Kinetic equations, distinguishability

As pointed out in the foregoing, there are two specific peculiarities qualitatively distinguishing these systems from the classical ones. These peculiarities are intramolecular chemical inhomogeneity of polymer chains and the dependence of the composition of macromolecules X on their length l. Experimental data for several nonclassical systems indicate that at a fixed monomer mixture composition x° and temperature such dependence of X on l is of universal character for any concentration of initiator and chain transfer agent [63,72,76]. This function X(l), within the context of the theory proposed here, is obtainable from the solution of kinetic equations (Eq. 62), supplemented by thermodynamic equations (Eq. 63). For heavily swollen globules, when vector-function F(X) can be presented in explicit analytical form... 

Before we can find the form of the concentration term in a rate expression, we must distinguish between different types of reactions. This distinction is based on the form and number of kinetic equations used to describe the progress of reaction. Also, since we are concerned with the concentration-dependent term of the rate equation, we hold the temperature of the system constant. 

Chemical reactions are classified usually as diffusion-controlled, whose rate is limited by a reactant spatial approach to each other, and reaction-controlled (kinetic stage), whose rate is limited by a reaction elementary event. For systems with ideal reactant mixing considered in Section 2.1.1, there is no mechanism of reactant mutual approach. On the other hand, the kinetic equations (2.1.40) distinguish between reaction in physically infinitesimal volumes and the distant reactant motion in a whole reaction volume. In the absence of reaction particle diffusion is described by equation... 

Specificity of a concrete system accounts for the source of the appearance of a small parameter and for its type. For homogeneous reactions, a small parameter is usually a ratio of rate constants for various reactions some reactions are much faster than the others. For just such a small parameter Vasiliev et al. [25] distinguished a class of chemical kinetic equations for which the application of the quasi-stationarity principle is correct (they considered a closed system). 

It is this calorimetric form of this equation (and the more complex equations that can be developed and used to describe sequential, parallel and more complex reacting systems) that can be analysed to yield values for the target parameters, n, k, H, K, G, S and E. References 6 and 7 provide details on the derivation of calorimetric kinetic equations and describe how to manipulate calorimetric data to determine equilibrium constants, Gibbs functions, entropies and activation energies. The approach described shows that, under normal storage conditions ( room temperature i.e. 298 K and ambient humidity), it is possible, from only 50 h of power time (the calorimetric output is of W vs. t) data, through these techniques, to distinguish between a... 

Two types of conditions for the process proceeding are distinguished by the authors of Refs [16, 17, 20-22] the first type, with a rapid and short routine, and a second type, which is slow and long (up to 1500 h for some samples) [18]. By evidence of two routines of radical trapping in Ref. [19] the integral dependency l/[R]-f, which is approximated by two straight lines with different slopes, is found. This means that radical trapping in the polymeric matrix is insufficiently described by the kinetic equation of the bimolecular chemical mechanism. 

Concentration variMes, fundamental variably primary variables those substances in an enzymatic system whose concentrations can be directly controlled by the experimenter, e.g. substrates, products and effectors. They are therefore distinguished from the enzyme species, whose concentrations can be calculated from the kinetic equations at steady state for the given values of the Cv. Usually, in kinetic experiments, one Cv. is varied and the others are held constant. 

For a surface reaction between two reactants A and B, there are two distinct mechanisms Langmuir-Hinshelwood and Rideal-Eley. In the former mechanism, the reaction occurs between A and B when both are adsorbed on the surface. In the latter, the reaction occurs between an adsorbed molecule and a molecule in the gas phase. These mechanisms may be distinguished on the basis of the corresponding kinetic equations. 

A kinetic model originally derived by Nyholm is distinguished from Monod s model by the fate of a hmiting substrate. Instead of immediate metabolism, the substrate in Nyholm s model is sequestered. The governing equations are ... 

In Chapter 1 we distinguished between elementary (one-step) and complex (multistep reactions). The set of elementary reactions constituting a proposed mechanism is called a kinetic scheme. Chapter 2 treated differential rate equations of the form V = IccaCb -., which we called simple rate equations. Chapter 3 deals with many examples of complicated rate equations, namely, those that are not simple. Note that this distinction is being made on the basis of the form of the differential rate equation. 

An important cautionary note must be inserted here. It may seem that the study of the salt effect on the reaction rate might provide a means for distinguishing between two kinetically equivalent rate terms such as k[HA][B] and k [A ][BH ], for, according to the preceding development, the slope of log k vs. V7 should be 0, whereas that of log k vs. V7 should be — 1. This is completely illusory. These two rate terms are kinetically equivalent, which means that no kinetic experiment can distinguish between them. To show this, we write the rate equation in the two equivalent forms, making use of Eq. (8-26) ... 

The dissociation constants are thermodynamic constants, independent of ionic strength. Equation (8-33), which was derived from (8-30), is, therefore, identical in its form, and its salt effect, with Eq. (8-31). Therefore, salt effects cannot be used to distinguish between Eqs. (8-30) and (8-31). Another way to express this is that if kinetically equivalent forms can be written, it is not possible to determine, on the... 

There are three different approaches to a thermodynamic theory of continuum that can be distinguished. These approaches differ from each other by the fundamental postulates on which the theory is based. All of them are characterized by the same fundamental requirement that the results should be obtained without having recourse to statistical or kinetic theories. None of these approaches is concerned with the atomic structure of the material. Therefore, they represent a pure phenomenological approach. The principal postulates of the first approach, usually called the classical thermodynamics of irreversible processes, are documented. The principle of local state is assumed to be valid. The equation of entropy balance is assumed to involve a term expressing the entropy production which can be represented as a sum of products of fluxes and forces. This term is zero for a state of equilibrium and positive for an irreversible process. The fluxes are function of forces, not necessarily linear. However, the reciprocity relations concern only coefficients of the linear terms of the series expansions. Using methods of this approach, a thermodynamic description of elastic, rheologic and plastic materials was obtained. 

It is sometimes found that a given set of a—time observations are obeyed with equal accuracy by two different rate equations and the kinetic analysis resolves itself into a test of distinguishing the applicability of the alternative functions of a. Four general approaches have been used in kinetic analyses. 

There have been few discussions of the specific problems inherent in the application of methods of curve matching to solid state reactions. It is probable that a degree of subjectivity frequently enters many decisions concerning identification of a best fit . It is not known, for example, (i) the accuracy with which data must be measured to enable a clear distinction to be made between obedience to alternative rate equations, (ii) the range of a within which results provide the most sensitive tests of possible equations, (iii) the form of test, i.e. f(a)—time, reduced time, etc. plots, which is most appropriate for confirmation of probable kinetic obediences and (iv) the minimum time intervals at which measurements must be made for use in kinetic analyses, the number of (a, t) values required. It is also important to know the influence of experimental errors in oto, t0, particle size distributions, temperature variations, etc., on kinetic analyses and distinguishability. A critical survey of quantitative aspects of curve fitting, concerned particularly with the reactions of solids, has not yet been provided [490]. 

Another term often used in reaction kinetics is molecularity . In order to distinguish this term from order of a reaction, present reference is drawn to the formation of N H3 from H2 and N2, as shown in the equation below ... 

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