19.10:
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'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'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.
19.10:
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."
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:
Lastly, recall the mean free time equation. By substituting the values, the mean free time can be determined as follows:
Suggested Reading
- Young, H.D and Freedman, R.A. (2012). University Physics with Modern Physics. San Francisco, CA: Pearson. Pp. 630
- OpenStax. (2019). University Physics Vol. 1. [Web version]. Retrieved from https://openstax.org/books/college-physics-2e/pages/13-4-kinetic-theory-atomic-and-molecular-explanation-of-pressure-and-temperature