\section{Derivations}
In this work, the implementation of the Kempers model, as well as the implementation of the Enskog solutions for a general spherical potential requires the derivation of various relationships. This section details these derivations. Implementation specific details and numerical methods are presented in Section \ref{sec:method}.

\input{derivations/eos_differentials}

\input{derivations/dynamics}

\input{derivations/effective_density}