Differential Equations from Chemical Kinetics

Chemical kinetics for an elementary reaction, according to the law of mass action, leads to a differential equation that is generally of the form 

For some reactions the rate constant kj can be very large, leading potentially to very rapid transients in the species concentrations (e.g., [A]). Of course, other species may be governed by reactions that have relatively slow rates. Chemical kinetics, especially for systems like combustion, is characterized by enormous disparities in the characteristic time scales for the response of different species. In a flame, for example, the characteristic time scales for free-radical species (e.g., H atoms) are extremely short, while the characteristic time scales for other species (e.g., NO) are quite long. It is this huge time-scale disparity that leads to a numerical (computational) property called stiffness. 

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On a more positive note, multimedia enable us to respond pedagogically to the shifting paradigms of science, where, as remarked above, numerical analysis is replacing analytical solutions of equations. Through it, we can present computational chemistry, for instance, and pursue the fascinating complexity that arises from systems of coupled differential equations in chemical kinetics. 

Chemistry has not been reduced to physics "How are the nonlinear differential equations of chemical kinetics derived from linear quantum mechanics "... 

Differential equations governing the kinetics of chemical reaction systems may be thought of as arising from statements of mass conservation. For example, consider the well mixed system illustrated in Figure 3.1, containing reactants A and B in a dilute system of constant volume, V. 

The drainage kinetics can be formally described using the equations of chemical kinetics. This yields expressions for the dependence of the volume of the liquid outflow on the time with respect to the volume of liquid in the foam [7,14,72], So Eq. (5.50) about the liquid volume in a foam can be derived from the following first order differential equation... 

This can be accomplished by setting up appropriate kinetic equations arid subsequent integration of the resulting differential equations, from which the population of the various states in which the positrons exist o-Ps and PsM can be found as a function of time. From these values and the positron annihilation constants for these states, an equation for the time dependent two photon annihilation rate can be obtained, which in turn allows the determination of the chemical reaction rate constants by utilizing sophisticated nuclear chemical lifetime measurement techniques. 

To describe the oxidation of methanol in supercritical wato, kinetic models were created partially independently by different groups. These models consist of elementary reactions [24,25,36,37] or lumped chemical reactions [26] in the mathematical form of ordinary differential equations. From a chemical point of view these models are very similar with small differences in single values of kinetic parameters. 

No adequate compilation of those solutions of the diffusion equation applicable to the diffusion of matter has as yet been made. The author has become aware of the difficulty of obtaining suitable solutions in a number of studies of diffusion, and it is hoped that this chapter will provide a source of reference for such solutions. Equations of chemical kinetics can be used readily in differential or integral form, but the equations of diffusion kinetics require for their treatment a special and often laborious technique. Many of the cases which can arise have not yet been solved rigorously, though the field is a rich one both from the mathematical and the experimental viewpoints. The present chapter aims at giving some solutions of the diffusion equation in a form in which they may be applied, together with cases of diffusion systems in which the boundary conditions are those of the diffusion equation solved ... 

It turns out that there is another branch of mathematics, closely related to tire calculus of variations, although historically the two fields grew up somewhat separately, known as optimal control theory (OCT). Although the boundary between these two fields is somewhat blurred, in practice one may view optimal control theory as the application of the calculus of variations to problems with differential equation constraints. OCT is used in chemical, electrical, and aeronautical engineering where the differential equation constraints may be chemical kinetic equations, electrical circuit equations, the Navier-Stokes equations for air flow, or Newton s equations. In our case, the differential equation constraint is the TDSE in the presence of the control, which is the electric field interacting with the dipole (pemianent or transition dipole moment) of the molecule [53, 54, 55 and 56]. From the point of view of control theory, this application presents many new features relative to conventional applications perhaps most interesting mathematically is the admission of a complex state variable and a complex control conceptually, the application of control teclmiques to steer the microscopic equations of motion is both a novel and potentially very important new direction. 

In chemical kinetics, one finds linked sets of differential equations expressing the rates of change of the interacting species. Overall, mathematical models have been exceedingly successfiil in depicting the broad outlines of an enormously diverse variety of phenomena in nature. Some scientists have even commented in surprise at how well mathematics works in describing nature. So successful have these mathematical models been that their use has spread from the hard sciences to areas as diverse as economics and the analysis of athletic performance [3]. 

Since the integral is over time t, the resulting transform no longer depends on t, but instead is a function of the variable s which is introduced in the operand. Hence, the Laplace transform maps the function X(f) from the time domain into the s-domain. For this reason we will use the symbol when referring to Lap X t). To some extent, the variable s can be compared with the one which appears in the Fourier transform of periodic functions of time t (Section 40.3). While the Fourier domain can be associated with frequency, there is no obvious physical analogy for the Laplace domain. The Laplace transform plays an important role in the study of linear systems that often arise in mechanical, electrical and chemical kinetic systems. In particular, their interest lies in the transformation of linear differential equations with respect to time t into equations that only involve simple functions of s, such as polynomials, rational functions, etc. The latter are solved easily and the results can be transformed back to the original time domain. 

Some economies are possible if equilibrium is assumed between selected compartments, an equal fugacity being assignable. This is possible if the time for equilibration is short compared to the time constant for the dominant processes of reaction or advection. For example, the rate of chemical uptake by fish from water can often be ignored (and thus need not be measured or known within limits) if the chemical has a life time of hundreds of days since the uptake time is usually only a few days. This is equivalent to the frequently used "steady state" assumption in chemical kinetics in which the differential equation for a short lived intermediate species is set to zero, thus reducing the equation to algebraic form. When the compartment contains a small amount of chemical or adjusts quickly to its environment, it can be treated algebraically. 

The solution of problems in chemical reactor design and kinetics often requires the use of computer software. In chemical kinetics, a typical objective is to determine kinetics rate parameters from a set of experimental data. In such a case, software capable of parameter estimation by regression analysis is extremely usefiil. In chemical reactor design, or in the analysis of reactor performance, solution of sets of algebraic or differential equations may be required. In some cases, these equations can be solved an-... 

In the case of classic chemical kinetics equations, one can get in a few cases analytical solution for the set of differential equations in the form of explicit expressions for the number or weight fractions of i-mcrs (cf. also treatment of distribution of an ideal hyperbranched polymer). Alternatively, the distribution is stored in the form of generating functions from which the moments of the distribution can be extracted. In the latter case, when the rate constant is not directly proportional to number of unreacted functional groups, or the mass action law are not obeyed, Monte-Carlo simulation techniques can be used (cf. e.g. [2,3,47-52]). This technique was also used for simulation of distribution of hyperbranched polymers [21, 51, 52],... 

The unambiguous identification of the extraction rate regime (diffusional, kinetic, or mixed) is difficult from both the experimental and theoretical viewpoints [12,13]. Experimental difficulties exist because a large set of different experimental information, obtained in self-consistent conditions and over a very broad range of several chemical and physical variables, is needed. Unless simplifying assumptions can be used, frequently the differential equations have no analytical solutions, and boundary conditions have to be detemtined by specific experiments. 

In practice, of course, it is rare that the catalytic reactor employed for a particular process operates isothermally. More often than not, heat is generated by exothermic reactions (or absorbed by endothermic reactions) within the reactor. Consequently, it is necessary to consider what effect non-isothermal conditions have on catalytic selectivity. The influence which the simultaneous transfer of heat and mass has on the selectivity of catalytic reactions can be assessed from a mathematical model in which diffusion and chemical reactions of each component within the porous catalyst are represented by differential equations and in which heat released or absorbed by reaction is described by a heat balance equation. The boundary conditions ascribed to the problem depend on whether interparticle heat and mass transfer are considered important. To illustrate how the model is constructed, the case of two concurrent first-order reactions is considered. As pointed out in the last section, if conditions were isothermal, selectivity would not be affected by any change in diffusivity within the catalyst pellet. However, non-isothermal conditions do affect selectivity even when both competing reactions are of the same kinetic order. The conservation equations for each component are described by... 

Another additional chemical complication can arise from the presence of quenching reagents which deactivate the reactive polymers. This kinetic quenching mechanism may also be included in the formalism through the addition of an additional differential equation. A more thorough treatment of these extensions and their applications to polymerization reactions is currently in progress. 

In Section 4.7, we discussed the relaxation process of SE s in a closed system where the number of lattice sites is conserved (see Eqn. (4.137)). A set of coupled differential equations was established, the kinetic parameters (v(x,iq,x )) of which describe the rate at which particles (iq) change from sublattice x to x. We will discuss rate parameters in closed systems in Section 5.3.3 where we deal with diffusion controlled homogeneous point defect reactions, a type of reaction which is well known in chemical kinetics. 

However, a question arises - could similar approach be applied to chemical reactions At the first stage the general principles of the system s description in terms of the fundamental kinetic equation should be formulated, which incorporates not only macroscopic variables - particle densities, but also their fluctuational characteristics - the correlation functions. A simplified treatment of the fluctuation spectrum, done at the second stage and restricted to the joint correlation functions, leads to the closed set of non-linear integro-differential equations for the order parameter n and the set of joint functions x(r, t). To a full extent such an approach has been realized for the first time by the authors of this book starting from [28], Following an analogy with the gas-liquid systems, we would like to stress that treatment of chemical reactions do not copy that for the condensed state in statistics. The basic equations of these two theories differ considerably in their form and particular techniques used for simplified treatment of the fluctuation spectrum as a rule could not be transferred from one theory to another. 

Section III discusses briefly (1) the relation between our logical equations and the differential equations as used in chemical kinetics (2) some aspects of logical versus numerical iteration methods (3) the possible application of our method (initially developed for genetic purposes) to other fields, and more particularly chemical kinetics and (4) the possibility of using this method al rovescio, that is, in a synthetic (inductive) way. In this perspective we assume that the essential elements of a system have been correctly identified and we ask to what extent one can proceed rationally from the observed behavior toward sets of interactions which account for this behavior. 

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