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Derive fick's second law

  • Thread starter jason177
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  • #1
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Homework Statement


Imagine a narrow pipe, filled with fluid, in which the concentration of a certain type of molecule varies only along the length of the pipe (in the x direction). By considering the flux of these particles from both directions into a short segment [tex]\Delta[/tex]x, derive Fick's second law, dn/dt = D d2n/dx2 (those should be partial derivatives not normal ones) where n is the particle concentration and D is the diffusion coefficient.


Homework Equations


Jx = -D dn/dx
where J is the particle flux

The Attempt at a Solution


I don't even know where to start
 

Answers and Replies

  • #2
vanesch
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The question is: can you take Fick's first law for granted ?
 
  • #3
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It doesn't say whether we can or not so I assume we can.
 
  • #4
vanesch
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Well, then this is not so hard. What you have to do is to consider a piece of medium with thickness [itex]\Delta X[/itex] and "do the bookkeeping" of what goes in, what goes out, and hence how things change locally (also called "mass conservation") during a time [itex]\Delta t[/itex].
 
  • #5
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Alright well after playing around with it for a while I still have no idea what to do. How would I do it if we couldn't take Fick's first law for granted?
 
  • #6
vanesch
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Consider a position x0, and a position a bit further, at x0 + [itex]\Delta x[/itex].

Consider a time t0 and a time t0 + [itex]\Delta t[/itex].

Consider a density n(x,t) that is function of x.

Now consider how much is "in" the box [itex] \Delta x [/itex] at time t.

Consider how much "comes in" at the "x" wall and how much "goes out" at the "x + [itex]\Delta x[/itex] side during the time [itex]\Delta t[/itex].

That should be more than enough to get you going...
 

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