Concentration-time curve from integrated rate equations

Consecutive reactions involving one first-order reaction and one second-order reaction, or two second-order reactions, are very difficult problems. Chien has obtained closed-form integral solutions for many of the possible kinetic schemes, but the results are too complex for straightforward application of the equations. Chien recommends that the kineticist follow the concentration of the initial reactant A, and from this information rate constant k, can be estimated. Then families of curves plotted for the various kinetic schemes, making use of an abscissa scale that is a function of c kit, are compared with concentration-time data for an intermediate or product, seeking a match that will identify the kinetic scheme and possibly lead to additional rate constant estimates. 

If, however, we actually integrate the reaction rate equations numerically using the rate constants in Table 1.1 we find that the system does not always stick to, or even stay close to, these pseudo-steady loci. The actual behaviour is shown in Fig. 1.10. There is a short initial period during which d and b grow from zero to their appropriate pseudo-steady values. After this the evolution of the intermediate concentrations is well approximated by (1.41) and (1.42), but only for a while. After a certain time, the system moves spontaneously away from the pseudo-steady curves and oscillatory behaviour develops. We may think of the. steady state as being unstable or, in some sense repulsive , during this period in contrast to its stability or attractiveness beforehand. Thus we have met a bifurcation to oscillatory responses . The oscillations... 

Kinetic methods can be classified according to how the measurement is made. Differential methods compute the rate of reaction and relate it to the analyte concentration. Rates are determined from the slope of the absorbance versus time curve. Integral methods use an integrated form of the rate equation and determine the concentration of analyte from the absorbance changes that occur over various time intervals. Curve-fitting methods fit a mathematical model to the absorbance versus time curve and compute the parameters of the model, including the analyte concentration. The most sophisticated of these methods use the parameters of the model to estimate the value of the equilibrium or steady-slate response. These methods can provide error compensation because the equilibrium position is... 

Equation 6.22, the integrated rate expression for a first order reaction, is a statement that the concentration of A diminishes as an exponential in time starting from the initial concentration [A]q. So, if we measure [A] at various times in the course of the reaction and plot that concentration against time, the resulting curve will be qualitatively different from that of a zero order reaction, and as we shall see, different from a higher order reaction. This is shown in Figure 6.7. 

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