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]

$$\frac{dc}{dt} = D \frac{d^2c}{dx^2}$$.

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

$$\frac{d^2c}{dx^2}=0$$

and consequently

$$\frac{dc}{dt}=0$$

That is, the concentration does not change with time.

 

[Mapes]

Setting

$$\frac{\partial^2c}{\partial x^2}=0$$

for nonzero times means that you're replacing diffusing particles with new particles to keep $c=2$ at $x=0$, and you're removing all the particles at $x=2$ to keep $c=0$. 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

$$c(x,0)=2-x$$

$$\frac{\partial c(0,t)}{\partial x}=\frac{\partial c(2,t)}{\partial x}=0$$

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