Diffusion is an everyday experience; a hot cup of tea distributes its thermal energy or an uncorked perfume bottle distributes its scent throughout a room. Since nonunifrom distributions tend to distribute themselves in such a way as to produce uniformity, it is reasonable to assume that the flux density, J, is proportional to the concentration gradient. For a one- dimensional system we can write
where n is the density of the physical quantity of interest. The
proportionality constant, K, is given various names depending on the
quantity that n represents. It is called the thermal
conductivity if n is heat and the coefficient of diffusion if
n is molecular concentration. For the purposes of this discussion, we
will consider 
to be a mass density along the x axis
and we will refer to the integral of this function, 
,
as the total mass.
The number of molecules between x and 
can be
approximated by 
. If we assume that the total number
of molecules is conserved (i.e., no source terms) then the number of
molecules within the interval 
will change due to the fluxes
at the two ends, x and 
.
The above equation is the differential form of the conservation of mass. A Taylor expansion about x of the last term allows the equation to be rewritten in the form
which can be combined with eq:Flux to produce the diffusion equation
