First-order reaction scheme

Concentration-time curves. Much of Sections 3.1 and 3.2 was devoted to mathematical techniques for describing or simulating concentration as a function of time. Experimental concentration-time curves for reactants, intermediates, and products can be compared with computed curves for reasonable kinetic schemes. Absolute concentrations are most useful, but even instrument responses (such as absorbances) are very helpful. One hopes to identify characteristic features such as the formation and decay of intermediates, approach to an equilibrium state, induction periods, an autocatalytic growth phase, or simple kinetic behavior of certain phases of the reaction. Recall, for example, that for a series first-order reaction scheme, the loss of the initial reactant is simple first-order. Approximations to simple behavior may suggest justifiable mathematical assumptions that can simplify the quantitative description. 

The simplest case is when two products are formed by two competing first-order or pseudo-first-order reactions (Scheme 2.16). 

The disappearance of melphalan from aqueous solutions has been shown to follow first-order kinetics [62,82,85] and the concentration profile during hydrolysis is consistent with a mechanism consisting of two consecutive pseudo first-order reactions (Scheme III A) [53]. 

Analytic solutions for reactant (A) conversion and desired product (B) yield for the first-order reaction schemes ... 

The observed first-order rate constant for carbanion formation may be controlled through the choice of the basic proton acceptor. Relatively strong carbon acids undergo detectable deprotonation by the weak base water in a pseudo-first-order reaction (Scheme I.IA), but stronger general bases (Scheme I.IB) or hydroxide ion (Scheme I.IC) are required to give detectable deprotonation of weaker carbon acids in bimolecular reactions. 

Considering coupled first order reaction schemes, Wei [1966] found that "for a reactor with distribution of residence times, all reactions are slowed down... 

Consider the following equilibrium for the interconversion of a reactant A to a product B in a first-order reaction (Scheme III) ... 

Scheme 10. Mechanislic possibililies for PF condensalion. Mechanism a involves an SN2-like attack of a phenolic ring on a methylol. This attack would be face-on. Such a mechanism is necessarily second-order. Mechanism b involves formation of a quinone methide intermediate and should be Hrst-order. The quinone methide should react with any nucleophile and should show ethers through both the phenolic and hydroxymethyl oxygens. Reaction c would not be likely in an alkaline solution and is probably illustrative of the mechanism for novolac condensation. The slow step should be formation of the benzyl carbocation. Therefore, this should be a first-order reaction also. Though carbocation formation responds to proton concentration, the effects of acidity will not usually be seen in the reaction kinetics in a given experiment because proton concentration will not vary.

Generalization of Scheme X to any number of consecutive irreversible first-order reactions is obviously possible, although the equations quickly become very cumbersome. However, Eqs. (3-42) and (3-44) reveal patterns in their form, and West-man and DeLury have developed a systematic symbolism that allows the equations to be written down without integration. 

Consecutive reactions involving one first-order reaction and one second-order reaction, or two second-order reactions, are very difficult problems. Chien has obtained closed-form integral solutions for many of the possible kinetic schemes, but the results are too complex for straightforward application of the equations. Chien recommends that the kineticist follow the concentration of the initial reactant A, and from this information rate constant k, can be estimated. Then families of curves plotted for the various kinetic schemes, making use of an abscissa scale that is a function of c kit, are compared with concentration-time data for an intermediate or product, seeking a match that will identify the kinetic scheme and possibly lead to additional rate constant estimates. 

Use of the isolation or pseudo-order technique. This approach is discussed in Chapter 2, where it was shown how a second-order reaction could be converted to a pseudo-first-order reaction by maintaining one of the reactant concentrations at an essentially eonstant level. The same method may be usefully applied to eomplex reactions. In this way, for example. Scheme XI can be studied under conditions such that it functions as Scheme IX. A corollary that must be kept in mind is that a reaction system that is observed to behave in accordance with (as an example) Scheme IX may actually be more complex than it appears to be, if an unsuspected reactant is present under pseudo-order conditions. 

One further system will be solved by the transform method. Scheme XV constitutes two consecutive reversible first-order reactions. 

First, we consider Scheme I, a single reversible first-order reaction Eyring et al." treated this case. 

To see the connection between this stochastic process and a chemically reacting system, consider the first step of Scheme IX. Each (real) molecule of A has an equal and constant probability of reacting in time t. In the simulation, each position in the grid has an equal and constant probability (p) of being selected. For this first-order reaction, the chemical system is described by... 

This peculiar form applies when a dimeric molecule dissociates to a reactive monomer that then undergoes a first-order or pseudo-first-order reaction. This scheme is considered in Section 4.3. Unless one can work at either of the limits, the form is such that a numerical solution or the method of initial rates will be needed, since the integrated equation has no solution for [A]r. 

Opposing first-order reactions. For the scheme A z P, show that... 

Consider a reaction scheme consisting of a sequence of two first-order (or pseudo-first-order) reactions in which intermediate I builds up and later falls. 

The fed-batch scheme of Example 14.3 is one of many possible ways to start a CSTR. It is generally desired to begin continuous operation only when the vessel is full and when the concentration within the vessel has reached its steady-state value. This gives a bumpkss startup. The results of Example 14.3 show that a bumpless startup is possible for an isothermal, first-order reaction. Some reasoning will convince you that it is possible for any single, isothermal reaction. It is not generally possible for multiple reactions. 

In an early work by Mertz and Pettitt, an open system was devised, in which an extended variable, representing the extent of protonation, was used to couple the system to a chemical potential reservoir [67], This method was demonstrated in the simulation of the acid-base reaction of acetic acid with water [67], Recently, PHMD methods based on continuous protonation states have been developed, in which a set of continuous titration coordinates, A, bound between 0 and 1, is propagated simultaneously with the conformational degrees of freedom in explicit or continuum solvent MD simulations. In the acidostat method developed by Borjesson and Hiinenberger for explicit solvent simulations [13], A. is relaxed towards the equilibrium value via a first-order coupling scheme in analogy to Berendsen s thermostat [10]. However, the theoretical basis for the equilibrium condition used in the derivation seems unclear [3], A test using the pKa calculation for several small amines did not yield HH titration behavior [13],... 

Entropic contributions to the acceleration of first-order reactions by microwaves should be negligible (AS = = 0). When ionization (SN1 or E,) or intramolecular addition (cyclizations) processes are involved a microwave effect could be viewed as resulting from a polarity increase from GS to TS, because of the development of dipolar intermediates (Scheme 3.5). 

Spectrophotometric study of the decomposition kinetics of nitronate (MeC>2C)2C=N(0)C)Me (73a) demonstrated that this compound decomposes at 25°C by a first-order reaction according to Equation 1 (Scheme 3.72) to give the... 

Figure 23.9 illustrates the model and kinetics scheme for these conditions. We confine our analysis to a single first-order reaction, based on the development of Kunii and Levenspiel (1990 1991, pp. 300-302). However, extension to other reaction orders is straightforward. 

FIGURE 2.32. First-order reaction product (C) and second-order product (D) yields for Scheme 2.17 as a function of the competition parameter, a Constant concentration-constant potential and constant-current electrolyses, b Exhaustive constant-potential electrolysis. 

When the reaction scheme involves first- or pseudo- first-order reactions, fast enough for pure kinetic conditions to be achieved D/k concentration profile of B is squeezed within a thin reaction layer adjacent to the electrode surface as represented in Figure 2.31 (bottom diagram). Starting from the electrode surface, the following relationships apply. 

Dealing now with Scheme 2.16, which involves two competing first-order reactions, the A concentration profile and gradients are not modified. The following differential equations govern the concentration profiles of the intermediate B and the two products C and D within the reaction layer ... 

Two possible approaches are indicated in Schemes 4 and 5. In the first, a reactive radical R> is spin-trapped in competition with its pseudo-first order reaction with a substrate SH, which occurs at a known rate to give RH and S. The growth of both spin-adducts (ST—R ) and (ST—S ) is monitored, and simple analysis leads to the trapping rate constant kT. In the second approach, R-does not react with a substrate, but undergoes unimolecular rearrangement or fragmentation at a known rate to give a new species R. This latter procedure... 

Starting with component A there is a relatively slow first order reaction to form the product B. The second reaction is of the order two, it opens another path for the formation of component B. As it is a second order reaction, the higher the concentration of B, the faster is the decomposition of A to form more B. The system of differential equations for this reaction scheme is given below (3.83) ... 

We develop the idea using a kinetic example. Any reaction scheme that consists exclusively of first order reactions, results in concentration profiles that are linear combinations of exponentials. There is no limit to the number of reacting components nc. 

The set of differential equations describing such a scheme of exclusively first order reactions can always be written in the following way. 

Kinetic studies of the thermolysis of diazomethane were carried out by various authors. These experiments demonstrated that the decomposition of diazomethane was a first order reaction 41-43) Similar investigations of the p5n olysis of diphenyl-diazomethane in xylene or 1-methylnaphthalene also showed that the disappearance of diphenyl-diazomethane is a first order process. It may be concluded that a free carbene is involved in these reactions, in accordance with the following scheme ). 

One does not often encounter the simple scheme involving only first-order reactions such as (1.152)... 

Eagland et al. ( 3) propose a different scheme. The helix formation is compared to a first order reaction concerning only individual chains. In the first step the helices are nucleated and stabilized by the solvent. Next, the chains slowly fold back and the helical sequences associate by hydrogen bonds. Van der Waals interactions or entanglements between the folded chains are responsible for the gel gormation (see Figure 1-b). 

When reversible steps occur in a reaction scheme, distinctions between consecutive and parallel reactions cannot always be made. For example, the consecutive first-order reactions... 

This argument shows that for the first-order reaction model the stationary state always has some sort of stability to perturbations. In fact, this is only a first step and will not reveal Hopf bifurcations or oscillatory solutions, should they occur-. A full stability analysis of typical flow-reaction schemes will appear in the next chapter. 

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