing so, it scales the temperature to the kinetic energy of gas molecules. This allows us to move from purely theoretical calculations to practical, observable results.
Root-Mean-Square Speed
The root-mean-square (RMS) speed is a statistical measure of the speed of particles in a gas that gives insight into their energy. Mathematically, it is given by the formula:υrms=3kTmwhere k is the Boltzmann constant, T is the temperature, and m is the mass of a single molecule of the gas.RMS speed considers each molecule’s speed squared, averages these values, and takes the square root of the result. This gives us an effective measure of speed that considers both the direction and magnitude of molecular velocities. Knowing the RMS speed helps us predict gas behaviors, such as diffusion rates and kinetic energies.
Average Speed
The average speed of gas molecules is another important component of molecular speeds in the kinetic theory of gases.This average speed is calculated using the formula:υav=8kTπmThe formula shows that the average speed is related to the temperature and mass of the gas molecules, similar to the other types of molecular speeds.• It takes into account all molecules, weighing the slower and faster ones.• While different from the root-mean-square speed, it provides a more intuitive measure of molecular motion.Understanding average speed is useful when evaluating how gases mix or transport energy through collisions.
Most Probable Speed
The most probable speed is the speed of a gas molecule that is most likely observed in a large sample of a gas.It is calculated by:υmp=2kTmHere’s why this speed matters:
  • It's the peak of the speed distribution curve for gases, meaning most molecules have this speed.
  • The formula highlights that the most probable speed depends on temperature and molecular mass.
  • The most probable speed is often lower than both the average and RMS speeds.
Knowing the most probable speed helps in predicting how gases behave under different conditions, such as pressure changes in a balloon.

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

Titan, the largest satellite of Saturn, has a thick nitrogen atmosphere. At its surface, the pressure is 1.5 earth-atmospheres and the temperature is 94 K. (a) What is the surface temperature in C? (b) Calculate the surface density in Titan's atmosphere in molecules per cubic meter. (c) Compare the density of Titan's surface atmosphere to the density of earth's atmosphere at 22C. Which body has denser atmosphere?

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 20.0C is 2.34 × 103 Pa. If the air temperature is 20.0C 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 m3 of air? (The molar mass of water is 18.0 g/mol. Assume that water vapor can be treated as an ideal gas.)

A cylindrical tank has a tight-fitting piston that allows the volume of the tank to be changed. The tank originally contains 0.110 m3 of air at a pressure of 0.355 atm. The piston is slowly pulled out until the volume of the gas is increased to 0.390 m3. If the temperature remains constant, what is the final value of the pressure?

A Jaguar XK8 convertible has an eight-cylinder engine. At the beginning of its compression stroke, one of the cylinders contains 499 cm3 of air at atmospheric pressure (1.01 × 105 Pa) and a temperature of 27.0C. At the end of the stroke, the air has been compressed to a volume of 46.2 cm3 and the gauge pressure has increased to 2.72 × 106 Pa. Compute the final temperature.

The atmosphere of Mars is mostly CO2 (molar mass 44.0 g/mol) under a pressure of 650 Pa, which we shall assume remains constant. In many places the temperature varies from 0.0C in summer to -100C in winter. Over the course of a Martian year, what are the ranges of (a) the rms speeds of the CO2 molecules and (b) the density (in mol/m3) of the atmosphere?

See all solutions

Recommended explanations on Physics Textbooks

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.

Sign-up for free