Kinetic rate equation, first-order

Singh and BuUin (1988) examined the kinetics of the reaction between COS and DGA over a temperature range of 307 to 322 K (93 to 120 F). They concluded that the reaction is controlled by hydrolysis and that the DGA has a catalytic effect on the hydrolysis reaction. The reaction was found to follow a second order rate equation first order in COS and first order in DGA. A competing reaction occurs to form NJ4 bis (hydroxyethoxyethyl) thiourea (BHEEU) which, foitunately, reacts with warn at high temperatures to regenerate DGA. 

From this expression, it is obvious that the rate is proportional to the concentration of A, and k is the proportionality constant, or rate constant, k has the units of (time) usually sec is a function of [A] to the first power, or, in the terminology of kinetics, v is first-order with respect to A. For an elementary reaction, the order for any reactant is given by its exponent in the rate equation. The number of molecules that must simultaneously interact is defined as the molecularity of the reaction. Thus, the simple elementary reaction of A P is a first-order reaction. Figure 14.4 portrays the course of a first-order reaction as a function of time. The rate of decay of a radioactive isotope, like or is a first-order reaction, as is an intramolecular rearrangement, such as A P. Both are unimolecular reactions (the molecularity equals 1). 

It was found that the substrate consumption rate followed first-order kinetics with respect to substrate concentration.21,22 The expression of substrate consumption with time is written in a first-order differential equation ... 

The growth of biomass in the reactor is assumed to follow Monod kinetics with a first-order death rate. A mass balance on the biomass in the reactor yields the following differential equation (assuming that no biomass enters the reactor in the feed) ... 

Decomposition Equation. The kinetics of decompn of EDNA were studied over the temp range 144.4 t o 163.5°C. At tha higher temp the sample darkened to a red brn with evolution of N02. At lower temp only si decompn was obssrved. The ttue decompn rate is first order. The data followed the Arrhenius equation and are tabulated (Ref 20b)... 

The predicted kinetics is still first-order, but the equation is simpler. Now the observed rate constant is identical with the microscopic constant kx. 

Since the base is regenerated in the fast step (2), each kinetic run is first-order in substrate. The first-order rate coefficients may be converted into second-order rate coefficients through the equation... 

For the former case (Equation (3)), which is environmentally more relevant for low contamination situations, the rate obeys first-order kinetics with respect to substrate and biomass (second-order overall), whereas in the latter case (Equation (4)), the kinetics have a first-order relationship to biomass but are independent of substrate concentration. Methods for measuring of biomass, B, have varied widely, and, for studies involving mixed populations, in which only a fraction of the organisms can degrade the substrate, a means for quantifying the responsible fraction is not available. 

The fits of the curves to the data are seen to display the rapid initial drop followed by the slower decline in endurance that is responsible for the curvature in the semilogarithmic plots. Scheme 2 can also potentially explain an initial increase in a measured property on aging by postulating that an initial concentration of intermediate X is present such that, with appropriate rate constants, the conversion of X into P will be faster than the transformation of P into X. A third feature of the above scheme is that when k12 and ki3 greatly exceed ki4, the property P will quickly reach a quasiequilibrium value of ki2/k z)F0 and then decay by first-order kinetics. When k 2 >> ki3, the kinetics will be first-order from the beginning, so that in limiting cases the conventional Equation 1 is obtained. 

The temperature dependence of the rate of reactions is particularly Important for the pyrolytic processes. Relation (5) can be used for the understanding of the common choices for the pyrolysis parameters. As an example, we can take the pyrolysis of cellulose [8]. Assuming a kinetics of the first order (pseudo first order), the activation energy in Arrhenius equation was estimated E = 100.7 kJ / mol. The frequency factor was estimated 9.60 10 s These values will lead to the following expression for the weight variation of a cellulose sample during pyrolysis ... 

In acidic conditions the tetracyclines undergo epimerisation at carbon atom 4 to form an equilibrium mixture of tetracycline and the epimer, 4-epi-tetracycline (Scheme 4.7). The 4-epi-tetracycline is toxic and its content in medicines is restricted to not more than 3%. The epimerisation follows the kinetics of a first-order reversible reaction (see equation (4.24)). The degradation rate is pH-dependent (maximum epimerisation occurring... 

The latter observation is confirmed by the studies of Gates et al. for various sulphonic acid resins and a number of alcohols. A kinetic study of t-butyl alcohol dehydration in the liquid phase between 35 and 77 °C for the -SO3H form revealed that (i) at low catalyst concentrations the reaction rate was first order in resin concentration and (ii) at high catalyst concentrations the order in resin concentration was four or five. A combined rate expression was written as (equation 5) ... 

The kinetics of a first-order reaction are very similar to those represented by the contracting volume equation [70], except in the final stages of reaction when a approaches 1.00. In measurements of reactivity, or in comparisons of properties of similar substances, the first-order expression can sometimes be used as a convenient empirical measurement of rate. The assumption of first-order behaviour is often made in the kinetic analyses of programmed temperature experiments (see Chapter 5). The software supplied with many commercial instruments often provides only order-based equations for kinetic analysis of data, whereas other equations more obviously applicable to solids, such as those given here, are not tested. 

As far as an IEMR is concerned, analytical closed-form solutions are available for limiting first and zero order kinetic rate equations.33... 

This simplified form illustrates the rate is first order in the acetone concentration. If, based on kinetic theories, we knew all the individual rate constants, we could calculate the rate of acetone production. Mternately, we can use Equation 5.95 as the basis for designing experiments to determine if the rate of production is first order in acetone and to determine the magnitude of the first-order rate constant, keff-... 

The apparent reaction order will lie somewhere between the true order and unity. For first-order reactions the diffusion process will have no effect on order, as is apparent from equation (7-22). It is convenient sometimes to recall that, kinetically, diffusion rates are first-order in concentration. 

Thus, the rate of diffusion kinetically obeys a first-order equation relative to the concentration Cx in the bulk of the solution, which was confirmed by experimental data. The rate of diffusion grows with the temperature according to a law similar to the Arrhenius equation ... 

Fig. 16 Ethylene uptake profiles symbols) in a batch reactor at 23°C, over (a) Cr(II) grafted on S948-500 (102.7 mg, 1.71 wt% Cr, 34.5 pmol Cr) and (b) Cr(II) grafted on S948-800 (314.3 mg, 0.62 wt% Cr, 38.1 pmol Cr). The lines are three-parameter fits to the first-order integrated kinetic rate equation...

The rate is first order with respect to both [oxidant] and [alcohol] and is catalysed by H+. A kinetic isotope effect knlk-D = 5.01 is consistent with a ratedetermining C—H cleavage from the alcohol carbon. A protonated Cr " species is proposed as the reactant and a mechanism involving hydride ion transfer within a chromate ester is invoked, e.g. equation (2). The possibility of direct... 

Ligand substitution on the d , 18-electron, tetrahedral complex Ni(CO) is a classic example of a dissociative ligand substitution. - Kinetic studies show that the rate is first order in nickel and independent of [L]. These data are consistent with rate-determining dissociation of CO (Equation 5.21). 

Each one of the fluid elements, which is a completely segregated cluster of fluid molecules, can be treated as a micro-batch reactor. The residence time 0 of a fluid element is taken as the batch reaction time to determine the conversion achieved in the fluid element. Consider a first-order reaction A—carried out in the laminar flow reactor, (-ni) = kCA is the kinetic rate equation. The rate of change of reactant concentration in a single fluid element (treated as a batch reactor) is given by... 

The rate of conversion of A at the active site of the solid catalyst is a function of concentrations of A and B (C and CgJ at the active site. If we assume the concentrations of A and B at the active site to be very small (C a 0 and Cg -> 0), then the kinetic rate equation for the solid-catalysed reaction takes the form of a first-order kinetic equation and can be written as... 

To simplify our discussion of kinetics, let s first assume that the reactions are irreversible and then consider the mathematical expressions or rate laws governing these classes of reaction. To do so, we need only examine the rate-determining step for the particular type of substitution. In the case of an Sjvjl reaction, this step is the formation of a carbocation from the precursor R-L (Eq. 14.2). The rate of the overall reaction is then proportional only to the concentration of substrate, as expressed in Equation 14.24. We see that the rate is first order in the concentration of R-L, expressed as [R-L], and zeroeth order in that of Nu , that is, [Nu ] , which means that the rate is independent of its concentration. Adding the two exponents for the concentration gives the overall order of the Sj l reaction, which is seen to be first order. A simplified version of the rate law is seen in Equation 14.25, and this is the form in which it is normally written. 

This is quite a complex integrated rate equation. However, if we study the kinetics of the reaction at points in time near the establishment of equilibrium, we make the assumption that the forward and reverse rates are becoming equal (as when equilibrium is really established). At equilibrium we define [x] as [x]e, where the extent of reaction is as far as it is going to go, which leads to W[ A]o - [x]c) = fcr([B]o + [x]e). Solving this equality for fcf[ A] - A r[B] , and substituting the result into Eq. 7.41, leads to Eq. 7.42. This tells us that as one approaches equilibrium, the rate appears first order with an effective rate constant that is the sum of the forward and reverse rate constants. This is an approximation because we defined [.v] as [. ]e to obtain this answer, but it is a very common way to analyze equilibrium kinetics. Chemists qualitatively estimate that the rate to equilibrium is the sum of the rates of the forward and reverse reactions. 

Note that the average heat capacity in the energy balance depends on the inlet feed only. For constant-pressure, steady-state operation, the terms on the RHS of the equation disappear. Under negligible shaft work, the Wj term also disappears. Substitution of the reaction rate for first-order kinetics, using the definition of enthalpy as C T and the definition of q from Equation 1.51 leaves us with... 

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