Q. 1.67 - An Introduction to Thermal Physics

Question

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

Step-by-Step Solution

Answer

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

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:

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

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