Mathematical modelling of the reaction of condensation telomerization and the investigation of the model
A mathematical model of the distribution of mixture components in the equilibrium condensation telomerization is developed depending on the ratio of the amounts of monomers and telogen, as well as the number of HX as regulating parameters, the computer implementation of the model is carried out, and its study is carried out by numerical simulation. The model is based on the well-known schematic diagram of the flow of the condensation telomerization process under the assumption of equal reactivity of the same functional groups (Flory principle). Based on the analysis of the flow pattern of the process, 6 structural elements are identified, reproducible at each stage associated with an increase in the degree of polymerization based on 4 basic components. It is proved that the equilibrium concentrations of these elements, depending on the polymerization degree, depend on the equilibrium concentration of products with a degree of polymerization 1 and are described by infinite geometric progression with the same denominator. According to the physical content of the task, this progression must be convergent. Equations of material balance of components are contained in the form of a system with 4 equations containing infinite sums. It is possible to minimize these sums using the properties of geometric progressions and to obtain a closed system with 4 nonlinear equations for the equilibrium concentrations of the base components.
The Monte Carlo method is used to study the features of the numerical solution of the system of equations of the model. It is found that with a random choice of initial approximations of solutions from an admissible region, the system contains 4 roots, of which 2 contain positive and negative components and are false, and 2 have completely positive components. A valid criterion for finding a real root has a physical meaning based on the calculation of the denominator of a geometric progression. The possibilities of practical use of the model are discussed
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