# Concept of mean free path

The mean free path λ of a gas molecule is its average path length between collisions and is given by,

λ = $\frac {1}{\sqrt{2} \pi d^2 \frac NV}$

Let’s look at the motion of a gas molecule inside an ideal gas, a typical molecule inside an ideal gas will abruptly change its direction and speed as it collides elastically with other molecules of the same gas. Though between the collisions the molecule will move in a straight line at some constant speed, this is applicable for all the molecules in the gas.

It is difficult to measure or describe this random motion of gas molecules thus we attempt to measure its mean free path λ.

As its name say λ is the average distance travelled by any molecule between collisions, we expect λ to vary inversely with N/V, which is number of molecules per unit volume or the density of molecules, because if there are more molecules more are the chances of them colliding with each other hence reducing the mean free path, and also λ would be inversely proportional to the diameter d of the molecules, because if the molecules were point masses then they would never collide with each other, thus larger the molecule smaller the mean free path, and it should be proportional to π times square of the diameter and not the  diameter itself because we consider the circular cross section and not the diameter itself.

## Derivation of Mean Free Path

We will derive the equation using the following assumptions, let’s assume that the molecule is spherical, and the collision occurs when one molecule hits other, and only the molecule we are going to study will be in motion and rest molecules will be stationary.

Let’s consider our single molecule to have a diameter of d and all the other molecules to be points this does not change our criteria for collision, as our single molecule moves through the gas, it sweeps out a short cylinder of cross section area $\pi d^2$ between successive collisions, for a small time t it will move a distance of vt where v is the velocity of the molecule, now if we sweep this cylinder we will get a volume of $\pi d^2$*vt so the number of point molecules inside this volume will give us the number of collisions the molecule might have,

Since N/V is the number of molecules per unit volume, the number of molecule in the cylinder will be N/V multiplied by the volume of cylinder i.e.$\pi d^2$vt, the mean free path can be derived as follows,

λ = $\frac {length~of~path~during~the~time~t}{number~of~collision~in~time~r} ~≈~ \frac {vt}{\pi d^2 vt \frac NV}~=~ \frac {1}{\pi d^2 \frac NV}$

Why we have approximated the equation is because we have assumed that all the particles are stationary with respect to the particle we are studying, in fact all the molecules are moving relative to each other, we have cancelled two velocities in the above equation but actually the v in the numerator is average velocity and v In the denominator is relative velocity hence they both differ from each other with a factor $\sqrt{2}$ therefore the final equation would be,

λ = $\frac {1}{\sqrt{2} \pi d^2 \frac NV}$

Mean free path at sea level is 0.1 micrometres.

### Mean Free Path Factors

Following are the mean free path factors:

• Density