Direct kinetics problem

A direct kinetic problem consists of calculating multi-component reaction mixture compositions and reaction rates on the basis of a given kinetic model (both steady-state and unsteady-state) with the known parameters. Reliable solution for the direct problem is completely dependent on whether these parameters, obtained either on theoretical grounds or from special experiments, have reliable values. Modern computers can solve high-dimensional problems. Both American and Soviet specialists have calculated kinetics for the mechanisms with more than a hundred steps (e.g. the reac-... 

Let us consider a direct kinetic problem for simple reactions in the closed exothermic system (the volume and the temperature are constant). If we assume the correspondence between kinetic and stoichiometric equations, the scheme of the simple reaction with sole reagent going in one stage could be written as ... 

Concentration Ca is called an initial concentration, and the values Ca(() in each moment of time - the current concentrations. An analytic solution of the direct kinetic problem is a definition of the functional cmmection between current concentration and time. 

In (1.1) variables are separated, therefore its solution could be accomplished in MathCAD (Fig. 1.1). Prior to the interpretation the results of the solution we need to examine document in Fig. 1.1 in detail. In the strict sense MathCAD does not have on-board sources for the analytic solution of the differential equations, therefore given solution is obtained in a little artificial way. Firstly, the variables were prehminarily separated, and the equation was represented in the form of equality, whose both parts were completely prepared for the integration. Secondly, both parts of the equation were written in such a way, that the names of the integration variables differed from the names of the variables, used as the limits of integration. However, we have obtained a solution of the direct kinetic problem, which allows writing a time-dependence of the reagent s current concentration ... 

Fig. 1.1 Analytic solution of the direct kinetic problem for the simple reaction by the means of...

Fig. 1.8 Solution of direct kinetic problem for second-order reversible reaction A + B C + D (on-line calculation http //twt.mpei.ac.ru/MCSAVorksheets/Chem/ChemKin-l-08-MCS.xmcd)...

We can also use special visually oriented interface elements, created by authors on the basis of bump pack Maplets, to solve direct kinetic problem, starting from Maple 9. Let us demonstrate some facilities of this package. Particularly, command... 

Even in the presence of two elementary stages kinetic equations of successive reaction become noticeably more complicated, if at least one of them passes due to the patterns of second-order reaction. Mathematic analysis of such mechanisms with the object of symbolic solution of direct kinetic problem in Mathcad is difficult, therefore in this case it is appropriate to use Maple s analytic facilities. It is still possible to get integrated forms of equations for some kinetic schemes in Maple. 

Fig. 1.18 Solution of direct kinetic problem for self-catalyzed reaction...

The Classical Matrix Method for Solving the Direct Kinetic Problem... 

Figure 2.5 demonstrates the Mathcad document designed to form all the vectors and matrices necessary for solving the direct kinetic problem. The built-in function e igenvals should be used to find eigenvalues of a constant rate matrix. A matrix of eigenvectors is calculated with the help of eigenvecs. 

Fig. 2.5 Solving the direct kinetic problem by the matrix method forming the necessary matrices...

Let us consider that before starting the reaction, the concentrations of A in the each reactor are the same and equal to Co mol L . Additionally, let us intentionally complicate the conditions of the direct kinetic problem. Assume that in the each reactor the reactimi mixture has different temperature and this temperature remains constant during the process (here, of course, we have to assume that new portions of the solution immediately takes a given reactor temperature). Thus, this means that the process is controlled by the different rate constants k, k2, k in the each reactor. 

Let us carry out a check of the steady-state principle. For this purpose, let us calculate the time dependence of the end product formation rate from the relationships obtained by accurate solving the direct kinetic problem (see Table 2.1). Next, let us compare the result with the calculations from obtained formula (2.9). The corresponding plots represented in Fig. 2.17 show that the behaviour of the both curves coincide after less than 0.5 s at given values of the rate constants satisfying the condition ki > k. This indicates applicability of the steady-state concentration method to the considered model of the consecutive reaction. 

The solution of the direct kinetic problem is shown in Fig. 3.21. We can see that in the case of the assumed rate constants the microorganism population is oscillating. The critical point type is the node, because all the Jacobian eigenvalues... 

So, the rate constant depends on temperature. In the case of the altering temperature the rate constant also becomes a function of time. Consequently, when solving the direct kinetics problem, we have to add the corresponding equations (the temperature over time relationships) to the reaction model. 

Here 8 is a molar extinction coefficient / is an optical path length (cuvette length). Behavior of the concentrations for the components A, B, and P over time is known from the direct kinetic problem solution (see Chap. 1). After some substitution and simplifications, one can obtain the following equation for optical density in time ... 

An extension pack DAEP (Data Analysis Extension Pack) includes some functions that might be useful for solving inverse problems requiring nonlinear fitting. This pack is automatically installed with the latest Mathcad versions. In Chap. 1, we obtained a solution for direct kinetic problem of the following consecutive second-order reaction ... 

Let s solve the problem in the Mathcad environment. One of the solutions for kinetic parameters is shown in Fig. 4.20. In fact, the beginning of the calculations corresponds to a solution of direct kinetic problem with arbitrary values of the constants (more precisely, logarithms of the constants) as initial guesses. Note an interesting approach implemented here to form the vector of the right parts of the differential equation system. Rate constants are used as the functions on time therefore, the vector includes four null elements (according to the number of constants) that represent the right parts of four differential equations ... 

Third-order raie-way reaction follows the scheme A + 7B Products. Rate constant equals to 0.005 mol s . Initial reactants concentrations equal to Ao = 1 molL andRo = 2 mol L . Build kinetic curves Aft) and R(i) based on the results of the analytic solution of the direct kinetic problem. "What are the concentrations of the reactants 500 s after the start of the reaction Assume that kinetics of the reaction fully follows its stoichiometiy. 

Solve direct kinetic problem for a second-order reaction A + B Products numerically and analytically. Initial concentrations Aq and Bq are 0.50 and 0.25 mol correspondingly. Rate constant equals to 0.05 L moP s ... 

Find values of rate constants for elementary stages of the process for a chosen temperature in the Internet (e.g. http //www.nist.gov) and solve direct kinetic problem, assuming arbitrary value for an initial concentration of the reactant. Verify the applicability of the quasistationery principle for a given kinetic scheme. 

These models satisfactorily describe the experimental data in qualitative level, however, the munerical test was not done, so we have posed the problem - to solve so-called direct kinetic problem and to compare experimental and calculated kinetic curves. 

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