Diffusion questoin

1. Feb 14, 2009

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.

2. Feb 15, 2009

Mapes

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

3. Feb 15, 2009

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.

4. Feb 15, 2009

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?