Diffusion Ques: Unintuitive Concentration Change Over Time?

[superwolf]

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.

 

[Mapes]

Me too. Can you go through the steps to show why you think the diffusion equation predicts that?

 

[superwolf]

[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.

 

[Mapes]

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?