Rate equations solution

First-order kinetics was chosen in writing Eq. (11-46), so that an analytical solution could be obtained. Numerical solutions for rj vs have been developed for many other forms of rate equations. Solutions include those for Langmuir-Hinshelwood equations with denominator terms, as derived in Chap. 9 [e.g., Eq. (9-32)]. To illustrate the extreme effects of reaction, Wheeler obtained solutions for zero- and second-order kinetics for a fiat plate of catalyst, and these results are also shown in Fig. 11-7. For many catalytic reactions the rate equation is approximately represented... 

Simple reactions rates, rate equations, solutions... 

Model Author Rate Equation Solution for Breakthrough Curve... 

The solution to the usual macroscopic kinetic rate equations for the reactant and product concentrations yields... 

To obtain an indication of the rate of solute transfer from the particle surface to the bulk of the Hquid, the concept of a thin film providing the resistance to transfer can be used (2) and the equation for mass transfer written as ... 

The clinical performance of a hemodialy2er is usually described in terms of clearance, a term having its roots in renal physiology, which is defined as the rate of solute removal divided by the inlet flow concentration as shown in equation 7, where Cl is clearance in ml,/min and all other terms are as defined previously except that, in deference to convention, flow rates are now expressed in minutes rather than seconds and feed side (/) is now synonymous with blood flow on the luminal side. 

Note that the numerator in each of the ratios in equation 7 represents the rate of solute removal from the patient. By mass balance, clearance is related to mass-transfer coefficient Kq as defined eadier in equations 3, 4, and 5, and where each of the three expressions equal rate of mass removal in g/s. 

A tank is charged initially with = 100 L of a solution of concentration Cio = 2 g moL/L. Another solution is then pumped in at V = 5 LVmin with concentration C q = 0.8 until a stoichiometric amount has been added. The rate equation is... 

A solution containing 0.5 lb moVft of reactive component is to be treated at 25 ftVh. The rate equation is... 

Initially a reactor contains 2 m of a solvent. A solution containing 2 kg moPm of reactant A is pumped in at the rate of 0.06 m /min nntrl the volume becomes 4 m . The rate equation is / = 0.25C , 1/min. Compare the time-composition profile of this operation with charging all of the feed instantaneously. 

In general, solutions are obtained by couphng the basic conservation equation for the batch system, Eq. (16-49) with the appropriate rate equation. Rate equations are summarized in Table 16-11 and 16-12 for different controlhng mechanisms. 

For noncoustaut diffusivity, a numerical solution of the conseiwa-tion equations is generally required. In molecular sieve zeohtes, when equilibrium is described by the Langmuir isotherm, the concentration dependence of the intracrystalline diffusivity can often be approximated by Eq. (16-72). The relevant rate equation is ... 

Asymptotic Solution Rate equations for the various mass-transfer mechanisms are written in dimensionless form in Table 16-13 in terms of a number of transfer units, N = L/HTU, for particle-scale mass-transfer resistances, a number of reaction units for the reaction kinetics mechanism, and a number of dispersion units, Np, for axial dispersion. For pore and sohd diffusion, q = / // p is a dimensionless radial coordinate, where / p is the radius of the particle, if a particle is bidisperse, then / p can be replaced by the radius of a suoparticle. For prehminary calculations. Fig. 16-13 can be used to estimate N for use with the LDF approximation when more than one resistance is important. 

Lenhoff, J. Chromatogr., 384, 285 (1987)] or by direct numerical solution of the conservation and rate equations. For the special case of no-axial dispersion with external mass transfer and pore diffusion, an explicit time-domain solution, useful for the case of time-periodic injections, is also available [Carta, Chem. Eng. Sci, 43, 2877 (1988)]. In most cases, however, when N > 50, use of Eq. (16-161), or (16-172) and (16-174) with N 2Np calculated from Eq. (16-181) provides an approximation sufficiently accurate for most practical purposes. 

Analytical solutions also are possible when T is constant and m = 0, V2, or 2. More complex chemical rate equations will require numerical solutions. Such rate equations are apphed to the sizing of plug flow, CSTR, and dispersion reactor models by Ramachandran and Chaud-hari (Three-Pha.se Chemical Reactors, Gordon and Breach, 1983). 

Here, we shall examine a series of processes from the viewpoint of their kinetics and develop model reactions for the appropriate rate equations. The equations are used to andve at an expression that relates measurable parameters of the reactions to constants and to concentration terms. The rate constant or other parameters can then be determined by graphical or numerical solutions from this relationship. If the kinetics of a process are found to fit closely with the model equation that is derived, then the model can be used as a basis for the description of the process. Kinetics is concerned about the quantities of the reactants and the products and their rates of change. Since reactants disappear in reactions, their rate expressions are given a... 

All other reversible seeond-order rate equations have the same solution with the boundary eonditions assumed in Equation 3-176. Table 3-4 gives solutions for some reversible reaetions. 

Now we set up the solution of the rate equations that express the reversible chemical process ... 

FIG. 18 Scaling plot of L t) from the numerical solution of the rate equations for quenches to different equilibrium states [64] from an initial exponential MWD with L = 2. The final mean lengths are given in the legend. The inset shows the original L t) vs t data. Here = (0.33Lqo). ... 

Linear differential equations with constant coefficients can be solved by a mathematical technique called the Laplace transformation . Systems of zero-order or first-order reactions give rise to differential rate equations of this type, and the Laplaee transformation often provides a simple solution. 

We now consider the solution of differential equations by means of Laplaee transforms. We have already solved one equation, namely, the first-order rate equation, but the technique is capable of more than this. It allows us to solve simultaneous differential equations. 

Complex reactions require the solution of simultaneous differential equations, and the Runge-Kutta procedure is applicable to these problems. To illustrate the method, Scheme XIV will be used. The rate equations are, in incremental form. 

Equations (4-21) are linear first-order differential equations. We considered in detail the solution of such sets of rate equations in Section 3-2, so it is unnecessary to carry out the solutions here. In relaxation kinetics these equations are always solved by means of the secular equation, but the Laplace transformation can also be used. Let us write Eqs. (4-21) as... 

Much of the study of kinetics constitutes a study of catalysis. The first goal is the determination of the rate equation, and examples have been given in Chapters 2 and 3, particularly Section 3.3, Model Building. The subsection following this one describes the dependence of rates on pH, and most of this dependence can be ascribed to acid—base catalysis. Here we treat a very simple but widely applicable method for the detection and measurement of general acid-base or nucleophilic catalysis. We consider aqueous solutions where the pH and p/f concepts are well understood, but similar methods can be applied in nonaqueous media. 

Throughout this section attention is restricted to rate equations that include concentrations of only the substrate, H, and OH. The observed first-order rate constant, therefore, contains concentrations of only H and OH (the quantity [OH ] is often replaced by A",v/[H j). The substrate (reactant) may be ionizable. Rate equations containing the concentration of additional solutes (especially catalytic additives) can be developed as shown in Section 6.3. 

It is not possible at present to provide an equation, or set of equations, that allows the prediction from fu st principles of the membrane permeation rate and solute rejection for a given real separation. Research attempting such prediction for model systems is underway, but the physical properties of real systems, both the membrane and the solute, are too complex for such analysis. An analogous situation exists for conventional filtration processes. The general... 

Numerical approaches for estimating reactivity ratios by solution of the integrated rate equation have been described.124 126 Potential difficulties associated with the application of these methods based on the integrated form of the Mayo-kewis equation have been discussed.124 127 One is that the expressions become undefined under certain conditions, for example, when rAo or rQA is close to unity or when the composition is close to the azeotropic composition. A further complication is that reactivity ratios may vary with conversion due to changes in the reaction medium. 

Both of the numerical approaches explained above have been successful in producing realistic behaviour for lamellar thickness and growth rate as a function of supercooling. The nature of rough surface growth prevents an analytical solution as many of the growth processes are taking place simultaneously, and any approach which is not stochastic, as the Monte Carlo in Sect. 4.2.1, necessarily involves approximations, as the rate equations detailed in Sect. 4.2.2. At the expense of... 

According to the definition given, this is a second-order reaction. Clearly, however, it is not bimolecular, illustrating that there is distinction between the order of a reaction and its molecularity. The former refers to exponents in the rate equation the latter, to the number of solute species in an elementary reaction. The order of a reaction is determined by kinetic experiments, which will be detailed in the chapters that follow. The term molecularity refers to a chemical reaction step, and it does not follow simply and unambiguously from the reaction order. In fact, the methods by which the mechanism (one feature of which is the molecularity of the participating reaction steps) is determined will be presented in Chapter 6 these steps are not always either simple or unambiguous. It is not very useful to try to define a molecularity for reaction (1-13), although the molecularity of the several individual steps of which it is comprised can be defined. 

For example, imagine that the reaction between BrOj and Br in acidic solutions, Eq. (1-11), is conducted with [H+]0 = 910 X [BrOj ]0 and [Br-]0 = 280 x [BrOj ]0. The effective concentrations of H+ and Br- being nearly constant, the rate equation would become... 

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