As a result of thermal motion there is an inherent transport of molecules through a gases and liquids in thermal equilibrium. This is as opposed to solids in which atoms generally stay but at a specific site of the lattice. Here we shall focus on transport in gases and shall set ourselves the task of calculating how fast atoms or molecules diffuse through a gas.
As we realized in a previous numerical example the motion of individual atoms
in a gas at room temperature is actually quite fast, typically 1000 m/s. However
the effective transport velocity is considerably slower because of the constant
collision between individual atoms. If we denote the cross sectional area
of an atom by 
then it can be shown the mean time between collisions
is
Correspondingly the mean distance traveled between collisions is
for the cross section we typically use
where r is a characteristic radius of the atom. The factor 2 comes from realizing that
the atoms can be separated up to a distance 2r and still collide!
Inserting
and
we get
So collisions occur often and distances traversed between collisions is small. Not however
that the distances traversed are very large compared to the size of a single
atom and this is what makes it possible to neglect collisions
for the purpose of calculating the pressure exerted by the gas.
The motion of an individual atom is diffusive and
it can be shown that in a time t it progresses a distance
Based on the values derived for 
and 
we calculate the
time taken for atoms to diffuse from one end of the auditorium to the other:
putting in numbers we get
A surprisingly long time indeed. Clearly we rely on convection to
spread the smells of perfume etc. Note however that even though no single
molecule moves very far, thermal equilibrium is spread at a speed close to the
speed of sound in the gas because energy travels through the system in waves at that
speed.