3.2 Special Cases with Analytical Solutions

For scenarios where the contaminant only exists in the aqueous or solid-sorbent phases and where no contaminant decays into another contaminant, the individual mass loss terms are given by Equations 3.5 and 3.10.  When these are substituted into Equation 3.4, the total mass loss rate can be expressed as



The analytical solution to Equation 3.11 is



where    M_io is the initial total mass or activity of contaminant i in the source zone (g or Ci), and exp[•] is the exponential function.

Equation 3.12 could be substituted into Equations 3.5 and 3.10 to produce analytical expressions for the individual mass loss rates as well.  Equations 3.5, 3.10, and 3.12 could then be used to verify the outputs of the source-term release module for these scenarios, if none of the contaminants decay/degrade into other species of concern (e.g., other contaminants in the initial source inventory).  If they do, this set of analytical solutions will not capture the entire decay-chain behavior (as the numerical procedure will).