Inverse kinetics problem

In short, geochemical kineticists do not have the luxury of chemical kineticists and must deal with real-world and more complicated systems. Geochemists developed the theories and concepts to deal with inverse kinetic problems, reaction kinetics during cooling, and other geologically relevant questions. These new scopes, especially the inverse theories, reflect the special need of Earth sciences, and make geochemical kinetics much more than merely chemical kinetic theories applied to Earth sciences. 

These dependences must always be taken into account in the solution of inverse kinetic problems. For example, when finding constants for eqns. (16) and (81) we must take into account that... 

We believe analysis such as has been demonstrated above will also prove to be useful in more general cases. It must be noted that this analysis places heavy demands on the inverse kinetic problem whose result is to restore summands of the steady-state kinetic equation. 

In the experiments of Hamielec et al.8S) and of Soviet researchers86 87 the method of solving an inverse kinetic problem has been used to find the dependence of values of effective constants on conversion in the thermo-initiated bulk polymerization of styrene. 

By solving the inverse kinetic problem for the conversion vs. time dependence, polymer concentration, [P], at the point of transition from the first to the second stage, and 0O are determined in... 

An important application of this approach is the radical polymerization of ethylene at high pressure. In101 an attempt is made to model a process which takes place in an industrial mixing reactor. It appears that by solving an inverse kinetic problem, the researchers have verified the pre-exponential factors and the E values for a series of elementary stages, although what these stages are is not mentioned in the paper. 

Nine elementary constants of the model were verified by solving an inverse kinetic problem. The values obtained are not cited. The investigators note that the calculated MWD values agree, though poorly, with those obtained experimentally by the GPC method. 

In commercial butadiene the presence of allenes slowly interacting with active centers is possible. By solving an inverse kinetic problem from a comparison of experimental and theoretical conversion — time curves, the values of the constants of chain termination on the allene impurity were found. The k values are not cited, but it was noted that the activation energy value depends on the type of solvent. 

This correlation is used for the solution of inverse kinetic problem. However, for such solution it is necessary to have specific information about equilibrium condition (values of equilibrium constant or equilibrium concentrations). 

The most significant stoichiometric matrix tool, which plays an important role in solving kinetic problems, is its rank. As it is known, a matrix rank defines the number of its linearly independent rows or columns. Using of the notion of a matrix rank allows to reduce the number of differential equations in a reaction mathematical model and, thereby, to make solving the direct and inverse kinetic problems easier. For example, let us consider a reaction scheme ... 

Consequently, the curve of a rate vs. a concentratiOT is linearized on rp — coordinates (Lineweaver-Burk coordinates) or on rp — rpJCs coordinates. The both types of coordinates can be used when treating experimental data for solving the inverse kinetic problem. The method of initial rates, when initial reaction rates are measured at different initial substrate concentrations, is used for this purpose. 

It is more difficult to solve the inverse problem if the mathematical model is not analytically integrable. In this case, concentrations of the compounds (or any property of the system proportional to the concentration) are not represented analytically. However, even in these cases one can use several approaches based solely on the built-in functions of Mathcad. In the following chapters of the book, we will try to show that these functions are applicable to indirect problems of rather high complexity. So, let us start with some specific examples of inverse kinetic problem. 

It is known that specialized computer software have been created for solution of different types of kinetic problems. For example, one can find a specialized software suite Dynafit that is capable of solving inverse kinetic problems. Surely, the developers of such programs usually include some tests or sample files to demonstrate advantages of their products. In particular, a real kinetic problem of a-pinene isomerization is given as one of the examples in a Dynafit tutorial. 

We want to stress out one syntax detail that is important for solving inverse kinetic problems in Mathcad. It concerns the way to define the target function. For example, the document in Fig. 4.10 shows a function that contains a vectorization operator. At the same time, the target function can be defined with index variables. This approach is not recommended - constructions with index variables cal lead to exceedingly long computation time, especially when experimental data sets contain many elements. 

Figure 4.14 shows a solution of inverse kinetic problem for a two-step consecutive reaction in Mathcad 11. The solution uses the known concentrations of an intermediate at different moments of time (this data is read fi om the corresponding data file and placed into vectors Be and te respectively). The analytical expression... 

Fig. 4.14 Solution of inverse kinetic problem in Mathcad 11 using genf it...

Fig. 4.16 Solution of the inverse kinetic problem in Mathcad 13 using genf it function...

It is interesting to check Mathcad ability to solve inverse kinetic problems with stoichiometric matrices of high dimensionality. Consider the following problem. 

In the end, we want to point out that all the examples of inverse kinetic problem given here, required calculations of kinetic parameters for the known kinetic scheme of the reaction. Real-life research often includes a much more complicated... 

Solve the inverse kinetic problem of calculating the rate constants for each stage of the reaction using given kinetic data for the current concentrations (mol L ) of the participants of a multistage reaction in time (s), as well as the kinetic scheme of the process. 

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