Definition of rate constants in terms

Definition of rate constants in terms of kinetic constants... 

Definition of Rate Constants in Terms of Kinetic Constants Ordered Ter Bi mechanism... 

If the interconversion of central complexes is taken into account (rate constants kg and fcio are present), the definition of kinetic constants in terms of rate constants becomes more complex (Plapp, 1973) ... 

Definition of kinetic constants in terms of rate constants KiA =... 

Definition of Kinetic Constants in Terms of Rate Constants... 

Using these definitions and assuming steady states and the equality of instantaneous and final diad and triad fractions, one can express ratios of all diads, triads, and tetrads in terms of rate constants (Table 1) or ratios of rate constants in terms of diad, triad, etc. fractions (Table 2). The rate constants themselves can be calculated if the rate constants of propagation are known for tire corresponding ejqjerimental conditions (1). 

If pt is used in the rate law instead of c,-, there are two ways of interpreting rt and hence kt. In the first of these, the definition of r, given in equation 1.4-2 is retained, and in the second, the definition is in terms of rate of change of p,. Care must be taken to identify which one is being used in a particular case. The first is relatively uncommon, and the second is limited to constant-density situations. The consequences of these two ways are explored further in this and the next section, first for the rate constant, and second for the Arrhenius parameters. 

The second-order rate-constants kp and kA for polymerisations in solution which we consider reliable are summarised in Table 12. The initiators used by the various investigators have not been listed, because by definition kp and Ep must be independent of these and there are insufficient data to permit any firm conclusions about the effects of the nature of the anion on and E. When considering the rate-constants in this Table it must be remembered that all of them, except those for isobutene, probably comprise a contribution from the polymer-complexed cation, p+p, greater or smaller according to circumstances (see Section 2.3), and correspondingly the activation energies would contain a term Ep+P and an enthalpy of complexation further, for the reason explained in Section 4.1.9, the kp from ionising radiation experiments are minimum values. 

In this chapter, we study networks of linear reactions. For any ordering of reaction rate constants we look for the dominant kinetic system. The dominant system is, by definition, the system that gives us the main asymptotic terms of the stationary state and relaxation in the limit for well-separated rate constants. In this limit any two constants are connected by the relation or... 

As indicated above, studying the reaction at constant volume in a single homogeneous phase greatly simplifies the analysis of the results (see also Chapter 4). There are, however, many industrial processes carried out in flow or stirred reactors in which the volume cannot be assumed constant. In such cases, the rigorous definition of the rate equations in terms of the extent of reaction must be used. 

The dynamical proposal can be expressed in terms of several alternative definitions. Yet, in order to exhibit dynamical contribution to catalysis, any definition must find a different magnitude of dynamical contributions to the rate constant in the enzyme and in water. All the reasonable definitions result with very small dynamical contributions.40 Rigorously speaking, it is agreed in the chemical physics community (see references in Ref. 4) that all the dynamical effects are contained in the transmission factor, that corrects the absolute rate theory for re-crossing of the reactive trajectories (see Ref. 40 for a clear definition). All reported simulation studies (e.g., Refs. 4,103) found that the transmission factors are similar in enzymes and in solution, and do not differ much from unity in the enzyme (e.g., Refs. 4,125). 

Reaction kinetics represented by the general form of Equation 1 have been employed in a number of quantitative chemical models of natural systems. Under ideal conditions, the four parameters, total mass transfer, kinetic rate constants, time, and the reactive surface area can be determined independently, permitting the unique definition of the model. In most cases, at least one of the variables, most often surface area, is treated as a dependent term. This nonuniqueness arises when the reactive surface area of a natural system cannot be estimated, or because such estimates made either from geometric or BET measurements do not produce reasonable fits to the other parameters. Most often the calculated total mass transfer significantly exceeds the observed transfer based on measured aqueous concentrations. 

Expression (67.Ill) can be considered as a "statistical formulation of the rate constant in that it represents a formal generalization of activated complex theory which is the usual form of the statistical theory of reaction rates. Actually, this expression is an exact collision theory rate equation, since it was derived from the basic equations (32.Ill) and (41. HI) without any approximations. Indeed, the notion of the activated complex has been introduced here only in a quite formal way, using equations (60.Ill) and (61.Ill) as a definition, which has permitted a change of variables only in order to make a pure mathematical transformation. Therefore, in all cases in which the activated complex could be defined as a virtual transition state in terms of a potential energy surface, the formula (67.HI) may be used as a rate equation equivalent to the collision theory expression (51.III). 

The concept of balanced growth was introduced by Campbell (1957) to describe a metabolic state of a culture in terms of the distributed concentration (xj) or total mass (X- ) of a metabolic variable. According to Campbell s definition, growth is balanced when the specific rate of change of all such variables is constant. That is. 

In practice, it is not uncommon to define the rate only in terms of the species whose concentration is being monitored. The consequences that can result from different definitions of the rate in relation to the stoichiometry are described below under the defrnition of the rate constant... 

By analogy with Eq. (3.1), we seek a description for the relationship between stress and strain. The former is the shearing force per unit area, which we symbolize as as in Chap. 2. For shear strain we use the symbol y it is the rate of change of 7 that is involved in the definition of viscosity in Eq. (2.2). As in the analysis of tensile deformation, we write the strain AL/L, but this time AL is in the direction of the force, while L is at right angles to it. These quantities are shown in Fig. 3.6. It is convenient to describe the sample deformation in terms of the angle 6, also shown in Fig. 3.6. For distortion which is independent of time we continue to consider only the equilibrium behavior-stress and strain are proportional with proportionality constant G ... 

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