The Set of Functions Corresponding to First-Order Chemical Kinetics Problems, Its Structure, and Related Inverse Problems

First-order chemical kinetics can be considered—at least mathematically—the most common ones since, under certain conditions, many other types of reactions also give a similar mathematical model, that is, a linear ODE system. In this chapter, we propose to study the set of functions that can be solutions to first-order chemical kinetics problems. First, we observe that this set of functions cannot have a vector space structure, although it is possible to define a norm. Also, the stability of the solutions is proved, and the importance of this result in chemistry is analyzed. Second, we move on to the central core of the proposal: the related inverse problems. This set of functions gives rise to several interesting inverse questions. For example: 1) Existence: Given a function of this set, is there always a mechanism that has it as a solution? 2) Uniqueness: Can two different mechanisms give the same solution? 3) Stability: Do small changes in the norm of the function always produce small changes in the corresponding mechanism? The chapter ends with a relevant conjecture and two open questions about other inverse problems.