Explicit solutions for some two-step kinetics problems involving reversible isomerization

We present explicit solutions to various two-step kinetics model systems involving reversible isomerization. The first problem consists of the isomerization followed by a zeroth-order process. Exact expressions for the area under the curve and the peak, termination and crossing times were also derived and found to involve the Lambert W function. We discuss in detail the passage to the irreversible limit and test it on the initial stages of alcohol elimination in humans. The solution of the generalized problem where both isomerizing species can simultaneously undergo a zeroth-order irreversible transformation is also presented. The exact solution for the consecutive problem that starts with a zeroth-order kinetics and is followed by a reversible isomerization is also obtained. In the second part of this work, we present the explicit solution for the two-step kinetics problem consisting of an irreversible dimerization followed by a reversible isomerization of the dimer complex. Both the homo- and cross-dimerization cases are presented and the solutions were found to involve the exponential integral and the hypergeometric functions, respectively. We comment on the situations where the obtained solutions could be helpful.