An Introduction to n-th Order and Autocatalysis Reactions

1. Introduction to Thermokinetics

Chemical kinetics is an important branch of modern physical chemistry. It

  • extracts rules from various actual chemical reactions,
  • studies the reactions from a reaction rate and reaction mechanism view,

concludes into a series of important parameters like

  • time,
  • temperature,
  • concentration,
  • pressure,
  • catalyst,
  • solution, etc.

Thermokinetics is a simplification of chemical kinetics. It:

  • correlates strongly with thermal analysis techniques like DSC & TGA,
  • simplifies or combines parameters which cannot be easily studied with thermal analysis methods,
  • and finally characterizes the reaction rate simply as a function of:
    • time,
    • temperature 
    • conversion.

The basic Thermokinetics equation in derivative form is:

This function is used to describe the following formal reaction:


  • t is time,
  • T is temperature,
  • α is the normalized conversion percentage.

dα/dt  is the rate of conversion change with time; in the scope of classical Thermokinetics. It only depends on the following two functions:

(1) k(T): rate constant, the dependency of the reaction rate on temperature. Normally it takes the form of the Arrhenius equation:

Here Ea is formal activation energy in kJ/mol. From a physical chemistry perspective, this parameter corresponds with the reaction’s energy barrier. It also corresponds with the change of reaction rate as a function of temperature. A is a direct proportion factor, which is called the pre-exponential factor or frequency factor. R is the gas constant, which has the value of 8.314 J/(mol*K).

(2) f(α): reaction type, also called mechanism function or reaction type. It represents the dependency of the reaction rate on conversion, and can be treated as a mathematical description of the reaction mechanism. This part is the most versatile. There are a lot of functions to mathematically describe different reaction mechanisms. The functions most commonly used can be classified into five categories:

  1. n-th order reaction,
  2. autocatalysis reaction,
  3. phase-boundary reaction,
  4. nucleation-growth reaction,
  5. diffusion barrier reaction.


Each category includes a few different functions to refine the description for different reaction mechanisms.

As for other parameters from chemical kinetics, either they are omitted1, or  normalized2, or combined into proportion factor A3, exponential factor Ea4 and mechanism function5.

e.g., most thermal analysis measurements are under normal pressure, so P is omitted here.

e.g., the relative change in concentration is normalized to “conversion” between 0…1.

the factors which influence the molecular-contact frequencies like mole concentration, viscosity of reaction system, cross-section-area of molecule, etc.

so Ea is called “apparent activation energy” and  could differ from the true activation energy in the chemical sense.

e.g., the geometry property of the reaction interface.


So, Thermokinetics is essentially a kind of phenomenological science which is used to mathematically abstract and treat data from various thermal analysis measurement results.

Take the TGA curve as an example: since the mass loss percentage (100% à x%) can be easily converted to conversion α (0 à 1), a TGA curve is essentially a certain (α,t,T) function, and by derivative calculation one can obtain the conversion rate  dα/dt (like DTG in shape).

Similarly to this, after certain corrections, the partial peak area of the DSC signal can be converted to conversion, and after derivative calculation one will obtain the conversion rate (like DSC in shape): see Fig.1.

Fig.1 Measured heat flow dQ/dt, degree of conversion α(t), and the formal reaction rate dα/dt.

So, no matter whether the curves are TGA or DSC, they can all be treated as(dα/dt, α,t,T)relation curves and can be put into equation(1) to solve out the kinetics parameters.

Classified by different solving methods, the  approaches are model-free analysis and model-based analysis. No matter which method is used, the objective is always to obtain the kinetics parameters including Ea, A & a certain f(α). The task is to find out a complete Thermokinetics function which only contains three variables (t,T,α). This set of parameters is called Kinetics Triplet.

Then the reaction progress (α & dα/dt) with the evolvement of time (t), temperature (T) & heating rate (β= dT/dt) can be regarded as known. Starting from this Thermokinetics function, we can predict the reaction behavior under different process temperature programs, or we can optimize the process temperature program following certain rate-control demands so as to provide guidance for actual process programs and obtain the expected reaction progress.

Above is a brief yet panoramic introduction for themokinetics. As a branching science starting from physical chemistry while combining real-world measurement techniques, there are many issues that can be discussed. In this short article, we would only like to discuss two kinds of mechanism functions: n-th order and autocatalysis reaction, which you may often encounter in homogeneous reaction systems.