Does the diffusion coefficient depend on units of concentration? I'm sure this is an elementary question, but I was struggling to explain the following concept to a (math) student recently and it exposed my own deficiencies in discussing units. 
In Fick's law(s), the diffusive flux $J$ is proportional to the gradient of concentration via 
$$
J = -D\nabla c
$$ The dimensions of $D$ are $L^2/T$, so for instance, a table might read that $D = 10^{-10}\text{m}^2/\text{s}$ for some particle in some medium.  My question is: it seems clear to me that the units of $c$ - for instance mass per unit volume versus number per unit volume do not matter for the value of $D$, only the choice of length and time units.  So for instance if I change from measuring $c$ in "moles per cubic meter" to "number per cubic meter" or "mass per cubic meter", I don't need to change my value for $D$; however if I change from "moles per cubic meter" to "moles per cubic centimeter", I clearly must change the value of $D$, for instance $10^{-10}\text{m}^2/\text{s} = 10^{-6}\text{cm}^2/\text{s}$. 
Is this correct?  If so, how should I answer such a question (if I change the units of some quantity in an equation, how do the constant values change) in a systematic way for myself and for students in the future?  
 A: Your intuition is correct, the diffusion constant $D$ has units of area per unit time regardless of how flux or concentration is defined (in terms of moles, mass, or number). Fick's law of course, must have the units balance on both sides of the equation.  As far as I know though, in Fick's law , the standard definition for flux and concentration is in terms of amount of substance (moles). 
Now as to the diffusion constant, (or diffusivity), the reason why it has units of area per unit time arises from more fundamental kinetic theory and statistical mechanics. 
For instance it can be shown (see for example the Wikipedia article on the Einstein relation and its references at here, for particles under an applied force obeying Maxwell -Boltzmann statistics, that the diffusivity takes  the form
$$D=\mu kT$$
Now mobility $\mu$ has units of  drift velocity (e.g m/s) divided by applied force (e.g Newtons). The Boltzmann constant $k$ has units of Joules/degrees Kelvin, and the temperature $T$ has units of degrees Kelvin.
Working out the units of diffusivity for this case , you find that they are $m^2/s$, i.e area per unit time. 
There are other expressions for diffusivity for other situations (e.g diffusion in viscous fluids, diffusion of charged particles in an electric field) but the general dimensionality of area per unit time holds.
