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