Rate Expressions, Difficulties with

The selectivity relationship merely expresses the proportionality between intermolecular and intramolecular selectivities in electrophilic substitution, and it is not surprising that these quantities should be related. There are examples of related reactions in which connections between selectivity and reactivity have been demonstrated. For example, the ratio of the rates of reaction with the azide anion and water of the triphenylmethyl, diphenylmethyl and tert-butyl carbonium ions were 2-8x10 , 2-4x10 and 3-9 respectively the selectivities of the ions decrease as the reactivities increase. The existence, under very restricted and closely related conditions, of a relationship between reactivity and selectivity in the reactions mentioned above, does not permit the assumption that a similar relationship holds over the wide range of different electrophilic aromatic substitutions. In these substitution reactions a difficulty arises in defining the concept of reactivity it is not sufficient to assume that the reactivity of an electrophile is related... 

The well-known difficulty with batch reactors is the uncertainty of the initial reaction conditions. The problem is to bring together reactants, catalyst and operating conditions of temperature and pressure so that at zero time everything is as desired. The initial reaction rate is usually the fastest and most error-laden. To overcome this, the traditional method was to calculate the rate for decreasingly smaller conversions and extrapolate it back to zero conversion. The significance of estimating initial rate was that without any products present, rate could be expressed as the function of reactants and temperature only. This then simplified the mathematical analysis of the rate fianction. 

Equation 6.19 is the basic equation relating the pressure drop to the flow rate. The difficulty that arises in the case of adiabatic flow is that the equation of state is unknown. The relationship, PVy = constant, is valid for a reversible adiabatic change but flow with friction is irreversible. Thus a difficulty arises in determining the integral in equation 6.19 an alternative method of finding an expression for dPIV is sought. 

Caution 1. In the special case where reactants are introduced in their stoichiometric ratio, the integrated rate expression becomes indeterminate and this requires taking limits of quotients for evaluation. This difficulty is avoided if we go back to the original differential rate expression and solve it for this particular reactant ratio. Thus, for the second-order reaction with equal initial concentrations of A and B, or for the reaction... 

The application of Absolute Rate Theory to the interpretation of catalytic hydrogenation reactions has received relatively little attention and, even when applied, has only achieved moderate success. This is, in part, due to the necessity to formulate precise mechanisms in order to derive appropriate rate expressions [43] and, in part, due to the necessity to make various assumptions with regard to such factors as the number of surface sites per unit area of the catalyst, usually assumed to be 10 5 cm-2, the activity of the surface and the immobility or otherwise of the transition state. In spite of these difficulties, it has been shown that satisfactory agreement between observed and calculated rates can be obtained in the case of the nickel-catalysed hydrogenation of ethylene (Table 3), and between the observed and calculated apparent activation energies for the... 

For nonlinear systems the solution of the governing equations must generally be obtained numerically, but such solutions can be obtained without undue difficulty for any desired rate expression with or without axial dispersion. The case of a Langmuir system with linear driving force rate expression and negligible axial dispersion is a special case that is amenable to analytical solution by an elegant nonlinear transformation. 

One difficulty with genetic-risk estimation is that the end point is diffuse, both in the diversity of effects and in their time distribution. Cancer-risk estimation differs sharply, in that there are distinct end points. The expectation of survival and death rates associated with different kinds of cancer are much better known, it is simpler for an increase in cancer incidence to be expressed in quantitative units, such as years of life lost, than for this to be done for genetic disorders, although there have been recent attempts to do it. Furthermore, there is a... 

In attempting to fit our concentration-time data to equations of this type, we have to find rate coefficients and orders which remain constant over the time scale of any particular experiment and which do not alter throughout a series of experiments in which the initial concentrations of the reactants are varied under otherwise constant conditions. Furthermore, if we take the view that the main purpose of processing the data is to cast the results into a form capable of interpretation, the values which a orb can take are limited to 0, O.S, 1, l.S and 2 with minor exceptions to include such simple fractions as etc. it is probably true to say that very often these special cases can be anticipated from the prior knowledge available to the investigator and so little difficulty is experienced in such situations. If values for the orders a and b other than small integers or simple fractions are found, it can be concluded that a rate expression more complicated than... 

The situation is much more complicated for multilayers of cells on microcarriers or for cells that grow in clumps or spheroids. The specific growth rate in such systems is normally a function of cell location. For example, cells at the outside of tumour cell spheroids generally prohferate rapidly, while those further in are quiescent. For spheroids with a radius larger than about ten cell diameters, there is normally a necrotic core of dead cells. In these complex systems it is only possible to determine apparent growth rates. These can be expressed in terms of total cell numbers, viable cell numbers or spheroid volume. Another difficulty with these systems is that accurate cell counts are hindered by cell clumps that resist dissociation with trypsin. One way around this is to lyse the cells with surfactant and count the nuclei released (Lin et al, 1991). However, analysis is complicated because the nuclei from recently dead cells are preserved, while those from cells dead for long periods are lost. 

Similarly the concentrations of individual components in the exit stream as analyzed by mass spectrometry may or may not be correctly represented by the intensity of the parent peak of the said component. Two steps are required to transform the raw data to kinetically useful quantities. The first step stems from the fact that each mass peak may contain contributions from a number of components in the effluent. To resolve this difficulty, a mathematical procedure called deconvolution is applied to the observed total mass spectrum of the exit stream from the reactor. Deconvolution allows us to separate the total intensities at individual mass numbers into their several components, each of which is due to a contribution arising from the spectrum of an individual constituent present in the effluent sample. The deconvoluted spectrum then reports the mol fractions of the individual components in the effluent. With this in hand, the next step is to transform the output mol fractions to concentrations so they can be used in rate expressions to correlate reaction rates. 

Another difficulty when n is large is the storage and computational burden of calculating F(x) and [F(x)] 1. This is less of a issue than it used to be and remains a difficulty only for problems with many hundreds of parameters. Fortunately, no problem involving fitting TSR data to rate expressions has anything like this number of parameters. 

Difficulties with Mechanistic Rate Expressions, Pages 61-70... 

We now have a direct expression for the nucleation rate, (11.11), with f given by (11.23). The difficulty in using this equation lies, as noted, in properly evaluating the evaporation coefficients y. ... 

The difficulty with this expression is that it involves the concentrations of H atoms and Br atoms, which are not readily measurable thus the equation is useless unless we can express the concentrations of the atoms in terms of the concentrations of the molecules, H2, Br2, and HBr. Since the atom concentrations are, in any case, very small, it is assumed that a steady state is reached in which the concentration of the atoms does not change with time the atoms are removed at the same rate as they are formed. From the elementary reactions, the rates of formation of bromine atoms and of hydrogen atoms are... 

Notwithstanding these difficulties, a number of detailed mechanistic schemes have been proposed and evaluated quantitively " " they contain up to eleven unit steps, and provide varying degrees of harmony with experiment. Alternatively, rate expressions based purely on Langmuir-Hinshelwood formalism... 

In the next five sections we shall discuss recently published work in the context of mechanism Ila (Figure 1) and the corresponding rate expression, equation (2). It has sometimes proved difficult to decide in which section a given paper should be placed, since the discussions frequently involve aspects which we have treated under various headings. The divisions were chosen with the intelligent layman in mind, and in future Reports it should be possible to avoid this difficulty by a different subdivision of the subject. Data published prior to 1965 have been collected by Eigen and Wilkins and between 1965 and 1970 by Hewkin and Prince. ... 

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