Calculating Mean Free Path of Molecules in a Test Tube

[n.hirsch1]

Homework Statement

A test tube of cylindrical shape having a length of 10 cm and a diameter of 2 cm contains 20 * 10 ^23 molecules (molecular size d = 3 * 10^-10 m). What is the mean free path of these molecules??

Homework Equations

λ = 1/ pi * d^2 * n

The Attempt at a Solution

This is an equation I have never used before and my textbook doesn't help with either. I tried to solve it by:
1 / pi * (2 cm)^2 * (20 * 10^23 / 3*10^-12 cm)
I haven't ever done something like it before, am I on the right track with this?

 

[ehild]

The mean free path is the distance between two subsequent collisions between the molecules. When the molecules touch each other, their centres are d distance apart (d is the diameter). The molecules are in motions and will collide with all molecules along their path get closer to centre -to centre than the diameter. Look at the blue molecule in the figure: When it travels along a path of length L it will collide with all molecules with centre confined in a cylinder of diameter 2d and length L.
If the density of molecules is n, the number of molecules in this cylinder is N= n*d^2*pi*L, there are N collisions along L length: the distance traveled between two subsequent collision is

$$\lambda = \frac{L}{N}= \frac{1}{d^2\pi n}$$

You know the number of the molecules in a known volume, so you can determine n.

ehild

 

[kerimek]

I can provide a response to this content by explaining the concept of mean free path and how it relates to the given scenario. The mean free path is the average distance a molecule travels between collisions with other molecules. It is an important concept in understanding the behavior of gases.

To calculate the mean free path of the molecules in the test tube, we can use the equation λ = 1/π * d^2 * n, where λ is the mean free path, d is the molecular size, and n is the number of molecules per unit volume.

Using the given values, we can substitute them into the equation as follows:

λ = 1/π * (3 * 10^-10 m)^2 * (20 * 10^23 molecules / 0.0002 m^3)

Simplifying this, we get:

λ = 1.5 * 10^-6 m or 1.5 micrometers

This means that on average, a molecule in the test tube will travel 1.5 micrometers before colliding with another molecule. This value may seem small, but it is significant in understanding the behavior of gases. It also shows that even in a seemingly empty space, there are still a large number of molecules present and constantly colliding with each other.

I hope this explanation helps you understand the concept of mean free path and how to calculate it. If you have any further questions, please feel free to ask.