Diffusion equation and fick law

In summary, the conversation discusses a problem with the resolution of Fick's second law of diffusion in one dimension. It begins with a discussion of the current density being proportional to the concentration of electroactive species and a solution for the initial condition. The conversation then moves on to finding the expression for i in the case of a spherical electrode and spherical diffusion. The speaker suggests using a similarity variable \eta and proposing an ordinary differential equation involving \eta. They also ask for confirmation and clarification on the proposed equations and their solvability.
  • #1
Chris-jap
5
0
Hello every body

I have previously post my question in this topic:
Physics Help and Math Help - Physics Forums > Science Education > Homework & Coursework Questions > Other Sciences > Fick and Cottrell Law

And after Goku suggestion I post my question here.

So my problem deal with the resolution with fick second law of diffusion (in one dimension)

In the case of a planar electrode (one dimension) the current density is proportinnal to the concentration of electroactive species: i=-nFkdC(x=0,t)/dt
From Fick law dC(x,t)/dt=Dd2C(x,t)/d2x

So in the case of initial condition C(x,t=0)=C0
I found this solution (not me, on internet) C(x,t)=C0erf(x/(Dt)1/2)

And so we can deduce Cottrell Law i=-nFAC0(D/Pit)1/2

Now I would like to found the expression of i in the case of spherical electrode and spherical diffusion, which species are inside the sphere (yes inside and not outside) of radius R
With C(R,t)=0 for t>0 and C(r,t=0)=C0
I would like to found the expression of C(r,t)

I think fick law in spherical diffusion is dC(r,t)/dt=D1/r2d/dr(r2d/dr(C(r,t)))

Is it right?

But now how can I find C(r,t) then dC(r,t)/dr for r=R ?

Do you have any suggestion?

Thank you for your attention
Chris
 
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  • #2
I think you're going to find a lot of standard methods for solving these equation, including Green Functions. Since there is no characteristic scale in your problem, I propose you to solve your problem in terms of a similarity variable [tex]\eta[/tex]. You should be able to work out, via scalement of the equation, that [tex]\eta=r/\sqrt{Dt}[/tex]. That is, you're scaling the radial coordinate with the diffusion length. Performing the change of variable you would end up with an ordinary equation for [tex]\eta[/tex], something like:

[tex]\frac{-1}{2}\eta\frac{\partial \C}{\partial \eta}=\frac{1}{\eta}\frac{\partial}{\partial \eta}\left(\eta\frac{\partial C}{\partial \eta}\right)[/tex]

Now it's on your own.
 
  • #3
Thank you for your help

I have done the change you propose
So now if Iunderstand C(r,t) becomes C(n)

I found a similar differential equation that the one you propose (n2 instead of n)

-1/2 n dC/n = 1/n2 d/dn(n2 dC/dn)

Are you agree with this one?

And then
nd2C/dn2 + (2-1/2*n2)dC/dn=0

Is it a non linear differential equation?

Is it possible to solve it?
 

What is the diffusion equation?

The diffusion equation is a mathematical model that describes the process of diffusion, which is the movement of particles from an area of high concentration to an area of low concentration. It is a partial differential equation that takes into account factors such as concentration, time, and the diffusion coefficient.

What is Fick's first law?

Fick's first law is a physical law that describes the rate at which particles diffuse through a medium. It states that the flux (rate of diffusion) is proportional to the concentration gradient (difference in concentration) and the diffusion coefficient. This law is often written as J = -D*∇C, where J is the flux, D is the diffusion coefficient, and ∇C is the concentration gradient.

How is the diffusion equation used in real-world applications?

The diffusion equation is used in a wide range of fields, including physics, chemistry, biology, and engineering. It can be used to model diffusion processes in gases, liquids, and solids, and is often used to understand and predict the behavior of particles in various systems. Some examples of real-world applications include drug delivery systems, pollution control, and the spread of diseases.

What are the assumptions underlying the diffusion equation and Fick's laws?

The diffusion equation and Fick's laws are based on several assumptions, including that the particles being diffused are small, the diffusion coefficient is constant, and there is no external force acting on the particles. These assumptions may not hold true in all situations, so it is important to analyze the specific conditions and make appropriate adjustments to the equations if necessary.

Can the diffusion equation and Fick's laws be applied to non-uniform systems?

Yes, the diffusion equation and Fick's laws can be applied to non-uniform systems. However, this may require more complex mathematical models and calculations to account for variations in concentration and diffusion coefficient. In some cases, it may also be necessary to use empirical data to determine the diffusion coefficient instead of assuming a constant value.

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