13B: Chemical Kinetics
Table of Contents
The earliest analytical methods based on chemical kinetics—which first appear in the late nineteenth century—took advantage of the catalytic activity of enzymes. In a typical method of that era, an enzyme was added to a solution containing a suitable substrate and their reaction monitored for a fixed time. The enzyme’s activity was determined by the change in the substrate’s concentration. Enzymes also were used for the quantitative analysis of hydrogen peroxide and carbohydrates. The development of chemical kinetic methods continued in the first half of the twentieth century with the introduction of nonenzymatic catalysts and noncatalytic reactions.
Despite the diversity of chemical kinetic methods, by 1960 they were no longer in common use. The principal limitation to their broader acceptance was a susceptibility to significant errors from uncontrolled or poorly controlled variables—temperature and pH are two examples—and the presence of interferents that activate or inhibit catalytic reactions. By the 1980s, improvements in instrumentation and data analysis methods compensated for these limitations, ensuring the further development of chemical kinetic methods of analysis.3
13B.1 Theory and Practice
Every chemical reaction occurs at a finite rate, making it a potential candidate for a chemical kinetic method of analysis. To be effective, however, the chemical reaction must meet three necessary conditions: the reaction must not occur too quickly or too slowly; we must know the reaction’s rate law; and we must be able to monitor the change in concentration for at least one species. Let’s take a closer look at each of these needs.
The material in this section assumes some familiarity with chemical kinetics, which is part of most courses in general chemistry. For a review of reaction rates, rate laws, and integrated rate laws, see the material in Appendix 17.
Reaction Rate
The rate of the chemical reaction—how quickly the concentrations of reactants and products change during the reaction—must be fast enough that we can complete the analysis in a reasonable time, but also slow enough that the reaction does not reach equilibrium while the reagents are mixing. As a practical limit, it is not easy to study a reaction that reaches equilibrium within several seconds without the aid of special equipment for rapidly mixing the reactants. We will discuss two examples of instrumentation for studying reactions with fast kinetics in Section 13B.3.
Rate Law
The second requirement is that we must know the reaction’s rate law—the mathematical equation describing how the concentrations of reagents affect the rate—for the period in which we are making measurements. For example, the rate law for a reaction that is first order in the concentration of an analyte, A, is
rate = − d[A] / dt = k[A] 13.1
where k is the reaction’s rate constant.
Because the concentration of A decreases during the reactions, d[A] is negative. The minus sign in equation 13.1 makes the rate positive. If we choose to follow a product, P, then d[P] is positive because the product’s concentration increases throughout the reaction. In this case we omit the minus sign; see equation 13.21 for an example.
An integrated rate law often is a more useful form of the rate law because it is a function of the analyte’s initial concentration. For example, the integrated rate law for equation 13.1 is
ln[A]t = ln[A]0 − kt 13.2
or
[A]t = [A]0e−kt 13.3
where [A]0 is the analyte’s initial concentration and [A]t is the analyte’s concentration at time t.
Unfortunately, most reactions of analytical interest do not follow a simple rate law. Consider, for example, the following reaction between an analyte, A, and a reagent, R, to form a single product, P
kf
A + R ⇋ P
kb
where kf is the rate constant for the forward reaction, and kb is the rate constant for the reverse reaction. If the forward and reverse reactions occur as single steps, then the rate law is
rate = −d[A]/ dt = kf [A] [R] − kb[P] 13.4
The first term, kf[A][R] accounts for the loss of A as it reacts with R to make P, and the second term, kb[P] accounts for the formation of A as P converts back to A and R.
Although we know the reaction’s rate law, there is no simple integrated form that we can use to determine the analyte’s initial concentration. We can simplify equation 13.4 by restricting our measurements to the beginning of the reaction when the product’s concentration is negligible. Under these conditions we can ignore the second term in equation 13.4, which simplifies to
rate = −d[A] / dt = kf[A] [R] 13.5
The integrated rate law for equation 13.5, however, is still too complicated to be analytically useful. We can further simplify the kinetics by carefully adjusting the reaction conditions.4 For example, we can ensure pseudo-first-order kinetics by using a large excess of R so that its concentration remains essentially constant during the time we are monitoring the reaction. Under these conditions equation 13.5 simplifies to
rate = −d[A] / dt = kf [A] [R]0 = k′[A] 13.6
where k′ = kf[R]0.
To say that the reaction is pseudo-first-order in A means that the reaction behaves as if it is first order in A and zero order in R even though the underlying kinetics are more complicated. We call k ′ the pseudo-first-order rate constant.
The integrated rate laws for equation 13.6 are
ln[A]t = ln[A]0 − k′t 13.7
or
[A]t = [A]0e−k′t 13.8
It may even be possible to adjust the conditions so that we use the reaction under pseudo-zero-order conditions.
rate = −d[A] / dt = kf[A]0[R]0 = k′′ 13.9
[A]t = [A]0 − k′′t 13.10
where k′′ = kf [A]0[R]0.
To say that a reaction is pseudo-zero-order means that the reaction behaves as if it is zero order in A and zero order in R even though the underlying kinetics are more complicated. We call k ′′ the pseudo-zero-order rate constant.
Equation 13.10 is the integrated rate law for equation 13.9.
Monitoring the Reaction
The final requirement is that we must be able to monitor the reaction's progress by following the change in concentration for at least one of its species. Which species we choose to monitor is not important.it can be the analyte, a reagent reacting with the analyte, or a product. For example, we can determine concentration of phosphate in a sample by first reacting it with Mo(VI) to form 12.molybdophosphoric acid (12-MPA).
H3PO4(aq) + 6Mo(VI)(aq)+ → 12-MPA(aq) + 9H+(aq) 13.11
Next, we reduce 12-MPA to form heteropolyphosphomolybdenum blue, PMB. The rate of formation of PMB is measured spectrophotometrically, and is proportional to the concentration of 12-MPA. The concentration of 12-MPA, in turn, is proportional to the concentration of phosphate.5 We also can follow reaction 13.11 spectrophotometrically by monitoring the formation of the yellow-colored 12-MPA.6
13B.2 Classifying Chemical Kinetic Methods
Figure 13.2 provides one useful scheme for classifying chemical kinetic methods of analysis. Methods are divided into two main categories: direct-computation methods and curve-fitting methods. In a direct-computation method we calculate the analyte’s initial concentration, [A]0, using the appropriate rate law. For example, if the reaction is first-order in analyte, we can use equation 13.2 to determine [A]0 if we have values for k, t, and [A]t. With a curve-fitting method, we use regression to find the best fit between the data—for example, [A]t as a function of time—and the known mathematical model for the rate law. If the reaction is first-order in analyte, then we fit equation 13.2 to the data using k and [A]0 as adjustable parameters.
Figure 13.2 Classification of chemical kinetic methods of analysis adapted from Pardue, H. L. “Kinetic Aspects of Analytical Chemistry,”Anal. Chim. Acta 1989, 216, 69–107.
Direct-Computation Fixed-Time Integral Methods
A direct-computation integral method uses the integrated form of the rate law. In a one-point fixed-time integral method, for example, we determine the analyte’s concentration at a single time and calculate the analyte’s initial concentration, [A]0, using the appropriate integrated rate law. To determine the reaction’s rate constant, k, we run a separate experiment using a standard solution of analyte. Alternatively, we can determine the analyte’s initial concentration by measuring [A]t for several standards containing known concentrations of analyte and constructing a calibration curve.