Name: gas collisions: path length and time
Uploaded: 2025-05-22T09:05:59.000Z
Duration: 1 min 22 s
Channel: JoVE

K12 Standards

Florida Department of Education

High School

Physical Science

SC.912.P.12

SC.912.P.12.10

Mean free path and Mean free time

Consider N gas molecules with radius r moving randomly with speed v in a cylindrical volume V. When one molecule collides with another molecule, the distance between their centers is 2r.
Imagine a cylinder with a radius of 2r, with an axis parallel to the molecule&#39;s velocity. When the molecule travels for a small time interval and collides inside the cylinder, the number of collisions per unit time can be determined.
Using the average relative velocity equation, the collisions for all the moving molecules per unit time can be determined.
The reciprocal of the equation gives the average time between collisions, known as the mean free time.
Meanwhile, the mean free path of a gas molecule is the product of the molecule&#39;s speed and the average time between collisions. By substituting the terms, the mean free path can be determined, which is inversely proportional to the number of molecules per unit volume and the cross-sectional area of the molecule.
Recalling the ideal-gas equation and substituting the terms, the macroscopic properties of the gas can be obtained.

mean free path and mean free time

Consider the gas molecules in a cylinder. They move in a random motion as they collide with each other and change speed and direction. The average of all the path lengths between collisions is known as the "mean free path."
<img alt="Lambda equation diagram, thermal de Broglie wavelength formula, physics and chemistry concept." src="/files/ftp_upload/14168/14168_Equation_1_new.svg" style="text-align: center; height: 40px;" data-altupdatedat="1754904545000" data-alt="λ formula for mean free path; equation in kinetic theory of gases.">
The mean free path varies inversely with the density of the molecules because when there are more molecules inside a volume, they have a greater chance of colliding with each other, thus reducing the mean free path. Additionally, the mean free path is inversely related to the diameter of the molecules because if they were point masses, they would never collide. Thus, larger molecules are associated with a shorter mean free path.
The gas expands when the temperature increases under constant pressure; thus, the average distance between molecules and the mean free path increases. However, when the pressure is increased at a constant temperature, the gas compresses, leading to a decrease in the mean free path. The mean free path can be defined as the product of the average speed and the mean free time, where the mean free time is the average time between collisions.
Consider argon atoms with a molar mass of 39.9 g/mol moving randomly in a cylinder at a temperature of 273 K and a pressure of 1 atm. Taking the radius of an argon atom to be 1.70 × 10-10m, determine the mean free time for argon atoms.
To solve the problem, first identify the known and unknown quantities, and convert them into SI units.
Secondly, recall the RMS speed equation for gas molecules. By substituting the values, the RMS speed can be determined as follows:
<img alt="Vrms formula, root-mean-square velocity equation, physics calculation diagram, molecular speed." src="/files/ftp_upload/14168/14168_Equation_2_new.svg" style="text-align: center; height: 45px;" data-altupdatedat="1754904545000" data-alt="Root mean square velocity formula: Vrms = √(3RT/M) = 413 m/s, kinetic theory concept.">
Lastly, recall the mean free time equation. By substituting the values, the mean free time can be determined as follows:
<img alt="Static equilibrium formula τ calculation diagram, thermodynamics, molecular dynamics." src="/files/ftp_upload/14168/14168_Equation_3_new.svg" style="text-align: center; height: 40px;" data-altupdatedat="1754904545000" data-alt="Relaxation time formula τ=kBT/(4π√2r²ρvrms); kinetic theory equation.">

Consider the gas molecules in a cylinder. They move in a random motion as they collide with each other and change speed and direction. The average of all ...

Education

JoVE Core

Physics

Unit 1

The Kinetic Theory of Gases

KEY TERMS AND CONCEPTS

gas collisions: path length and time

Gas collisions in a cylinder help explain the mean free path and mean free time. Gas molecules move in random motion. They collide with each other and change speed and direction. The mean free path is the average path length between collisions.

The mean free path depends on how many molecules are packed into a volume. It varies inversely with density. When more molecules are present, collisions happen more often, so the mean free path becomes shorter. It also depends on molecule size. Larger molecules have a shorter mean free path, while point masses would never collide.

Temperature and pressure also affect the mean free path. When temperature increases at constant pressure, a gas expands. The average distance between molecules increases, and the mean free path increases too. When pressure increases at constant temperature, the gas compresses. That reduces the mean free path.

The mean free path can also be written as the product of average speed and mean free time. Mean free time is the average time between collisions. A sample problem uses argon atoms with a molar mass of 39.9 g/mol at 273 K and 1 atm. The radius of an argon atom is taken as 1.70 × 10 -10 m. To find the mean free time, first identify the known and unknown values and convert them to SI units. Then use the RMS speed equation for gas molecules. After that, substitute the values into the mean free time equation to calculate the result.

Gas collisions: path length and time

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Gas collisions in a cylinder help explain the mean free path and mean free time. Gas molecules move in random motion. They collide with each other and change speed and direction. The mean free path is the average path length between collisions. The mean free path depends on how many molecules are packed into a volume. It varies inversely with density. When more molecules are present, collisions happen more often, so the mean free path becomes shorter. It also depends on molecule size. Larger...

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key terms and concepts

Learn how gas molecule collisions determine mean free path and mean free time, and see how density, size, temperature, and pressure affect them in argon.

Gas collisions: path length and time | JoVE K12

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A. Motion can be measured and described qualitatively and quantitatively. Net forces create a change in motion. When objects travel at speeds comparable to the speed of light, Einstein's special theory of relativity applies.

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Gas collisions: path length and time

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