Question

Make a rough estimate of how far food coloring (or sugar) will diffuse through water in one minute.

Short answer

In one minute, the food coloring (or sugar) will diffuse through the water.$\mathrm{\Delta }x=1.732\times {10}^{-4}\mathrm{m}$

Step-by-step solution

Step1:Find food coloring (or sugar) will diffuse through water in one minute. 

D is the diffusion constant for sucrose (table sugar) in water at room temperature.$D=5\times {10}^{-10}{\mathrm{m}}^2·{\mathrm{s}}^{-1}$We can estimate how far a sucrose molecule will diffuse by taking $∆x$the distance over which diffusion has occurred. This region's volume is:.

$V=A\mathrm{\Delta }x$

where A denotes the container's cross sectional area If N is the total number of molecules in this region, the particle density is as follows:

$n=\frac{N}{V}=\frac{N}{A\mathrm{\Delta }x}$

The flux can be calculated by taking $∆x$the time required for the volume$A∆x$ to acquire the N molecules, yielding:

$J_x=\frac{N}{A\mathrm{\Delta }t}$

Step2:Ficks law

$J_x=-D\frac{dn}{dx}$

substitute from equations $J_x=\frac{N}{A\mathrm{\Delta }t}$and$n=\frac{N}{V}=\frac{N}{A\mathrm{\Delta }x}$into equation $J_x=-D\frac{dn}{dx}$, so:

$$\begin{gathered} \frac{N}{A\mathrm{\Delta }t}=-D\frac{d}{dx}\left[\frac{N}{A\mathrm{\Delta }x}\right] \\ =D\left[\frac{N}{A(\mathrm{\Delta }x{)}^2}\right] \end{gathered}$$

$\begin{array}{r}\to \mathrm{\Delta }t=\frac{(\mathrm{\Delta }x{)}^2}{D}\\ \to \mathrm{\Delta }x=\sqrt{D\mathrm{\Delta }t}\end{array}$

substitute with,$\Delta t=60$s and $D=5\times {10}^{-10}{\mathrm{m}}^2·{\mathrm{s}}^{-1}$, so:

$\mathrm{\Delta }x=\sqrt{5\times {10}^{-10}\times 60}=1.732\times {10}^{-4}\mathrm{m}$

$\to \mathrm{\Delta }x=1.732\times {10}^{-4}\mathrm{m}$