Integrated rate equation/expression

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. 

The rate of change of C has been given already as Equation (8.42). Equations (8.42) and (8.43) show why the derivation of integrated rate equations can be difficult for consecutive reactions while we can readily write an expression for the rate of forming C, the rate expression requires a knowledge of [B], which first increases, then decreases. The problem is that [B] is itself a function of time. 

This sort of analysis is very important in the formulation of the steady state approximation, developed to deal with kinetic schemes which are too complex mathematically to give simple explicit solutions by integration. Here the differential rate expression can be integrated. The differential and integrated rate equations are given in equations (3.61)—(3.66). 

You must decide whether to use the rate-law expression or the integrated rate equation. Remember that... 

When you need to find the rate that corresponds to particular concentrations, or the concentrations needed to give a desired rate, you should use the rate-law expression. When time is involved in the problem, you should use the integrated rate equation. 

You must choose the form of the rate-law expression or the integrated rate equation —zero, first, or second order—that is appropriate to the order of the reaction. These are summarized in Table 16-2. One of the following usually helps you decide. 

Plot the appropriate concentration expressions against time to find the order of the reaction. Find the rate constant of the reaction from the slope of the line. Use the given data and the appropriate integrated rate equation to check your answer. 

At equilibrium, the concentration of alloisoleucine does not equal that of isoleucine (58). The rate expression for Equation 13 is similar to that given in Equation 9 except k (D-amino acid) is replaced by k (D-allo-isoleucine). Since the solutions initially contained only L-isoleucine, when the amount of racemization is small, the term k (D-alloisoleu-cine) can be neglected in the rate expression and in this case the integrated rate equation is... 

The integrated form of the left-hand side of this equation depends of course on the form of the rate equation expressing (—r). Equation (4-41) is the basic design equation of the plug-flow reactor (PER) model and may be encountered in a number of different forms. The most familiar of these employs the reactor volume, V, instead of length total mass or molal flow rate, F conversion, x. This is commonly written... 

The integrated rate equation can be expressed in terms of a, the fraction reacted, because... 

A kinetic equation can be written in two distinct forms, either as an expression that shows how the concentration of reactants change with time, or as one that shows how the rate of reaction varies with the concentrations of the reactants. The enzymologist normally considers the second of these to be the usual form, whereas to the chemist the first is the usual form. Chemists have continued to prefer integrated rate equations, which have the merit to express what is actually measured. Michaehs andMenten (1913) showed that the behavior of enzymes could be studied much more simply by measuring initial rates, when the complicating effects of product accumulation and substrate depletion did not interfere. 

A differential rate equation such as equation 6.19 is generally not as useful to us as is the integrated rate equation, which allows us to compare experimental concentration data with that predicted by the rate expression. For a first-order reaction, the integrated rate expression is... 

The solution of the integrated rate laws for even this very simple equilibrium reaction between A and B is lengthy as it involves a system of differential equations. However, integrated rate law expressions for the reactant (A) and product (B) concentrations as a function of time can eventually be obtained ... 

Time The Integrated Rate Equation 16-5 Collision Theory of Reaction Rates 16-6 Transition State Theory 16-7 Reaction Mechanisms and the Rate-Law Expression... 

By integration of the rate equations, it is possible to obtain expressions that describe changes in the concentration of reactants or products as a function of time. As described below, integrated rate equations are extremely useful in the experimental determination of rate constants and reaction order. 

A special application of the first-order integrated rate equation is in the determination of decimal reduction times, or D values, the time required for a one-logio reduction in the concentration of reacting species (i.e., a 90% reduction in the concentration of reactant). Decimal reduction times are determined from the slope of logio([A(]/[Ao]) versus time plots (Fig. 1.4). The modified integrated first-order integrated rate equation can be expressed as... 

Equation (20.9) is the rate law for a zero-order reaction. Another useful equation, called an integrated rate law, expresses the concentration of a reactant as a function of time. This equation can be established rather easily... 

Equation 39 can often be simplified by adopting the concept of a mass transfer unit. As explained in the film theory discussion eadier, the purpose of selecting equation 27 as a rate equation is that is independent of concentration. This is also tme for the Gj /k aP term in equation 39. In many practical instances, this expression is fairly independent of both pressure and Gj as increases through the tower, increases also, nearly compensating for the variations in Gj. Thus this term is often effectively constant and can be removed from the integral ... 

Stoichiometric Balances The amounts of aU participants in a group of reactions can be expressed in terms of a number of key components equal to the number of independent stoichiometric relations. The independent rate equations will then involve only those key components and will be, in principle, integrable. 

It has often been observed that the plot of ln(L) versus L results in curvature rendering the method of determining the growth rate from the slope strictly inappropriate, but ways to accommodate such deviations have also been proposed. Thus, if G = G(L) integration of equation 3.15 leads to the following expression for determining crystal growth rates (Sikdar, 1977)... 

The differential rate equations of a complex reaetion, expressing rates as functions of concentrations, are usually simpler in form than are the corresponding integrated equations, whieh express concentrations as funetions of time moreover, it is always possible to write down the differential rate equations for a postulated kinetie seheme, whereas it may be difficult or impossible to integrate them. Of course, we usually measure concentration as a funetion of time. If, however, we can measure rates, we may use the differential equations directly. 

Rate equations of the form f(a) = kt are derived through integrations of specific forms of the generalized expression [ 28] representing the summation of the growth of all nuclei, so that the volume of product at time f, V(t), is given by... 

An irreversible, elementary reaction must have Equation (1.20) as its rate expression. A complex reaction may have an empirical rate equation with the form of Equation (1.20) and with integral values for n and w, without being elementary. The classic example of this statement is a second-order reaction where one of the reactants is present in great excess. Consider the slow hydrolysis of an organic compound in water. A rate expression of the form... 

Equation (8) is the differential rate expression for a first-order reaction. The value of the rate constant, k, could be calculated by determining the slope of the concentration versus time curve at any point and dividing by the concentration at that point. However, the slope of a curved line is difficult to measure accurately, and k can be determined much more easily using integrated rate expressions. 

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