Diffusion problem with concentration

[tigger88]

Homework Statement

Show that concentration of a substance obeys the diffusion equation if the rate at which substance leaves a region is proportional to the concentration gradient.

Homework Equations

Diffusion equation: \nabla²\phi = (1/a) (d(phi)/dt)
where Phi is a function of r, t.

The Attempt at a Solution

I'm having trouble even starting this question. I think I have to integrate concentration (C(r,t)) over a volume to get total amount of substance, but there might be some other constants I have to throw in as well... and I'm not sure what they would be...

I have a vaguely similar solved example to do with temperature, but I'm having trouble relating it to concentration.

I don't really know where to begin! I'm not looking for a full worked solution, but could someone just give me a push in the right direction?

Thanks very much!

 

[Mapes]

The equations for temperature and concentration are exactly equivalent (at this level), so feel free to adapt the temperature problem.

Perhaps you could perform a mass balance on some arbitrary volume element.

 

[tigger88]

In my temperature problem, density and spec. heat capacity are used. What are the corresponding properties I should use for concentration? Also in that problem the solution integrates the product of density, Cp and the temperature function over a small volume V. Obviously this gives Q (heat energy), but what would it give for the concentration question?

I should probably know this, but I get a mental block whenever concentration is involved with anything.

Thanks for your help!

 

[Mapes]

You should probably review http://en.wikipedia.org/wiki/Fick%27s_law_of_diffusion" concepts. Regarding the integration question: the units are a good indication of the physical interpretation (e.g., if you end up with moles, you're dealing with the total amount of the substance).

 

[tigger88]

I'm sorry, but I'm still completely in the dark about how to start this question. Forget the fact that I have a vaguely similar temperature question, I don't understand it anyway and I'm getting overly confused about where the similarities/symmetries lie.

My best guess of a vague procedure for this (which I can't actually attempt because I don't know precisely what to begin with) is to take the integral of something involving concentration (maybe some other stuff too?) over some volume V, then differentiating wrt time, using the divergence theorem to relate it to a volume integral, and equating this final expression with the initial one. Is that the right idea?

Do I start off by finding the integral (over volume V) of concentration, or do I find the integral of the product of concentration and some other stuff (if so, what's the other stuff?)?

Also, in order to find the derivative wrt time, is there something similar to the heat flow equation: dQ/dt = kA(dT/dx) that I could use?

Thanks again for the help, clearly my brain isn't working.

 

[Mapes]

Sure, you could integrate and then use the divergence theorem. You could also (perhaps more easily) consider just a differential element, assume the rate law specified in the problem description, and equate the input flow rate to the change in concentration within the element.

The matter equivalent to Fourier's Law is Fick's First Law.