Exact and Approximate Lumping

Consider first a discrete monomolecular system in which every species present can transform to any other one by a first-order reaction. In such a system, the kinetic equation at all possible compositions is given by Eq. (70), which is repeated here  

The matrix K must satisfy several constraints, which are best illustrated by considering a simple three-component system. The rate of formation of compo- 

Conversely, the rate of consumption of component 1 is (ki2 + k -i)tni, and hence for the diagonal elements  

Finally, there must exist at least one stationary point m (corresponding to chemical equilibrium) such that 

We now wish to substitute for this system a lower order one of NX N) pseudocomponents, the masses of which are linear combinations of the masses of the original components. In general, we want to introduce anN X N lumping matrix L such that 

---