# Fick and Cottrell Law

#### Chris-jap

Hello everybody
I have got an electrochemistry probleme
In the case of a planar electrode (one dimension) the current density is proportinnal to the concentration of electroactive species: i=-nFkC C depending of time
From Fick law dC/dt=Dd2C/d2x we can found Cottrell law: i=-nFAC0(D/PIt)1/2

Do tou know How?

And my second question is
What is the fick lack for spherical coordinate and what are the expression of C(t, spheric coordinate)
And also what is the new expression of Cottrell law?

Thank You for your attention and I apologize for my bad english.
Christophe

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#### Astronuc

Staff Emeritus
With respect to Fick's Law - see http://en.wikipedia.org/wiki/Fick's_law_of_diffusion

As for spherical coordinates, see -
http://en.wikipedia.org/wiki/Nabla_in_cylindrical_and_spherical_coordinates

For a purely radial dependency, use only the $$\frac{\partial^2}{\partial{r^2}}$$ term.

With regard to this question -
From Fick law dC/dt=Dd2C/d2x we can found Cottrell law: i=-nFAC0(D/PIt)1/2

Do tou know How?
Please clarify what one is asking. "Do you know how . . . ?"

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#### Gokul43201

Staff Emeritus
Gold Member
Chris-jap said:
Hello everybody
I have got an electrochemistry probleme
In the case of a planar electrode (one dimension) the current density is proportinnal to the concentration of electroactive species: i=-nFkC C depending of time

I thought the current density was proportional to the concentration gradient dC(x,t)/dx

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#### Chris-jap

I made a mistake for the expression of current density, Goku you are right.

Thanks to you I solve one part of my problem

I found this solution for planar diffusion (not me, in a chemistry book)
C(x,t)=C0erf(x/(Dt)1/2)
and dC(x,t)/dx=C0/(PiDt)1/2 for x=0 replacing this term in the expression of current we found Cottrell law.

Now I try to found the expression of C(r,t) for a purely radial dependence and dC(r,t)/dr for r=R (R is the radius of the sphere, particles are inside the sphere and diffuse for the center to the border of the the sphere)

Do you have any other suggestion?

Chris

#### Gokul43201

Staff Emeritus
Gold Member
Chris-jap said:
I made a mistake for the expression of current density, Goku you are right.

Thanks to you I solve one part of my problem

I found this solution for planar diffusion (not me, in a chemistry book)
C(x,t)=C0erf(x/(Dt)1/2)
and dC(x,t)/dx=C0/(PiDt)1/2 for x=0 replacing this term in the expression of current we found Cottrell law.
Correct (if your initial condition is C(x,t=0) = C0 ).

Now I try to found the expression of C(r,t) for a purely radial dependence and dC(r,t)/dr for r=R (R is the radius of the sphere, particles are inside the sphere and diffuse for the center to the border of the the sphere)
This just looks like the 3D generalization of the previous 1D problem (with boundary conditions that require spherical symmetry). (ie : you are looking at 3D diffusion from a point source)

Perhaps the solution would look like dC(r,t)/dr = Kr2exp(r2/Pi*D*t) ?

You should probably ask a question in the math section (Calc and beyond) on methods for solving the differential equation for diffusion. They will be able to help better.