Mean free path is the distance traveled by a particle such as electron, in between two successive collisions. It could be an atom, molecule in gases or liquids, or electrons in any medium or photons too.
l = 1 / sigma * n
where sigma = effective cross sectional area offered by each particle, for collision
n = number of particles per unit volume = concentration of particles
One can prove this by estimating probability of a particle hitting or being stopped by collision in a small volume dV of area A & thickness dx:
From a beam many particles are travelling parallelly through a vertical slab of area A and thickness dx. In this volume we have n A dx number of particles. Each particle has an effective surface area for collision = sigma. It is because only one side may be available for collision etc. Total surface area of these that is available for collision is n A dx sigma. A particle that entered into this vertical slab of area A can encounter a surface area n A dx sigma as obstruction or wall. So the expression for probability :
P (particle stopped or collision in volume dV) = n A dx sigma / A = n sigma dx
Now the Intensity I of a beam of particles is decreased by an amount dI. dI is proportional to Intensity and the probability P.
dI = - I n sigma dx -ve because intensity decreases due to collisions
dI / I = - n sigma dx In I = - n sigma x => I(x) = I0 exp (- n sigma x)
I0 is I at x=0
Find the probability of a particle being stopped from x to x+dx. This is
P1(x) = [ I(x) - I(x+dx) ] / I0 = (n sigma) exp(-n sigma x) dx
Expected value of x is Integral from 0 to infinity of x * P1(x)
integral of x (n sigma) exp ( - n sigma x ) dx = 1/n sigma
We call 1 / n * sigma = mean free path length = l