Pseudo-first order example

The integrated form of the rate law for equation 13.4, however, is still too complicated to be analytically useful. We can simplify the kinetics, however, by carefully adjusting the reaction conditions. For example, pseudo-first-order kinetics can be achieved by using a large excess of R (i.e. [R]o >> [A]o), such that its concentration remains essentially constant. Under these conditions... 

Fixed-time integral methods are advantageous for systems in which the signal is a linear function of concentration. In this case it is not necessary to determine the concentration of the analyte or product at times ti or f2, because the relevant concentration terms can be replaced by the appropriate signal. For example, when a pseudo-first-order reaction is followed spectrophotometrically, when Beer s law... 

In a curve-fitting method the concentration of a reactant or product is monitored continuously as a function of time, and a regression analysis is used to fit an appropriate differential or integral rate equation to the data. Eor example, the initial concentration of analyte for a pseudo-first-order reaction, in which the concentration of a product is followed as a function of time, can be determined by fitting a rearranged form of equation 13.12... 

Flooding and Pseudo-First-Order Conditions For an example, consider a reaction that is independent of product concentrations and has three reagents. If a large excess of [BJ and [CJ are used, and the disappearance of a lesser amount of A is measured, such flooding of the system with all components butM permits the rate law to be integrated with the assumption that all concentrations are constant except A. Consequentiy, simple expressions are derived for the time variation of A. Under flooding conditions and using equation 8, if x happens to be 1, the time-dependent concentration... 

The points that we have emphasized in this brief overview of the S l and 8 2 mechanisms are kinetics and stereochemistry. These features of a reaction provide important evidence for ascertaining whether a particular nucleophilic substitution follows an ionization or a direct displacement pathway. There are limitations to the generalization that reactions exhibiting first-order kinetics react by the Sj l mechanism and those exhibiting second-order kinetics react by the 8 2 mechanism. Many nucleophilic substitutions are carried out under conditions in which the nucleophile is present in large excess. When this is the case, the concentration of the nucleophile is essentially constant during die reaction and the observed kinetics become pseudo-first-order. This is true, for example, when the solvent is the nucleophile (solvolysis). In this case, the kinetics of the reaction provide no evidence as to whether the 8 1 or 8 2 mechanism operates. 

The intermediate diphenylhydroxymethyl radical has been detected after generation by flash photolysis. Photolysis of benzophenone in benzene solution containing potential hydrogen donors results in the formation of two intermediates that are detectable, and their rates of decay have been measured. One intermediate is the PhjCOH radical. It disappears by combination with another radical in a second-order process. A much shorter-lived species disappears with first-order kinetics in the presence of excess amounts of various hydrogen donors. The pseudo-first-order rate constants vary with the structure of the donor with 2,2-diphenylethanol, for example, k = 2 x 10 s . The rate is much less with poorer hydrogen-atom donors. The rapidly reacting intermediate is the triplet excited state of benzophenone. 

The unit of the veloeity eonstant k is see Many reaetions follow first order kineties or pseudo-first order kineties over eertain ranges of experimental eonditions. Examples are the eraeking of butane, many pyrolysis reaetions, the deeomposition of nitrogen pentoxide (NjOj), and the radioaetive disintegration of unstable nuelei. Instead of the veloeity eonstant, a quantity referred to as the half-life iyj is often used. The half-life is the time required for the eoneentration of the reaetant to drop to one-half of its initial value. Substitution of the appropriate numerieal values into Equation 3-33 gives... 

A parameter such as a rate constant is usually obtained as a consequence of various arithmetic manipulations, and in order to estimate the uncertainly (error) in the parameter we must know how this error is related to the uncertainties in the quantities that contribute to the parameter. For example, Eq. (2-33) for a pseudo-first-order reaction defines k, which can be determined by a semilogarithmic plot according to Eq. (2-6). By a method to be described later in this section the uncertainty in itobs (expressed as its variance associated with cb. Thus, we need to know how the errors in fcobs and cb are propagated into the rate constant k. 

If pseudo-first-order conditions can be established, for example, by setting Cb Ca, then Scheme II collapses to Scheme III,... 

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. 

Considering the attention that we have given in this chapter to concentrationtime curves of complex reactions, it may seem remarkable that many kinetic studies never generate a comprehensive set of complicated concentration-time data. The reason for this is that complex reactions often can be studied under simplified conditions constituting important special cases for example, whenever feasible one chooses pseudo-first-order conditions, and then one studies the dependence of the pseudo-first-order rate constant on variables other than time. This approach is amplified below. 

In this scheme the reversible conversion of A to O is the reaction whose rate is to be studied, whereas the reduction of O to R is the electrode process. Scheme XIV can also represent a pseudo-first-order formation of O. A specific example is the acid-base equilibrium of pyruvic acid, shown in Scheme XV. 

First-order and second-order rate constants have different dimensions and cannot be directly compared, so the following interpretation is made. The ratio intra/ inter has the units mole per liter and is the molar concentration of reagent Y in Eq. (7-72) that would be required for the intermolecular reaction to proceed (under pseudo-first-order conditions) as fast as the intramolecular reaction. This ratio is called the effective molarity (EM) thus EM = An example is the nu-... 

The overall catalytic rate constant of SNase is (see, for example, Ref. 3) kcat — 95s 1 at T = 297K, corresponding to a total free energy barrier of Ag at = 14.9 kcal/mol. This should be compared to the pseudo-first-order rate constant for nonenzymatic hydrolysis of a phosphodiester bond (with a water molecule as the attacking nucleophile) which is 2 x 10 14 s corresponding to Ag = 36 kcal/mol. The rate increase accomplished by the enzyme is thus 101S-1016, which is quite impressive. 

Within the strictly chemical realm, sequences of pseudo-first-order reactions are quite common. The usually cited examples are hydrations carried out in water and slow oxidations carried out in air, where one of the reactants... 

Example 2.8 Most polymers have densities appreciably higher than their monomers. Consider a polymer having a density of 1040 kg/m that is formed from a monomer having a density of 900 kg/m. Suppose isothermal batch experiments require 2h to reduce the monomer content to 20% by weight. What is the pseudo-first-order rate constant and what monomer content is predicted after 4h ... 

Example 2.9 Repeat Example 2.8 assuming that the polymerization is second order in monomer concentration. This assumption is appropriate for a binary polycondensation with good initial stoichiometry, while the pseudo-first-order assumption of Example 2.8 is typical of an addition polymerization. 

Example 4.3 Suppose a pure monomer polymerizes in a CSTR with pseudo-first-order kinetics. The monomer and pol5Tner have different... 

Example 5.5 Ingredients are quickly charged to a jacketed batch reactor at an initial temperature of 25°C. The jacket temperature is 80°C. A pseudo-first-order reaction occurs. Determine the reaction temperature and the fraction unreacted as a function of time. The following data are available ... 

Example 5.7 A CSTR is commonly used for the bulk pol5anerization of styrene. Assume a mean residence time of 2 h, cold monomer feed (300 K), adiabatic operation UAgxt = ), and a pseudo-first-order reaction with rate constant... 

Phthalic anhydride will, in the presence of the V2O5 catalyst of Example 9.1, undergo complete oxidation with A772 = — 760kcal/mol. Suppose the complete oxidation is pseudo-first-order in phthalic anhydride concentration and that ln( k//) = 12.300—10,000/T. 

Example 11.10 Determine phase concentrations for a liquid-liquid reaction in a packed-bed reactor. The reactive component is dilute in both phases. It enters the reactor in one phase but undergoes a pseudo-first-order reaction in the other phase. All parameters are constant. 

FIGURE 11.7 A pseudo-first-order reaction in one phase with reactant supplied from the other phase. See Example 11.10. 

Example 11.12 Solve Equations (11.31) and (11.32) for the simple case of constant parameters and a pseudo-first-order reaction occurring in the liquid phase of a component supplied from the gas phase. The gas-phase film resistance is negligible. The inlet concentration of the reactive component is... 

Another example of performance enhancement using a zeolite/TUD-1 catalyst is shown in n-hexane cracking using a series of zeolite-Beta-embedded TUD-1 catalysts (29) 20, 40 and 60 wt% zeolite Beta in Al-Si-TUD-1 (Si/Al = 150). These are compared to pure zeolite Beta, and to a physical mixture of 40% zeolite Beta and 60% Al-Si-TUD-1. These catalysts were tested in a fixed bed reactor, at atmospheric pressure, with constant residence time at 538°C. The pseudo-first-order rate constants are shown in Figure 41.8. Note that the zeohte-loaded catalysts were clearly superior to both the pure zeolite Beta catalyst and the zeohte-TUD-1 physical mixture. Again, this is evidence that catalyst performance benefits from a hierarchical pore stracture such as zeolite embedded in TUD-1. 

Figure 3. An example plot of CC concentration versus decarboxylation time for pseudo first order reaction at 196°C.

Fortunately, most ADME processes behave as pseudo-first-order processes—not because they are so simple, but because everything except the drug concentration is constant. For example, the elimination of a drug from the body may be written as follows ... 

Advection. Advection to and from a compartment of volume V at flow rate G m3/h by, for example, air or water flow can be expressed as a pseudo first order rate process with a rate constant k equal to G/V. The rate is also given by... 

The EC mechanism (Scheme 2.1) associates an electrode electron transfer with a first-order (or pseudo-first-order) follow-up homogeneous reaction. It is one of the simplest reaction schemes where a heterogeneous electron transfer is coupled with a reaction that takes place in the adjacent solution. This is the reason that it is worth discussing in some detail as a prelude to more complicated mechanisms involving more steps and/or reactions with higher reaction orders. As before, the cyclic voltammetric response to this reaction scheme will be taken as an example of the way it can be characterized qualitatively and quantitatively. 

The preceding approach applies to all linear systems that is, those involving mechanisms in which only first-order or pseudo-first-order homogeneous reactions are coupled with the heterogeneous electron transfer steps. As seen, for example, in Section 2.2.5, it also applies to higher-order systems, involving second-order reactions, when they obey pure kinetic conditions (i.e., when the kinetic dimensionless parameters are large). If this is not the case, nonlinear partial derivative equations of the type... 

The reactions, in which molecularity and order are different due to the presence of one of the reactant in excess, are known as pseudo-order reactions. The word (pseudo) is always followed by order. For example, inversion of cane sugar is pseudo-first order reaction. 

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