Finding steady state concentration in diffusion without initial conditions

[superwolf]

Homework Statement

2. The attempt at a solution

a) The steady state concentration is the concentration when t --> infitine, right? How can I find that when I don't know c(x, 0)?

 

[CarlB]

You have two fluxes, one due to diffusion, the other due to the potential. In steady state, these cancel each other. So try writing those equations down and solving them.

 

[superwolf]

Flux due to diffusion = $$D\frac{dc}{dx}$$, right?

How about the flux due to the potential? Is that simply $$ax^2$$?

It that is true, I get $$c=\frac{a}{3D}x^3$$

I still don't know what to do with b) and c) though...

 

[CarlB]

No, you need to convert the potential into a force. You do that by taking the gradient.

Now is it the case that $$D\frac{dc}{dx}$$ is a force? If it is, then maybe the next step would be to set these forces equal, diffusion and that from the potential. As far as getting a force from diffusion, would that have something to do with "pressure"? I hope there is something in your text or notes that will further you on this.

By the way, I think Einstein was the one who originally solved this question for Brownian motion. One of the applications is to calculate the density of the Earth's atmosphere as a function of altitude. In this case, the potential is due to gravity, which is balanced against diffusion. (I.e. that's why we can breathe at even high altitude places a couple miles above the lowest points on the planet.) Here's an article:
http://psas.pdx.edu/RocketScience/PressureAltitude_Derived.pdf