An Introduction to n-th Order and Autocatalysis Reactions

3. n-th order reaction

3. n-th order reaction

The n-th order reaction is the simplest and also most commonly used homogeneous reaction model. Here, the consumption of reactant molecules is taken as the only influencing factor for reaction rate. The common function is:

Here, the relative change in reactant concentration is normalized to conversion α:

where c0 is the initial concentration, c1 is the final concentration and c is the current concentration of reactant.

For example, the reactant’s initial concentration is 0.7 mol/L; after the reaction finishes, this is reduced to 0.2 mol/L (in the real world, the reactant is often not 100% consumed after the reaction). Then the relative change in concentration will be normalized to “conversion” between 0...1, i.e. in Table1:

Mole concentration  mol/L Conversion α 1 - α
0.7 0 1
0.6 0.2 0.8
0.5 0.4 0.6
0.4 0.6 0.4
0.3 0.8 0.2
0.2 1.0 0

Table1. Recalculation from mol/L concentration to conversion α and relative amount of reactant (1-a)


Here 1-α corresponds to the relative amount of reactant at any given point in time during the reaction, but it seems that the information about absolute mole concentration has been lost. However if the reactant concentration is 7mol/L instead of 0.7 mol/L  then the higher concentration has a strong impact on the reaction rate. But this influence has been detached and ascribed into the proportional factor A. For the reaction system with a higher mole concentration, the probability of molecule-contact will be higher within a certain time period. The frequency of chemical reaction is higher, so the frequency factor A will also be larger. So, if one uses the classical Thermokinetics method to build a model for the same reaction with a different mole concentration, the frequency factor may differ. The researcher should pay attention to this.

For a homogeneous reaction system, there are two commonly used n-th order reactions with integer reaction order and clear physical/chemical sense:

First order reaction (F1): n = 1, f(α)=1-α; i.e., if the temperature is fixed, the reaction rate will be directly proportional to the relative remaining amount of reactant. In other words, as the reactant is consumed, the reaction decelerates at the same ratio. This can often be seen for the monomolecular reaction A -> B inside of a homogeneous system:

  • molecular structural rearrangement,
  • spontaneous decay of radioactive atoms,
  • some liquidus decomposition reactions, etc.

Second order reaction (F2): n=2, f(α)=(1-α)2. If the temperature is fixed, the reaction rate will be proportional to the square of the relative remaining amount of reactant; this is often seen for the dual-molecular reaction in solvents, e.g., 2A -> B.

Now let’s talk about some mathematics. For the general equation of n-th order reaction (Fn):

Let’s discuss the change in f(α) versus α with different reaction orders.

Fig.2 Dependence of reacton type function f(α) vs α for n-th order reaction Fn with different values of n

In Figure 2 we can see:

1. For all curves, the maximum value always appears at the starting point. This means for n-th order reactions that if the temperature is fixed, the reaction rate is highest at the beginning, then decreases as the reaction progresses.

2. If n=1 is taken then f(α) is a diagonal line. If n increases then f(α) will decrease faster with α. This indicates with the transition of reactant that if the reaction has a higher reaction order, then the reaction rate will decrease faster. In this Figure the small values of n correspond to phase-boundary reactions of contracting geometry in heterogenius materials.

From a physical chemistry point of  view, the reaction order is always an integer and seldom exceeds 3 (a synthetic reaction participating with more than 3 molecules simultaneously is very rare). But from a formal kinetics  view, by mathematical fitting, the reaction order can be non-integer; the value can be higher than 3, or smaller than 1. This often indicates that the inner reaction mechanism is inhomogeneous or not really chemical.

For example, if you use an n-th order function to do a curve fit and the reaction order is higher than 3, this means that the reaction will rapidly decelerate with the transition of the reactant. It could be either a diffusion barrier reaction in which the product accumulates on the interface or not pure chemical reaction. If the reaction order is less than 1, it may be a phase boundary contraction reaction. In such cases: 

  • n=2/3 might be a three-dimensional phase boundary reaction in which the ball-shaped interface contracts;
  • n=1/2 might be a two-dimensional phase boundary reaction with a column-shaped contracting interface;
  • n=0 (zero order reaction) might correspond to a one-dimensional phase boundary reaction in which the interface area never changes.