Fick's Law (Schroeder, Thermal Physics Page 47)

[torquemada]

This isn't a HW problem - I am just having a hard time following one of his examples.

On page 47 near Eq. 1.71 he says "As a quick example, consider a drop of food coloring added to a glass of water. Imagine that the dye has already spread uniformly through half of the glass. How long would it take to diffuse into the other half? According to Fick's law (J_x= -Ddn/dx; J=net flux, n=particle concentration), I can write very roughly



N/(AΔt)=D(N/V)/Δx (1.71)

where N is the total number of dye molecules, Δx is about 0.1m and V≈A*Δx. I've written the particle flux in terms of the same N to indicate that I want Δt to be the time for approximately all (that is, half) of the molecules to cross from one side of the glass to the other. I don't know how big a molecule of food coloring is, but it can't be too different in size from sucrose so I'll guess D=10^-9 m^2/s. Solving for Δt then gives 10^7 seconds, or almost 4 months."

I'm not following if Δx is the length of the whole glass, or half the glass? Similarly is the volume A*Δx the whole volume or half of the volume?

I am not sure how he got his rough dn/dx? He doesn't really elaborate on that.

The most confusing part though is how he relates it to the left side of the equation.

He says "I've written the particle flux in terms of the same N to indicate that I want Δt to be the time for approximately all (that is, half) of the molecules to cross" - is it all? Or is it half?

Which surface is he evaluating the flux across? I'm assuming the surface at halfway into the glass, but I just want to make sure.

Any elaboration here is greatly appreciated. Thanks

 

[Nate810]

in advance!The Δx he is referring to is half the length of the glass, since he is considering the time it takes for the dye molecules to diffuse from one side of the glass to the other. The volume A*Δx is also half of the total volume as it is the volume of the half of the glass. He is referring to the total number of dye molecules, N, to indicate that he wants Δt to be the time it takes for approximately all, or half, of the molecules to cross from one side of the glass to the other. He is therefore evaluating the flux across the surface at the halfway point of the glass.