Reactions Higher Order

Expressions similar to equation (17) may easily be derived for various second-, third-, and higher-order reactions. These expressions are readily integrated for all second-order reactions and for many third- and higher-order reactions, yielding (in many cases) relations analogous to equation (18), which define useful concentration-time graphs. The dimensions of the rate constant k for an nth order reaction are (concentration) (time)  

If some of the reactant or product species are present in excessive quantities, then the fractional changes in their concentrations over the entire duration of the reaction may be immeasurably small. In such cases the concentrations of the reactants present in excess remain approximately constant and may be absorbed into the rate constant fe. A measurement of the order of the reaction from concentration-time plots then does not reveal the dependence of the rate on the concentrations of the overabundant species the measurement yields the pseudo molecularity of the reaction, that is, the sum of the orders with respect to the species that are not present in excess. Thus a number of higher-order reactions are found to be pseudounimolecular under certain conditions. This observation provides the basis for the isolation method of determining the order of a complex reaction with respect to a particular reactant in this method, the apparent overall order (pseudo-molecularity) of the reaction is measured under conditions in which all of the reactants except the one of interest are present in excess. 

Unfortunately, the explicit solutions to these systems usually involve fairly complicated functions such as Bessel, Haenkel, theta, and incomplete gamma functions in which the arguments depend on the rate constants. The result is that these solutions are of use only if the ratios of the rate constants are known. Otherwise an inordinate amount of arithmetical labor is involved. 

A minus sign has been introduced into every term for mathematical convenience. Multiplying each side by — y% and integrating from (y = 0, t = 0) to (y, t) gives 

When we replace the y s by their equivalents in concentrations, this equation becomes 

This equation strongly resembles the first-order law in Eq. (32.16) and reduces to it in limiting circumstances. For example, suppose B is present in very great excess—so great that yg — Ja yl and Cg/cS 1 throughout the course of the reaction. Equation (32.44) then reduces to 

If both A and B are present in comparable concentrations, we can plot the quantity on the left-hand side of Eq. (32.44) against t to determine the rate constant. This plot should be a straight line with a slope of —k(y c% — VA g). All other quantities are known, so fc can be obtained from the slope. 

Reactions of order higher than second are occasionally important. A third-order rate law may have any of the forms 

For any reaction of the type nA — products, where n 1, the integrated rate equation has the general form 

For an nth-order reaction, a plot of l/[(n — 1)[A ] versus time yields a straight line with positive slope kr. 

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The catalytic effect on unimolecular reactions can be attributed exclusively to the local medium effect. For more complicated bimolecular or higher-order reactions, the rate of the reaction is affected by an additional parameter the local concentration of the reacting species in or at the micelle. Also for higher-order reactions the pseudophase model is usually adopted (Figure 5.2). However, in these systems the dependence of the rate on the concentration of surfactant does not allow direct estimation of all of the rate constants and partition coefficients involved. Generally independent assessment of at least one of the partition coefficients is required before the other relevant parameters can be accessed. 

The activation parameters for an initiator can be deterrnined at normal atmospheric pressure by plotting In vs 1/T using initiator decomposition rates obtained in dilute solution (0.2 M or lower) at several temperatures. Rate data from dilute solutions are requited in order to avoid higher order reactions such as induced decompositions. The intercept for the resulting straight line is In and the slope of the line is —E jR therefore both and E can be calculated. 

The ratio goes up sharply as the conversion increases and down sharply as the number of stages increases. For higher-order reactions the numbers are of comparable magnitudes. 

One facet of kinetic studies which must be considered is the fact that the observed reaction rate coefficients in first- and higher-order reactions are assumed to be related to the electronic structure of the molecule. However, recent work has shown that this assumption can be highly misleading if, in fact, the observed reaction rate is close to the encounter rate, i.e. reaction occurs at almost every collision and is limited only by the speed with which the reacting entities can diffuse through the medium the reaction is then said to be subject to diffusion control (see Volume 2, Chapter 4). It is apparent that substituent effects derived from reaction rates measured under these conditions may or will be meaningless since the rate of substitution is already at or near the maximum possible. 

The treatment here is restricted to first-order irreversible reactions under steady-state conditions. Higher order reactions are considered by ARJS(30). 

A similar procedure can be employed for subsequent and parallel chemical reactions and for higher-order reactions. 

This illustration has provided us with a concrete example that indicates in quantitative form the validity of the general rule of thumb that we have stated for analyzing parallel reactions. High concentrations favor the higher-order reaction, and low concentrations favor the lower-order reaction. 

Bromination can be a second-, third- or higher-order reaction, first-order in olefin but first-, second- or higher-order in bromine. Most of the early kinetic studies were focused on this complex situation (De la Mare, 1976). It is now known that bromine concentrations less than 10 3 m are necessary to obtain simple or workable kinetic equations. This limit varies slightly with the solvent for instance, in methanol 10 2 m bromine leads to convenient rate equations (Rothbaum et al, 1948) but in acetic acid 10 3 m is the highest that can be used (Yates et al, 1973). 

Third, for minerals with binary or higher order reactions, there is no assurance that the reactants are available in stoichiometric proportions. We could prepare solutions equally supersaturated with respect to gypsum by using differing Ca++ to SO4 ratios. A solution containing these components in equal amounts would precipitate the most gypsum. Solutions rich in Ca++ but depleted in SO4 , or rich in SO4 but depleted in Ca++, would produce lesser amounts of gypsum. 

Results for enhanced absorption with reversible and higher order reactions have been obtained numerically. Some of them are quoted in problems P8.02.02 and P8.02.08. 

The overwhelming majority of reactions are bimolecular. Some reactions are unimolecular and a mere handful of processes proceed as a trimolecular reactions. No quadrimolecular (or higher order) reactions are known. 

For higher-order reactions, the fluid-element concentrations no longer obey (1.9). Additional terms must be added to (1.9) in order to account for micromixing (i.e., local fluid-element interactions due to molecular diffusion). For the poorly micromixed PFR and the poorly micromixed CSTR, extensions of (1.9) can be employed with (1.14) to predict the outlet concentrations in the framework of RTD theory. For non-ideal reactors, extensions of RTD theory to model micromixing have been proposed in the CRE literature. (We will review some of these micromixing models below.) However, due to the non-uniqueness between a fluid element s concentrations and its age, micromixing models based on RTD theory are generally ad hoc and difficult to validate experimentally. 

Thus, the ASM scalar flux in a first-order reacting flow will decrease with increasing reaction rate. For higher-order reactions, the chemical source term in (3.102) will be unclosed, and its net effect on the scalar flux will be complex. For this reason, transported PDF methods offer a distinct advantage terms involving the chemical source term are closed so that its effect on the scalar flux is treated exactly. We look at these methods in Chapter 6. 

For higher-order reactions, a model must be provided to close the covariance source terms. One possible approach to develop such a model is to extend the FP model to account for scalar fluctuations in each wavenumber band (instead of only accounting for fluctuations in In any case, correctly accounting for the spectral distribution of the scalar covariance chemical source term is a key requirement for extending the LSR model to reacting scalars. 

For first-order reactions then, there is no compressibility term in the expression for In k, no matter what concentration scale is used. For higher order reactions involving molar concentrations, Eq. (22) could be applied when accurate rate data are available. Whether Eq. (27) should be applied depends on the method used for obtaining the data. If a spectrophotometric determination of the relative decrease in [A] is used, a relative measure of (d In k/dp)T is obtained from Eq. (27). If an absolute determination of [A] can be made at various times, Eq. (24) can be used directly, and k and (d In k/dp)T can be immediately obtained. The situation is easily generalized to higher order kinetics. In some cases, where AVf < 0 and the method of measurement detects [A] but not [X ], there may be a slight displacement of the quasi-equilibrium with pressure which leads to different initial concentrations of A. When AVf can be determined from Eq. (22), it may appear pressure-dependent, i.e.,... 

An activated complex containing three species (other than solvent or electrolyte), which attends a third-order reaction, is not likely to arise from a single termolecular reaction involving the three species. Third- (and higher-) order reactions invariably result from the combination of a rapid preequilibrium or preequilibria with a rds, often unidirectional. Such reactions are... 

For these reactions the concentration-time curves are of little generality for they are dependent on the concentration of reactant in the feed. As with reactions in parallel, a rise in reactant concentration favors the higher-order reaction a lower concentration favors the lower-order reaction. This causes a shift in and this property can be used to improve the product distribution. 

If reactions are not (pseudo) unimolecular but bimolecular, data analysis becomes considerably more complicated (higher order reactions will not be discussed here, but kinetic schemes can be derived following similar approaches). Two limiting cases can be discerned (1) the second reactant is a counterion to the surfactant or (2) the second reactant is a neutral molecule. 

An excellent review of higher order reactions, opposing reactions, consecutive reactions, etc, is given in Chapts II III of Ref. 10 and in Ref 11... 

Illustrative Example 21.3 Higher-Order Reaction of Nitrilotriacetic Acid (NTA) in Greifensee (Advanced Topic)... 

The CSTR is not an integral reactor. Since the same concentration exists everywhere, and the reactor is operating at steady state, there is only one reaction rate at the average concentration in the tank. Since this concentration is low because of the conversion in the tank, the value for the reaction rate is also low. This is particularly significant for higher-order reactions compared with integral reactor systems. 

It is clear that (3-picoline formation is a higher order reaction than pyridine formation, since the reactions involve 5 and 4 molecules respectively. Since a lower order reaction is favored in a more shape-selective environment, pyridine production is highest on ZSM-5 zeolites. Alternatively, one might try to maximize the fraction of (3-isomers in the picoline products. With a H-Beta zeolite, more than 98% of the picolines consist of (3-picoline, which highly simplifies the product purification (12) ... 

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