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Diffusion questoin

  1. Feb 14, 2009 #1
    If the gas particles in a box are uniformly distributed in the y and z directions, and linearly distributed in the x direction, is it true that the concentration won't change with time, according to the diffusion equation? I find this very unintuitive.
     
  2. jcsd
  3. Feb 15, 2009 #2

    Mapes

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    Me too. Can you go through the steps to show why you think the diffusion equation predicts that?
     
  4. Feb 15, 2009 #3
    [tex]
    \frac{dc}{dt} = D \frac{d^2c}{dx^2}
    [/tex].

    If c(x,0) = 2-x, then

    [tex]
    \frac{d^2c}{dx^2}=0
    [/tex]

    and consequently

    [tex]
    \frac{dc}{dt}=0
    [/tex]

    That is, the concentration does not change with time.
     
  5. Feb 15, 2009 #4

    Mapes

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    Setting

    [tex]\frac{\partial^2c}{\partial x^2}=0[/tex]

    for nonzero times means that you're replacing diffusing particles with new particles to keep [itex]c=2[/itex] at [itex]x=0[/itex], and you're removing all the particles at [itex]x=2[/itex] to keep [itex]c=0[/itex]. In other words, you're maintaining the linear relationship.

    For a constant amount of the diffusing species, try solving the equation for the boundary conditions

    [tex]c(x,0)=2-x[/tex]

    [tex]\frac{\partial c(0,t)}{\partial x}=\frac{\partial c(2,t)}{\partial x}=0[/tex]

    which implies impermeable boundaries. You'll find that at long times the solution approaches [itex]c=1[/itex] everywhere. Make sense?
     
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