Rate equations integrated

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... 

If the same measurement is repeated for different [BJ it should be possible to extractjy by plotting log vs log [BJ. This should be a straight line with slopejy. In a similar manner, can be obtained by varying [CJ. At the same time the assumption that x equals 1 is confirmed. Ideally, a variety of permutations should be tested. Even if xis not 1, and the integrated rate equation is not a simple exponential, a usefiil simplification stiU results from flooding all components except one. 

An alternative to a graphical display is to calculate the rate constant for each data point using the integrated rate equation, seeking constancy in the calculated k values. 

A reaction order determined by plotting the integrated rate equation is sometimes called the order with respect to time f this order has an unambiguous meaning only if the order is independent of time, which means that the plotted function is linear... 

The isolation experimental design can be illustrated with the rate equation v = kc%CB, for which we wish to determine the reaction orders a and b. We can set Cb >>> Ca, thus establishing pseudo-oth-order kinetics, and determine a, for example, by use of the integrated rate equations, experimentally following Ca as a function of time. By this technique we isolate reactant A for study. Having determined a, we may reverse the system and isolate B by setting Ca >>> Cb and thus determine b. 

The functional dependence of the half-life on reactant concentration varies with the reactant order. From the integrated rate equations we obtain these results ... 

A reading of Section 2.2 shows that all of the methods for determining reaction order can lead also to estimates of the rate constant, and very commonly the order and rate constant are determined concurrently. However, the integrated rate equations are the most widely used means for rate constant determination. These equations can be solved analytically, graphically, or by least-squares regression analysis. 

The most widely used method for fitting a straight line to integrated rate equations is by linear least-squares regression. These equations have only two variables, namely, a concentration or concentration ratio and a time, but we will develop a more general relationship for use later in the book. 

Find the integrated rate equation for a third-order reaction having the rate equation —dc/ ldt = kCf,. ... 

There is no general explicit mathematical treatment of complicated rate equations. In Section 3.1 we describe kinetic schemes that lead to closed-form integrated rate equations of practical utility. Section 3.2 treats many further approaches, both experimental and mathematical, to these complicated systems. The chapter concludes with comments on the development of a kinetic scheme for a complex reaction. 

After separation of variables, Eq. (3-3) is integrated to give Eq. (3-4) as the integrated rate equation. 

Choice of initial conditions. To give a very obvious example, in Chapter 2 we saw that a second-order reaction A -I- B —> products could be run with the initial conditions Ca = cb, thus permitting a very simple plotting form to be used. For complex reactions, it may be possible to obtain a usable integrated rate equation if the initial concentrations are in their stoichiometric ratio. 

A final comment on the interpretation of stochastic simulations We are so accustomed to writing continuous functions—differential and integrated rate equations, commonly called deterministic rate equations—that our first impulse on viewing these stochastic calculations is to interpret them as approximations to the familiar continuous functions. However, we have got this the wrong way around. On a molecular level, events are discrete, not continuous. The continuous functions work so well for us only because we do experiments on veiy large numbers of molecules (typically 10 -10 ). If we could experiment with very much smaller numbers of molecules, we would find that it is the continuous functions that are approximations to the stochastic results. Gillespie has developed the stochastic theory of chemical kinetics without dependence on the deterministic rate equations. 

Sections 3.1 and 3.2 considered this problem Given a complex kinetic scheme, write the differential rate equations find the integrated rate equations or the concentration-time dependence of reactants, intermediates, and products and obtain estimates of the rate constants from experimental data. Little was said, however, about how the kinetic scheme is to be selected. This subject might be dismissed by stating that one makes use of experimental observations combined with chemical intuition to postulate a reasonable kinetic scheme but this is not veiy helpful, so some amplification is provided here. 

Kinetic studies at several temperatures followed by application of the Arrhenius equation as described constitutes the usual procedure for the measurement of activation parameters, but other methods have been described. Bunce et al. eliminate the rate constant between the Arrhenius equation and the integrated rate equation, obtaining an equation relating concentration to time and temperature. This is analyzed by nonlinear regression to extract the activation energy. Another approach is to program temperature as a function of time and to analyze the concentration-time data for the activation energy. This nonisothermal method is attractive because it is efficient, but its use is not widespread. ... 

Using calculus, it is possible to develop integrated rate equations relating reactant concentration to time. We now examine several such equations, starting with first-order reactions. 

Throughout Section 11.3, balanced chemical equations are written in such a way that the coefficient of die reactant is 1. In general, if the coefficient of the reactant is a, where a may be 2 or 3 or..., then k in each integrated rate equation must be replaced by the product ak. (See Problem 101.)... 

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. 

Integrated Rate Equations Time Dependence of Concentrations in Reactions of Different Orders... 

Deriving the integrated rate equation of a second-order reaction is a little more complicated. Let us assume the second-order reaction... 

Finally, although rare, we mention the occurrence of zero-order reactions. The special case of a pseudo-zero order reaction arises if a reactant is present in large excess, and the reaction does not noticeably change the concentration of the reactant. The differential and integral rate equations for a zero-order reaction R —> P are... 

What is the difference between a differential and an integral rate equation ... 

This is the integrated rate equation for a first-order reaction. When dealing with first-order reactions it is customary to use not only the rate constant, k for the reaction but also the related quantity half-life of the reaction. The half-life of a reaction refers to the time required for the concentration of the reactant to decrease to half of its initial value. For the first-order reaction under consideration, the relation between the rate constant k and the half life t0 5 can be obtained as follows ... 

The integrated rate equation for a reaction that is second order with respect to N02 as the only reactant ... 

Substituting into the integrated rate equation, we have... 

Since rate = [N205], we must use the first order integrated rate equation ln( ) = akt with tm = where a is the stoichiometric coefficient of N2Os... 

We can use the integrated rate equation to find the ratio of the final and initial concentrations. This ratio equals the fraction of the initial concentration that remains. 

B We use the integrated rate equation to determine the rate constant, realizing that one-third remains when two-thirds have decomposed. 

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