Chapter 24, Problem 6P

Fick's first diffusion law states that

$\text{Mass flux}=-D\frac{dc}{dx}\left(\text{P24}\text{.6}\text{.1}\right)$

where mass flux 5 the quantity of mass that passes across a unit area per unit time $\left(g/{\text{cm}}^2\text{/s}\right),D=\text{a}$ diffusion coefficient $\left({\text{cm}}^2/\text{s}\right),c=$ concentration, and $x=\text{distance }\left(\text{cm}\right)$. An environmental engineer measures the following concentration of a pollutant in the sediments underlying a lake $(x=0$ at the sediment-water interface and increases downward):

x, cm	0	1	3
$c,10°\text{g}/{\text{cm}}^3$	0.06	0.32	0.6

Use the best numerical differentiation technique available to estimate the derivative at $x=0$. Employ this estimate in conjunction with Eq. $\left(\text{P24}\text{.6}\text{.1}\right)$ to compute the massflux of pollutant out of the sediments and into the overlying waters to compute the mass flux of pollutant out of the sediments and into the overlying waters $(D=1.52\times {10}^{-6}{\text{cm}}^2\text{/s)}$. For a lake with $3.6\times {10}^6{\text{m}}^2$ of sediments, how much pollutant would be transported into the lake over a year's time?