Four Reactions in a Circle

The equivalent Laplace transformed expression for concentration of reactant A for a system of four reactions in circle, assuming that all the reactions in the cycle obey simple, hrst-order kinetics, can be derived as 

The conditions where the concentration can be expected to exhibit subcritical damped oscillations can be found when the roots of the following equation becomes complex  

It can be seen that a is the sum of all four reaction rate constants, P is the sum of products of aU possible pairs of the reaction rate constants, and % is the sum of products of all possible triple products of rate constants in the system of reactions in circle. Equation (12.76) can be converted to the depressed cubic equation by use of the substitution  

This method was developed in the Renaissance period [6]. The depressed cubic without the quadratic term will then be 

The complex roots to Equation (12.83) shall occur when D 0, where 

---

Thermal polymerization kinetics that obey the Diels-Alder mechanism were studied. Expressions for rate of propagation and molecular weight were derived. The rate of propagation varies as the cube of monomer concentration. Subcritical damped oscillations can be expected under certain conditions. Reaction in a circle representation of free radical reactions was used to analyze the stability of reactions. Oscillations in concentration were discussed for systems of three and four reactions in a circle and the results generalized to n reactions in a circle. 

---