Why is the diffusion coefficient in Fick's law squared?

[jjmclell]

Hi there,

I'm trying to wrap my head around Fick's first law of diffusion (for one dimension):

$$J_{x} = -D \frac{\partial \phi}{\partial x}$$

I understand that $$\phi$$ is the concentration in units amount/volume and that $$x$$ is position on the gradient in units length. What I don't understand is why $$-D$$ is in units area/time. If we're talking one dimensional diffusion, why do we bring area into the equation? Or, put another way, why do we square length in the diffusion coefficient?

Thanks,

jjmclell

 

[Andy Resnick]

It's in units of area/time because it represents the flux of material through a (control) surface

 

[sir-pinski]

an

The reason for the squared diffusion coefficient in Fick's law is because it is related to the rate of diffusion, which is dependent on both the concentration gradient and the distance traveled by the diffusing particles. The concentration gradient is measured in units of concentration/length, while the distance traveled is measured in units of length. When we multiply these two quantities together, we get units of concentration*length/length, which simplifies to just concentration. However, since the diffusion coefficient is a constant that is specific to the material and conditions, it must have units that cancel out with the concentration term. This is why the diffusion coefficient is squared, as it cancels out the length unit in the concentration gradient term, leaving us with just concentration.

In other words, the squared diffusion coefficient accounts for the fact that the rate of diffusion is a function of both the concentration gradient and the distance traveled. This ultimately helps us understand how quickly or slowly a substance will diffuse through a given medium.

I hope this helps clarify your understanding of Fick's law and the squared diffusion coefficient. Let me know if you have any further questions.