Chapter 2.P2, Problem 1P

Derive Eq. $\left(3\right)$ from Eqs. $\left(1\right)$ and $\left(2\right)$ and show that $\alpha =v_d{A}_s{c}_0/m_0^{\gamma }$.

Equation $\left(1\right)$: The equation describing the rate of DNAPL mass discharge from the source region is

$\frac{dm}{dt}=-A_s{v}_d{c}_s\left(t\right)$, $\left(1\right)$

Whereas an algebraic relationship between $c_s\left(t\right)$ and $m\left(t\right)$ is postulated in the form of a power law,

$\frac{c_s(t)}{c_0}={\left[\frac{m(t)}{m_0}\right]}^{\gamma }$, $\left(2\right)$

in which $\gamma >0$ is empirically determined. Combining Eqs. $\left(1\right)$ and $\left(2\right)$(Problem $1$) yields a first order differential equation

$\frac{dm}{dt}=-\alpha m^{\gamma }$ $\left(3\right)$

That models the dissolution of DNAPL into the groundwater flowing through the source region.