4.2 Special Cases with Analytical Solutions

For contamination scenarios where the contaminant only exists in either the aqueous or solid-sorbent phases, where no contaminant decays into another contaminant, and when time-averaging of the volatilization term is not needed, the individual mass loss terms are given by Equations 4.5, 4.9, 4.12, and 4.14.  When these are substituted into Equation 4.4, it becomes



The analytical solution to Equation 4.18, for an initial contaminant mass of M_io, is



Equation 4.19 could be substituted into Equations 4.5, 4.9, 4.12, and 4.14 to produce analytical expressions for the individual mass loss rates as well.  Equations 4.5, 4.9, 4.12, 4.14, and 4.19 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).