# An Introduction to n-th Order and Autocatalysis Reactions

## 6. Combined auto-catalysis reaction

The single auto-catalysis functions are not so frequently used in real applications. The reason becomes clear if you substitute the Prout-Tompkins reaction type into the Thermokinetics equation:

At the start of the reaction, α=0, so dα/dt=0. But if the conversion rate is zero, this means the reaction will not occur; α will always stay zero!

We’ll arrive at the same conclusion if we take the chemical model of the Prout-Tompkins reaction type:

To start the reaction, the participation of product B is required. Thus, before the reaction, one must mix a small amount of B into the system; otherwise, if starting from 100% A without B’s participation, a first B product will never be generated, which means that the reaction will never occur.

Actually, in a real reaction system, what often occurs is the parallel existence of two reactive paths:

i.e., A can convert to B by the first pathway (Fn), just before the time where the second autocatalytical pathway (Bna), under the “catalysis” of B , becomes to be significant.

This kind of reaction can be called a combined Kamal-Sourour auto-catalysis reaction. Under the assumption that the activation energies of the two paths are the same, we got the partial case of Eq12, the function Cnm:

If we look further into the function, we’ll see that the reaction rate is presented as the sum of two items, namely Fn and Bna. Additionally, there’s a weight factor (auto-catalysis factor) Kcat to represent the contribution of Bna, or it can also be said that the frequency factors of the two paths are different.

Simplified versions of Cnm include C1 (both exponents n and m = 1, i.e., the combination of F1 and B1) and Cn (m=0, namely the reaction order of A is n while B plays the role as first-order). Cn is more commonly used.

If assuming the activation energies of the two paths are different, the general Kamal-Sourour reaction type is used:

This reactaion type is the sum of reactions Fn and Bna with a different values of Ea and a certain weight factor (or different frequency factor).

Reactions C1, Cn, Cmn are the simplified case of the general Kamal-Sourour reaction with two competitive pathways. Being the combination of an n-th order and an auto-catalysis reaction, a combined auto-catalysis reaction will exhibit accelerating performance in between that of a pure n-th order reaction and a pure auto-catalysis reaction; i.e., there will be a certain induction period, and after that, the reaction’s acceleration will be more significant than n-th order, but not as dramatic as the pure auto-catalysis Prout-Tompkins reaction. Of course, the actual accelerating behavior depends on the combination weight of the two paths.