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I'm reading the book "An Introduction to thermal physics" by D. Schroeder and in section 1.7 about Diffusion he presents Fick's Law for 1 dimension: $$J_{x}=-D\frac{dn}{dx}$$ where n is the density of the particles.

He then gives the following example: A drop of dye is added to a glass of water. Assuming that the dye has already spread through half the glass, how long would it take for it to diffuse into the other half? He then sets up, as an approximation, the equation to get $\Delta t$: $$\frac{N}{A\Delta t} = D\frac{N/V}{\Delta x}$$ where N is the total number of particles and $V = A\cdot \Delta x$. How does one arrive at this? I would have done it by calculating $$J_{x} \cdot A \cdot \Delta t = N/2$$ and using the same expression for $J_{x}$ but then I get an extra factor of $1/2$. I am also not sure if V is the volume of the whole glass or just the upper half.

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2 Answers 2

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A drop of dye is added to a glass of water. Assuming that the dye has already diffused through half the glass, how long would it take for it to diffuse into the other half?

I beg your pardon but that's quite an oxymoronic way of looking at things. What does "has already diffused through half the glass" really mean, for instance?

In essence one is looking for the time ($t$) evolution of the distribution in space ($x$) of the solute. That's a function of the type $c(x,t)$ ($c$ for concentration)

And that requires the application of Fick's Second Law (and not the First Law):

$$\frac{\partial c(x,t)}{\partial t}=D\frac{\partial^2 c(x,t)}{\partial x^2}$$

Here's an example that solves the Second Law (from my own work) for a couple of configurations.

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  • $\begingroup$ I changed it to "spread thorugh the glass" now, thats the way it's written in the book. But in this case I am just trying to understand the approximation that was made using Ficks first law. $\endgroup$
    – eloatas
    Feb 27, 2021 at 22:04
  • $\begingroup$ That doesn't really change anything. The concentrations (that vary from one $x$ to another) change exponentially with time. The book is trying to do the impossible and use the wrong law for this problem. It's silly, frankly but not unusual (no shortage of crappy textbooks) $\endgroup$
    – Gert
    Feb 27, 2021 at 22:11
  • $\begingroup$ Thanks for the upvote! $\endgroup$
    – Gert
    Feb 27, 2021 at 22:12
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Fick's 1st Law is useless here. Only its corollary, Fick's 2nd Law helps. $$\frac{\partial c}{\partial t}=D\frac{\partial^2c}{\partial x^2}$$ Or in PDE shorthand: $$c_t=Dc_{xx}$$ So we're looking for a function of the type: $$c(x,t)$$ Boundary conditions: $$c(0,t)=c_0$$ which assumes the concentration to be constant at $c_0$ at $x=0$ $$c_x(L,t)=0$$ which assumes a 'hard' boundary at $x=L$ (no diffusion)

Initial condition: $$c(x=0 \to a)=c_0$$ which assumes the initial concentration to be $c_0$ in a narrow layer $x=0 \to x=a$ and $0$ everywhere else.

Using the Ansatz: $$c(x,t)=X(x)\Theta(t)$$ and separation of variables (not shown here) we get: $$c(x,t)=\displaystyle\sum_{n=1}^{\infty}C_n\exp(-n^2D t)c_0(\tan nL \sin nx+\cos nx)$$

$$\text{for }n=1,2,3,...$$ The coefficients $C_n$ are to be determined by Fourier expansion (not shown here)

Below are shown the general curves, $c$ v. $t$, for a related (but not identical) diffusion problem:

Diffusion

Assuming that the dye has already spread through half the glass, how long would it take for it to diffuse into the other half?

The curves clearly show that it is impossible for "the dye has already spread through half the glass" or for "it to diffuse into the other half". The question makes no sense.

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