Application of Fick's First law (diffusion problem)

[jokkon]

Homework Statement

Molecules with diffusion coefficient of 1.0 x 10^-10 m^2s^-1 are released at a constant rate of 10^10 molecules/s in the middle of a large pool and dffuse away ( assume the 3-dimensional pool is of infinite size). What is the steady state concentration 1 cm away from the source? [Hint: Consider molecules diffusing out through a spherical surface of radius r, with a source at the centre of the sphere].

Homework Equations

Fick's First law: Flux = -Dgradient(n) with n being the concentration of the molecules

The Attempt at a Solution

I took flux to be equal to the rate at which molecules are being added/surface area of a sphere. Plug it into Fick's first law and then isolate dn/dr (since the source is a point phi and theta should be trivial) and integrate. I have no idea if the assumption that flux = rate/surface area is correct especially since the rate is particles added to the system. If this is not right how should I find the flux? Thank you in advance.

 

[willem2]

I took flux to be equal to the rate at which molecules are being added/surface area of a sphere. Plug it into Fick's first law and then isolate dn/dr (since the source is a point phi and theta should be trivial) and integrate. I have no idea if the assumption that flux = rate/surface area is correct especially since the rate is particles added to the system.

It is correct. If n molecules/second are released at the center of the sphere, n molecules/second
must go out through the surface of the sphere after the steady state condition is reached.
you can convert the amount of molecules entereing, the concentration and the flux to mol/second
mol/m^3 and mol/m^2 s, instead of particles/second if you want.

 

[jokkon]

thank you for your reply! I am happy that I got it right :D
I can see that all molecules going into the center has to go out of the sphere at some point in time, but the fact that they diffuse out of the sphere at the same rate as the particles entering the system doesn't seem very intuitive to me. Is there any way to show that must be the case?

 

[willem2]

thank you for your reply! I am happy that I got it right :D
I can see that all molecules going into the center has to go out of the sphere at some point in time, but the fact that they diffuse out of the sphere at the same rate as the particles entering the system doesn't seem very intuitive to me. Is there any way to show that must be the case?

well in the steady state, it seems obvious that they must become the same in the long term, because otherwise the particles would pile up inside the sphere to infinite density, or there would be more coming out then going in.