Chapter 30 Kinetic Methods of Analysis

In kinetic methods, measurements are made under dynamic conditions in which the concentrations of reactants and products are changing as a function of time. A + R  P where A represents the analyte, R the reagent, and P the product. Kinetic methods are carried out during the time interval from 0 to t e when reactant and product concentrations are changing continuously. Selectivity in kinetic methods is achieved by choosing reagents and conditions that produce differences in the rates at which the analyte and potential interferences react. Many kinetic methods are based on catalyzed reactions. In one type of catalytic method, the analyte is the catalyst and is determined from its effect on an indicator reaction with reactants or products that are determined conveniently.

30 A Rates of chemical reactions The mechanism by which a chemical reaction proceeds consists of a series of chemical equations describing the individual elementary steps that lead to products being formed from reactants. The rate law for a reaction is an experimentally determined relationship between the rate of a reaction and the concentration of reactants, products, and other species such as catalysts, activators, and inhibitors.

Rate laws are algebraic expressions consisting of concentration terms and constants. Molar concentrations, symbolized with square brackets, change with time. Thus, [A] t, [A] 0, and [A] ∞ indicate the concentration of A at time t, time zero, and infinite time, respectively. Infinite time is regarded as any time greater than that required for equilibrium to be achieved. The empirical rate law can be expressed as: Rate = - d[A]/dt = - d[R]/dt = d[P]/dt = k[A] m [R] n Because A and R are being depleted, the rates of change of [A] and [R] with respect to time are negative.

Units for Rate Constants Reaction rates are always expressed in terms of concentration per unit time. The units of the rate constant are determined by the overall order p of the reaction according to the relation: concentration/time = (units of k)(concentration) p Where p = m + n Rearranging gives, units of k = (concentration) 1-p  time -1 The units for a first-order rate constant are s -1, and the units for a second-order rate constant are M -1 s -1.

The Rate Law for First-Order Reactions For a spontaneous irreversible decomposition of a species A: A - k  P The reaction is first order in A, and the rate is rate = - d[A]/dt = k[A] Pseudo-First-Order Reactions Usually, the rate law for a reaction involving two species is sufficiently complex that simplifications are needed for analytical purposes. A higher-order reaction that is executed such that a simplification is feasible is termed a pseudo-first-order reaction.

Mathematics Describing First-Order Behavior Rearrangement of earlier equation gives: d[A]/[A] = - kdt Upon integration, we get Evaluation of the integrals gives

The concentration of A at any time is equal to its original concentration minus the concentration of product (when 1 mol of product forms for 1 mol of analyte). Thus, [A] t = [A] 0 – [P] t Substitution and rearrangement gives [P] t = [A] 0 (1 - e -kt ) over the time interval t =  = 1/k (also called natural lifetime of species A) [A]  = [A] 0 e -k  = [A] 0 e -k/k = (1/e)[A] 0

Rate Laws for Second-Order and Pseudo-First-Order Reactions Consider the reaction A + R - k  P If the reaction occurs in a single elementary step, the rate is proportional to the concentration of each of the reactants, and the rate law is -d[A]/dt = k[A][R] If the concentration of R is chosen such that, [R] >> [A] -d[A]/dt = k’[A] which is identical in form to the first-order case and hence the reaction is said to be pseudo-first order.

In a second-order reaction, the rate of the reaction is the difference between the forward rate and the reverse rate: -d[A]/dt = k 1 [A][R] – k -1 [P] where k 1 is the second-order rate constant for the forward reaction and k –1 is the first-order rate constant for the reverse reaction.

Catalyzed Reactions 1.Enzyme catalyzed reactions: Enzymes are high-molecular-mass protein molecules that catalyze reactions of importance. The species acted on by an enzyme is called a substrate. Species that enhance the rate of a reaction but do not take part in the stoichiometric reaction are called activators. Species that do not participate in the stoichiometric reaction but decrease the reaction rate are called inhibitors.

The behavior of many enzymes is consistent with the general mechanism E + S  ES – k2  P+ E This is the Michaelis-Menten mechanism, where the enzyme E reacts reversibly with the substrate S to form an enzyme-substrate complex ES. This complex then decomposes irreversibly to form the product(s) and the regenerated enzyme. The rate law corresponding to the mechanism is obtained by using the steady-state approximation. d[ES]/dt = k 1 [E][S] – k -1 [ES] – k 2 [ES] = 0 the concentrations of enzyme [E] and substrate refer to the free concentrations at any time t.

By mass balance, the total (initial) enzyme concentration [E] 0 is given by [E] 0 = [E] + [ES] The rate of formation of product is given by d[P]/dt = k 2 [ES] Solving for [ES], we get [ES] = k 1 [E][S] k -1 + k 2 [ES] = k 1 [E] 0 [S] k -1 + k 2 + k 1 [S]

The term K m is known as the Michaelis constant. The Michaelis constant is similar to the equilibrium constant for the dissociation of the enzyme-substrate complex and is sometimes referred to as a pseudo- equilibrium constant. If [S] >> Km, d[P]/dt = k 2 [E] 0 When [S] << Km,

It is necessary to measure d[P]/dt at the beginning of the reaction, where [S] < [S]0, so that d[P]/dt  k’[S] 0

When the conversion of ES to products is slow compared to the reversible first step, the first step is essentially at equilibrium throughout. Mathematically, this occurs when k 2 <<< k –1.

30 B Determining reaction rates Experimental Methods The way rates are measured depends on whether the reaction of interest is fast or slow. A reaction is generally regarded as fast if it proceeds to 50% of completion in  10s. Analytical methods that use fast reactions generally require special equipment. If a reaction is sufficiently slow, conventional methods of analysis can be used to determine the concentration of a reactant or product as a function of time. The most convenient approach for obtaining kinetic data is to monitor the prog- ress of the reaction continuously by spectrophotometry, conductometry, potenti- ometry, amperometry, or some other instrumental technique.

Types of Kinetic Methods The Differential Method In the differential method, concentrations are computed from reaction rates by means of a differential form of a rate expression. Rates are determined by measuring the slope of a curve relating analyte or product concentration to reaction time.

The differential method is used to determine the concentration of an analyte [A] 0 from experimental rate measurements.

Integral Methods In contrast to the differential method, integral methods take advantage of integrated forms of rate laws. Graphical Methods A plot of the natural logarithm of experimentally measured concentrations of A (or P) as a function of time should yield a straight line with a slope of -k and a y intercept of ln [A] 0. ln[A] t = -kt + ln[A] 0

Fixed-time Methods Fixed-time methods are based on the equation[A] 0 = [A] t /e -kt For the situation where [P] is measured experimentally rather than [A], the equation can be rearranged to solve for [A] 0 [A] 0 = [P] t /1 - e -kt Or we can measure [A] or [P] at two times t 1 and t 2 [P] t1 = [A] 0 (1 – e -kt1 ) [P] t2 = [A] 0 (1 – e -kt2 ) Subtracting the first from the second and rearranging gives,

30 C Applications of kinetic methods The reactions used in kinetic methods fall into two categories: 1.Catalyzed reactions are widely used because of their superior sensitivity and selectivity. 2.Uncatalyzed reactions are used to advantage when high-speed, automated measurements are required.

Catalytic methods have been used to determine both inorganic and organic compounds.

Many different enzyme substrates have been determined with enzyme-catalyzed reactions.