Kinetics reversible first-order

Study of reversible reactions close to equilibrium. This possibility was discussed in eonnection with Scheme II and is further treated in Chapter 4. It turns out that if the displacement from equilibrium is small, the kinetics approach first-order behavior. 

An interesting method, which also makes use of the concentration data of reaction components measured in the course of a complex reaction and which yields the values of relative rate constants, was worked out by Wei and Prater (28). It is an elegant procedure for solving the kinetics of systems with an arbitrary number of reversible first-order reactions the cases with some irreversible steps can be solved as well (28-30). Despite its sophisticated mathematical procedure, it does not require excessive experimental measurements. The use of this method in heterogeneous catalysis is restricted to the cases which can be transformed to a system of first-order reactions, e.g. when from the rate equations it is possible to factor out a function which is common to all the equations, so that first-order kinetics results. 

We have examined the effects of concentration, temperature, solvent and added electrolyte on the kinetics of this structural interconversion. In all instances, the kinetics are well described by the rate law for a reversible first-order reaction [Equation 1] ... 

The iodine-catalyzed photoisomerization of all-trans- a- and (3-carotenes in hexane solutions produced by illumination with 20 W fluorescence light (2000 lux) and monitored by HPLC with diode-array detection yielded a different isomer distribution (Chen et al. 1994). Four cis isomers of [3-carotene (9-cis, 13-cis, 15-cis, and 13,15-cli-r/.v) and three cis isomers of a-carotene (9-cis, 13-cis, and 15-ri.v) were separated and detected. The kinetic data fit into a reversible first-order model. The major isomers formed during the photosensitized reaction of each carotenoid were 13,15-di-d.v- 3-carotene and 13-ds-a-carotene (Chen et al. 1994). 

Figure 8.22 Kinetic graph for a reversible first-order reaction with the axes for an integrated rate equation ln([A], — A fcq ) (as 3/ ) against time (as V). The gradient is —5.26 x 10 3 min 1...

The conversion of second-order reversible reactions to reversible first-order kinetics by using all but one of the reactants and all but one of the products in excess is a valuable ploy. The reversible reaction... 

With the system of Example 9.2 and starting with an R-free solution, kinetic experiments in a batch reactor give 58.1% conversion in 1 min at 65°C, 60% conversion in 10 min at 25°C. Assuming reversible first-order kinetics, find the rate expression for this reaction and prepare the conversion-temperature chart with reaction rate as parameter. 

Most of the kinetic studies made have been spectrophotometric and have employed acidic or neutral solutions. For the ammine and ethylenediamine complexes pseudo first-order rate constants were obtained for solutions initially containing either monohydroxo-bridged species [ obs(M)] or dihydroxo-bridged species [ obs(D)]. It was found that kobs(M) = obs(D), which is consistent with reversible first-order kinetics. 

Therefore, cp = /T. There was an opinion that a2(oJD c [or a(a>a/D c)1/2] is a generally applicable criterion of the attainment of the kinetic region. However, this is not always true. In the case of a reversible first-order reaction ... 

Mehran and Tanji (1974) Proposed irreversible first-order kinetics for nitrification, denitrification, mineralization, immobilization, and plant uptake and reversible first-order kinetics for NH4 ion exchange. Model verified with published incubation data. 

An example of a two-site model for pesticide desorption kinetics from soils was presented by McCall and Agin (1985). Using a reversible first-order equation to describe picloram desorption from soil, the authors plotted ln(CB - CBeq) versus t, where CB is the bound form of picloram and CBeq is the bound form at equilibrium, in Fig. 9.4. It is obvious that desorption conforms to a two-step process where a fast step occurs for about a 5-h period, and then a slow process as shown from the linear portion of Fig. 9.4 occurs from 5 to 300 h. McCall and Agin (1985) used the following two-step model to describe the data shown in Fig. 9.4 ... 

Two prototype reaction examples (reversible first-order and irreversible second-order kinetics) were discussed to address issues of rounding when switching from deterministic variables to stochastic (i.e., conversion of real numbers to integers), as well as the thresholds of population sizes and transition probabilities to control accuracy in the first two moments of the population (mean and variance). Other more complex examples were also mentioned. The... 

Racemization (1,2-oxygen migration) occurred between 2a and 2b (97T12203). The racemization took place slowly in solution at room temperature and obeyed reversible first-order kinetics krac = 4.3 x 10 6 s-1 (25°C, CH2C12). Tlie AH and AS, calculated from rate constants at temperatures between 20° and 40°C, were 24.3 kcal/mol and -2.0 eu, respectively. The rate of the racemization was hardly dependent on either the concentration of the solution or the polarity of the solvent examined. The low activation energy and the analogy of the mechanism to that for the dispro-... 

Although the first demonstration that amino acid racemization took place in fossils used mollusc shells (23), the application of this reaction in dating these materials has been extensively investigated only recently (19,20). Work on Mercenaria (19), Chione (20), and other species (24) has tested the application of racemization dating to fossil mollusc shells from geological contexts and Indian shell middens. These and other studies have shown that there are problems with amino acid racemization dating of carbonaceous fossils which are not encountered with bone. Reversible first-order racemization kinetics which are observed in bone... 

A sampling of the type of data obtained from this experiment is given in Figure 4.3.1b. Kinetic constants can be calculated from these data using analyses like those presented above for the simple reversible, first-order system [Equation (4.3.1)]. 

When pyrolyzed, p-hydroxy alkenes cleave to give alkenes and aldehydes or ketones." Alkenes produced this way are quite pure, since there are no side reactions. The mechanism has been shown to be pericyclic, primarily by observations that the kinetics are first order" and that, for ROD, the deuterium appeared in the allylic position of the new alkene." This mechanism is the reverse of that for the oxygen analog of the ene synthesis (16-54). p-Hydroxyacetylenes react similarly to give the corresponding allenes and carbonyl compounds." " The mechanism is the same despite the linear geometry of the triple bonds. 

Characteristic features of this mechanism are that (i) the rate of the reaction does not depend on the concentration of the base and the kinetics are first order (in substrate) (ii) the reaction may not be stereospecific (iii) the elimination/substitution ratio is mostly independent of the leaving group (but in solvents of low ionization energy ion pairs are formed and then the ratio depends upon the leaving group) (iv) by-products are formed via rearrangements (v) the reaction is reversible (vi) generally the most stable alkene is formed (Zaitsev orientation see Section 5.1.2.5)... 

The isomerization process of the fulvene titanium allyl complex Cp (Fv)Ti(773-C3Fl5)1293 (Fv = CsMe4CH2) (Scheme 506 Section 4.05.4.2.1) to the 1-propenyl Cp FvTi( 71-CH=CHMe) has been investigated. Mechanistic, kinetic, and thermodynamic aspects suggest that the reaction proceeds via reversible first-order steps with the participation of four intermediates.1323... 

We have already considered in detail the reversible first order reaction. The kinetics might be quite different (as illustrated in Exercise 6.2.4) and similar expressions might be derived, but the algebraic labor would be much greater. If the adsorption and desorption are rapid in comparison with reaction we can always substitute the equilibrium adsorbed concentrations in any rate law. Thus if the reaction A B really second order in both... 

It was proved in Example 11-6 that the curve for first-order kinetics in Fig. 11-7 could be used for a reversible first-order reaction, provided... 

Because the reduction potentials for the a5Ru (His) and Fe heme centers are closely matched = 19 mV), the observed rate constant for intramolecular ET over the 12.7 A from Ru to the heme followed reversible first-order kinetics (26) ... 

Some data are available about catalysis in 1,2-cycloadditions. Tributyl phosphine catalyses dimerisation of phenyl isocyanate to uretidinedione in toluene . The reaction is kinetically of first order with respect to catalyst and overall third order the reverse process is first order with respect to catalyst and overall second order. The mechanism is complex, as revealed by the value of the apparent activation energy of the forward reaction (E= l.l 0.7 kcal.mole" ), which presumably results from the combined temperature dependence of two or more steps, including formation of an isocyanate-phosphine complex (see eqn. (13), p. 113). 

HYSYS provides default values for the Forward Order and Reverse Order based on the reaction stoichiometry. The kinetic data for this case is based on an excess of water, so the kinetics are first order in Propylene Oxide only. 

Finally, an important theoretical result emerges from the preceding treatment. The formulation discussed in this chapter shows clearly that purely kinetic oscillations are impossible in the case of a network of reversible first-order reactions. This was first demonstrated by Jost (1947) and is a consequence of the principle of microscopic reversibility. 

As we saw in Chapter 10, most reactions are, to some degree, reversible. To study the kinetics of a reversible reaction, it is necessary to take into account both the forward and the reverse rates. Consider, for example, a reversible first-order reaction of molecule A to form molecule B ... 

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