Reaction extent or advancement of reaction: A definition for complex chemical reactions

The concept of reaction extent (the progress of a reaction, advancement of the reaction, conversion, etc.) was introduced around 100 years ago. Most of the literature provides a definition for the exceptional case of a single reaction step or gives an implicit definition that cannot be made explicit. There are views that the reaction extent somehow has to tend to 1 when the reaction goes to completion as time tends to infinity. However, there is no agreement on which function should tend to 1. Starting from the standard definition by IUPAC and following the classical works by De Donder, Aris, and Croce, we extend the definition of the reaction extent for an arbitrary number of species and reaction steps. The new general, explicit definition is also valid for non-mass action kinetics. We also studied the mathematical properties (evolution equation, continuity, monotony, differentiability, etc.) of the defined quantity, connecting them to the formalism of modern reaction kinetics. Our approach tries to adhere to the customs of chemists and be mathematically correct simultaneously. To make the exposition easy to understand, we use simple chemical examples and many figures, throughout. We also show how to apply this concept to exotic reactions: reactions with more than one stationary state, oscillatory reactions, and reactions showing chaotic behavior. The main advantage of the new definition of reaction extent is that by knowing the kinetic model of a reacting system one can now calculate not only the time evolution of the concentration of each reacting species but also the number of occurrences of the individual reaction events.