Determination of the Reaction Rate Expression

In the preceding two sections, we discussed how to apply the design equation when the rate expression is known. In this section, we will describe methods to determine the reaction rate from operating data, and then determine the parameters of the rate expression. 

The main difficulty in determining the reaction rate r is that the extent is not a measurable quantity. Therefore, we have to derive a relationship between the reaction rate and the appropriate measurable quantity. We do so by using the design equation and stoichiometric relations. Also, since the characteristic reaction time is not known a priori, we write the design equation in terms of operating time rather than dimensionless time. Assume that we measure the concentration of species j, Cj(t), as a function of time in an isothermal, constant-volume batch reactor. To derive a relation between the reaction rate, r, and Cj(t), we divide both sides of Eq. 6.2.4, by obtain 

Since the determination of an r involves differentiating of experimental data with respect to time and applying the differential form of the design equation, it is commonly called the differential method. 

Before we illustrate how to apply Eq. 6.2.40, a comment on numerical differentiation is in order. To achieve higher accuracy of the derivatives, we use second-order differentiation relations (see Appendix C). Therefore, when the data points are equally spaced, we calculate the slope of each point, except the two endpoints, using the central differentiation equation. For the first point, we use the backward differentiation equation, and, for the last point, we use the forward differentiation equation. When the points are not equally spaced, we use the central differentiation equation for the midpoint between any two adjacent data points. Hence, for n data points on species concentrations, we obtain n — 1 derivative values for the midpoints concentrations. 

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Basic Concepts in Chemical Kinetics—Determination of the Reaction Rate Expression... 

The determination of the reaction rate expression involves a two-step procedure. First, the concentration dependence is determined at a fixed temperature. Then the temperature dependence of the reaction rate constant is evaluated to give a complete reaction rate expression. The form of this temperature dependence is given by equation 3.0.14, so our present problem reduces to that of determining the form of the concentration dependence and the value of the rate constant at the temperature of the experiment. 

It is always preferable to use as much of the data as possible to determine the reaction rate expression. This principle often implies that one should use some sort of graphical procedure to analyze the data. Visual inspection of such plots may indicate that certain points are seriously in error and should not be weighted heavily in the determination of the reaction rate expression. The consistency and precision of the data can also be evaluated visually by observing the deviation of the data points from a smooth curve (ideally, their deviations from a straight line). 

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