Modern Analytical Cheymistry - Chapter 13 pot

622 \\CChhaapptteerr 1 3 Kinetic Methods of Analysis Asystem under thermodynamic control is in a state of equilibrium, and its signal has a constant, or steady-state value (Figure 13.1a). When a system is under kinetic control, however, its signal changes with time (Figure 13.1b) until equilibrium is established. Thus far, the techniques we have considered have involved measurements made when the system is at equilibrium. By changing the time at which measurements are made, an analysis can be carried out under either thermodynamic control or kinetic control. For example, one method for determining the concentration of NO 2 – in groundwater involves the diazotization reaction shown in Figure 13.2. 1 The final product, which is a reddish-purple azo dye, absorbs visible light at a wavelength of 543 nm. Since the concentration of dye is determined by the amount of NO 2 – in the original sample, the solution’s absorbance can be used to determine the concentration of NO 2 – . The reaction in the second step, however, is not instantaneous. To achieve a steady-state signal, such as that in Figure 13.1a, the absorbance is measured following a 10-min delay. By measuring the signal during the 10-min development period, information about the rate of the reaction is obtained. If the reaction’s rate is a function of the concentration of NO 2 – , then the rate also can be used to determine its concentration in the sample. 2 There are many potential advantages to kinetic methods of analysis, perhaps the most important of which is the ability to use chemical reactions that are slow to reach equilibrium. In this chapter we examine three techniques that rely on measurements made while the analytical system is under kinetic rather than thermodynamic control: chemical kinetic techniques, in which the rate of a chemical reaction is measured; radiochemical techniques, in which a radioactive element’s rate of nuclear decay is measured; and flow injection analysis, in which the analyte is injected into a continuously flowing carrier stream, where its mixing and reaction with reagents in the stream are controlled by the kinetic processes of convection and diffusion. 1400-CH13 9/8/99 4:32 PM Page 622 Figure 13.1 Plot of signal versus time for an analytical system that is under (a) thermodynamic control; and (b) under kinetic control. Chapter 13 Kinetic Methods of Analysis 623 Signal Time Signal Time H 2 NO 3 S Step 1 sulfanilamide diazonium ion NH 2 + NO 2 – + 2H + H 2 NO 3 SNN + N + 2H 2 O Step 2 diazonium ion H 2 NO 3 S C 2 H 4 NH 3 NN N -(1-naphthyl)-ethylenediamine dihydrochloride N + + NNH 2 + + azo dye H 2 NO 3 S C 2 H 4 NH 3 NN N + H + NNH 2 + + (a) (b) Figure 13.2 Analytical scheme for the analysis of NO 2 – in groundwater. 1 3 A Methods Based on Chemical Kinetics The earliest examples of analytical methods based on chemical kinetics, which date from the late nineteenth century, took advantage of the catalytic activity of en- zymes. Typically, the enzyme was added to a solution containing a suitable sub- strate, and the reaction between the two was monitored for a fixed time. The en- zyme’s activity was determined by measuring the amount of substrate that had reacted. Enzymes also were used in procedures for the quantitative analysis of hy- drogen peroxide and carbohydrates. The application of catalytic reactions contin- ued in the first half of the twentieth century, and developments included the use of nonenzymatic catalysts, noncatalytic reactions, and differences in reaction rates when analyzing samples with several analytes. 1400-CH13 9/8/99 4:32 PM Page 623 Despite the variety of methods that had been developed, by 1960 kinetic meth- ods were no longer in common use. The principal limitation to a broader accep- tance of chemical kinetic methods was their greater susceptibility to errors from un- controlled or poorly controlled variables, such as temperature and pH, and the presence of interferents that activate or inhibit catalytic reactions. Many of these limitations, however, were overcome during the 1960s, 1970s, and 1980s with the development of improved instrumentation and data analysis methods compensat- ing for these errors. 3 1 3 A.1 Theory and Practice Every chemical reaction occurs at a finite rate and, therefore, can potentially serve as the basis for a chemical kinetic method of analysis. To be effective, however, the chemical reaction must meet three conditions. First, the rate of the chemical reac- tion must be fast enough that the analysis can be conducted in a reasonable time, but slow enough that the reaction does not approach its equilibrium position while the reagents are mixing. As a practical limit, reactions reaching equilibrium within 1 s are not easily studied without the aid of specialized equipment allowing for the rapid mixing of reactants. A second requirement is that the rate law for the chemical reaction must be known for the period in which measurements are made. In addition, the rate law should allow the kinetic parameters of interest, such as rate constants and concen- trations, to be easily estimated. For example, the rate law for a reaction that is first order in the concentration of the analyte, A, is expressed as 13.1 where k is the reaction’s rate constant. As shown in Appendix 5,* the integrated form of this rate law ln [A] t =ln[A] 0 – kt or [A] t = [A] 0 e –kt 13.2 provides a simple mathematical relationship between the rate constant, the reac- tion’s elapsed time, t, the initial concentration of analyte, [A] 0 , and the analyte’s concentration at time t, [A] t . Unfortunately, most reactions of analytical interest do not follow the simple rate laws shown in equations 13.1 and 13.2. Consider, for example, the following re- action between an analyte, A, and a reagent, R, to form a product, P where k f is the rate constant for the forward reaction, and k b is the rate constant for the reverse reaction. If the forward and reverse reactions occur in single steps, then the rate law is Rate = k f [A][R] – k b [P] 13.3 Although the rate law for the reaction is known, there is no simple integrated form. We can simplify the rate law for the reaction by restricting measurements to the A+R P b f t k k rate = A] A]−= d dt k [ [ 624 Modern Analytical Chemistry rate constant In a rate law, the proportionality constant between a reaction’s rate and the concentrations of species affecting the rate (k). rate The change in a property’s value per unit change in time; the rate of a reaction is a change in concentration per unit change in time. rate law An equation relating a reaction’s rate at a given time to the concentrations of species affecting the rate. *Appendix 5 provides a general review of kinetics. 1400-CH13 9/8/99 4:32 PM Page 624 beginning of the reaction when the product’s concentration is negligible. Under these conditions, the second term in equation 13.3 can be ignored; thus Rate = k f [A][R] 13.4 The integrated form of the rate law for equation 13.4, however, is still too compli- cated to be analytically useful. We can simplify the kinetics, however, by carefully adjusting the reaction conditions. 4 For example, pseudo-first-order kinetics can be achieved by using a large excess of R (i.e. [R] 0 >> [A] 0 ), such that its concentration remains essentially constant. Under these conditions 13.5 ln [A] t =ln[A] 0 – k′t or [A] t = [A] 0 e –k′t 13.6 It may even be possible to adjust conditions such that measurements are made under pseudo-zero-order conditions where 13.7 [A] t = [A] 0 – k″t 13.8 A final requirement for a chemical kinetic method of analysis is that it must be possible to monitor the reaction’s progress by following the change in concentra- tion for one of the reactants or products as a function of time. Which species is used is not important; thus, in a quantitative analysis the rate can be measured by moni- toring the analyte, a reagent reacting with the analyte, or a product. For example, the concentration of phosphate can be determined by monitoring its reaction with Mo(VI) to form 12-molybdophosphoric acid (12-MPA). H 3 PO 4 + 6Mo(VI) + 9H 2 O → 12-MPA + 9H 3 O + 13.9 We can monitor the progress of this reaction by coupling it to a second reaction in which 12-MPA is reduced to form heteropolyphosphomolybdenum blue, PMB, 12-MPA + nRed → PMB + nOx where Red is a suitable reducing agent, and Ox is its conjugate form. 5,6 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. Reaction 13.9 also can be followed spectropho- tometrically by monitoring the formation of 12-MPA. 6,7 Classifying Chemical Kinetic Methods A useful scheme for classifying chemical ki- netic methods of analysis is shown in Figure 13.3. 3 Methods are divided into two main categories. For those methods identified as direct-computation methods, the concentration of analyte, [A] 0 , is calculated using the appropriate rate law. Thus, for a first-order reaction in A, equation 13.2 is used to determine [A] 0 , provided that values for k, t, and [A] t are known. With a curve-fitting method, regression is used to find the best fit between the data (e.g., [A] t as a function of time) and the known mathematical model for the rate law. In this case, kinetic parameters, such as k and [A] 0 , are adjusted to find the best fit. Both categories are further subdivided into rate methods and integral methods. Rate A] A] R] 00 =− = = ′′ d dt kk [ [[ Rate A] RA] A]=− = = ′ d dt kk [ [][ [ 0 Chapter 13 Kinetic Methods of Analysis 625 1400-CH13 9/8/99 4:32 PM Page 625 Figure 13.3 Classification of chemical kinetic methods of analysis. 626 Modern Analytical Chemistry Chemical kinetic methods Direct-computation methods Curve-fitting methods Integral methods Rate methods Fixed-time One-point Two-point Variable-time One-point Two-point Initial rate Intermediate rate Linear response Nonlinear response Integral methods Rate methods Direct-Computation Integral Methods Integral methods for analyzing kinetic data make use of the integrated form of the rate law. In the one-point fixed-time integral method, the concentration of analyte is determined at a single time. The initial con- centration of analyte, [A] 0 , is calculated using equation 13.2, 13.6, or 13.8, depend- ing on whether the reaction follows first-order, pseudo-first-order, or pseudo-zero- order kinetics. The rate constant for the reaction is determined in a separate experiment using a standard solution of analyte. Alternatively, the analyte’s initial concentration can be determined using a calibration curve consisting of a plot of [A] t for several standard solutions of known [A] 0 . EXAMPLE 1 3 .1 The concentration of nitromethane, CH 3 NO 2 , can be determined from the kinetics of its decomposition in basic solution. In the presence of excess base the reaction is pseudo-first-order in nitromethane. For a standard solution of 0.0100 M nitromethane, the concentration of nitromethane after 2.00 s was found to be 4.24 × 10 –4 M. When a sample containing an unknown amount of nitromethane was analyzed, the concentration remaining after 2.00 s was found to be 5.35 × 10 –4 M. What is the initial concentration of nitromethane in the sample? SOLUTION The value for the pseudo-first-order rate constant is determined by solving equation 13.6 for k′ and making appropriate substitutions; thus Equation 13.6 can then be solved for the initial concentration of nitromethane. This is easiest to do using the exponential form of equation 13.6. [[ [ [. . ( . )( . A] A] A] A] M M 0 0 s s) t kt t kt e e e = == × = − ′ − ′ − − − 535 10 0 0126 4 158 200 1 ′ = − = −× = − − k t t ln[ ln[ ln( . ) ln( . ) . . A] A] s 0 0 0100 4 24 10 200 158 4 1 s 1400-CH13 9/8/99 4:32 PM Page 626 Chapter 13 Kinetic Methods of Analysis 627 In Example 13.1 the initial concentration of analyte is determined by measur- ing the amount of unreacted analyte at a fixed time. Sometimes it is more conven- ient to measure the concentration of a reagent reacting with the analyte or the con- centration of one of the reaction’s products. The one-point fixed-time integral method can still be applied if the stoichiometry is known between the analyte and the species being monitored. For example, if the concentration of the product in the reaction A+R→ P is monitored, then the concentration of the analyte at time t is [A] t = [A] 0 – [P] t 13.10 since the stoichiometry between the analyte and product is 1:1. Substituting equa- tion 13.10 into equation 13.6 gives ln([A] 0 – [P] t )=ln[A] 0 – k′t 13.11 which is simplified by writing in exponential form [A] 0 – [P] t = [A] 0 e –k′t and solving for [A] 0 . 13.12 EXAMPLE 1 3 .2 The concentration of thiocyanate, SCN – , can be determined from the pseudo- first-order kinetics of its reaction with excess Fe 3+ to form a reddish colored complex of Fe(SCN) 2+ . The reaction’s progress is monitored by measuring the absorbance of Fe(SCN) 2+ at a wavelength of 480 nm. When a standard solution of 0.100 M SCN – is used, the concentration of Fe(SCN) 2+ after 10.0 s is found to 0.0516 M. The analysis of a sample containing an unknown amount of SCN – results in a concentration of Fe(SCN) 2+ of 0.0420 M after 10.0 s. What is the initial concentration of SCN – in the sample? SOLUTION The pseudo-first-order rate constant is determined by solving equation 13.11 for k′ and making appropriate substitutions Equation 13.12 then can be used to determine the initial concentration of SCN – . The one-point fixed-time integral method has the advantage of simplicity since only a single measurement is needed to determine the analyte’s initial con- centration. As with any method relying on a single determination, however, a [ [. . ( . )( . A] P] M M 0 s s) = − = − = − ′ − − t kt e e 1 0 0420 1 0 0814 0 0726 10 0 1 ′ = −− = −− = − k t t ln[ ln([ [ ) ln( . ) ln( . . ) . . A] A] P] s 00 0 1 0 1 0 0516 10 0 0 0726 1 s [ [ A] P] 0 = − − ′ t kt e1 1400-CH13 9/8/99 4:32 PM Page 627 one-point fixed-time integral method cannot compensate for constant sources of determinate error. Such corrections can be made by making measurements at two points in time and using the difference between the measurements to determine the analyte’s initial concentration. Constant sources of error affect both measurements equally, thus, the difference between the measurements is independent of these er- rors. For a two-point fixed-time integral method, in which the concentration of an- alyte for a pseudo-first-order reaction is measured at times t 1 and t 2 , we can write [A] t 1 = [A] 0 e –k′t l 13.13 [A] t 2 = [A] 0 e –k′t 2 Subtracting the second equation from the first equation and solving for [A] 0 gives 13.14 The rate constant for the reaction can be calculated from equation 13.14 by measur- ing [A] t 1 and [A] t 2 for a standard solution of analyte. The analyte’s initial concentra- tion also can be found using a calibration curve consisting of a plot of ([A] t 1 – [A] t 2 ) versus [A] 0 . Fixed-time integral methods are advantageous for systems in which the signal is a linear function of concentration. In this case it is not necessary to determine the concentration of the analyte or product at times t 1 or t 2 , because the relevant con- centration terms can be replaced by the appropriate signal. For example, when a pseudo-first-order reaction is followed spectrophotometrically, when Beer’s law (Abs) t = εb[A] t is valid, equations 13.6 and 13.14 can be rewritten as (Abs) t = [A] 0 (e –k′t )εb = c[A] 0 where (Abs) t is the absorbance at time t, and c is a constant. An alternative to a fixed-time method is a variable-time method, in which we measure the time required for a reaction to proceed by a fixed amount. In this case the analyte’s initial concentration is determined by the elapsed time, ∆t, with a higher concentration of analyte producing a smaller ∆t. For this reason variable- time integral methods are appropriate when the relationship between the detector’s response and the concentration of analyte is not linear or is unknown. In the one- point variable-time integral method, the time needed to cause a desired change in concentration is measured from the start of the reaction. With the two-point vari- able-time integral method, the time required to effect a change in concentration is measured. One important application of the variable-time integral method is the quantita- tive analysis of catalysts, which is based on the catalyst’s ability to increase the rate of a reaction. As the initial concentration of catalyst is increased, the time needed to reach the desired extent of reaction decreases. For many catalytic systems the rela- tionship between the elapsed time, ∆t, and the initial concentration of analyte is 1 ∆t FF=+ cat 0 uncat A][ [ (( [( ( ]A] Abs) Abs) Abs) Abs) 0 = − − ×= − − ′ − ′ tt kt kt tt ee b c 12 12 12 1 ε [ [[ A] A] A] 0 = − − − ′ − ′ tt kt kt ee 12 12 628 Modern Analytical Chemistry 1400-CH13 9/8/99 4:32 PM Page 628 Figure 13.5 Determination of reaction rate from a tangent line at time t. Chapter 13 Kinetic Methods of Analysis 629 where F cat and F uncat are constants that are functions of the rate constants for the catalyzed and uncatalyzed reactions, and the extent of the reaction during the time span ∆t. 8 EXAMPLE 1 3 . 3 Sandell and Kolthoff 9 developed a quantitative method for iodide based on its catalytic effect on the following redox reaction. As 3+ + 2Ce 4+ → As 5+ + 2Ce 3+ Standards were prepared by adding a known amount of KI to fixed amounts of As 3+ and Ce 4+ and measuring the time for all the Ce 4+ to be reduced. The following results were obtained: ∆t Micrograms I – (min) 5.0 0.9 2.5 1.8 1.0 4.5 How many micrograms of I – are in a sample for which ∆t is found to be 3.2 min? SOLUTION The relationship between the concentration of I – and ∆t is shown by the calibration curve in Figure 13.4, for which Substituting 3.2 min for ∆t in the preceding equation gives 1.4 µg as the amount of I – originally present in the sample. 1 8 67 10 0 222 9 ∆t =− × + −− ()µg I 012345 0.0 1.2 1.0 0.8 0.6 0.4 0.2 µg I – 6 1 D t Figure 13.4 Calibration curve for the variable-time integral determination of I – . Direct-Computation Rate Methods Rate methods for analyzing kinetic data are based on the differential form of the rate law. The rate of a reaction at time t, (rate) t , is determined from the slope of a curve showing the change in concentration for a reactant or product as a function of time (Figure 13.5). For a reaction that is first- order, or pseudo-first-order in analyte, the rate at time t is given as (rate) t = k[A] t Time [P] t ∆[ P ] (rate) t = ∆[ P ] ∆ t ∆ t 1400-CH13 9/8/99 4:32 PM Page 629 630 Modern Analytical Chemistry Substituting an equation similar to 13.13 into the preceding equation gives the fol- lowing relationship between the rate at time t and the analyte’s initial concentration. (rate) t = k[A] 0 e –kt If the rate is measured at a fixed time, then both k and e –kt are constant, and a cali- bration curve of (rate) t versus [A] 0 can be used for the quantitative analysis of the analyte. The use of the initial rate (t = 0) has the advantage that the rate is at its maxi- mum, providing an improvement in sensitivity. Furthermore, the initial rate is measured under pseudo-zero-order conditions, in which the change in concentra- tion with time is effectively linear, making the determination of slope easier. Finally, when using the initial rate, complications due to competing reactions are avoided. One disadvantage of the initial rate method is that there may be insufficient time for a complete mixing of the reactants. This problem is avoided by using a rate mea- sured at an intermediate time (t > 0). EXAMPLE 1 3 . 4 The concentration of aluminum in serum can be determined by adding 2-hydroxy-1-naphthaldehyde p-methoxybenzoyl-hydrazone and measuring the initial rate of the resulting complexation reaction under pseudo-first-order conditions. 10 The rate of reaction is monitored by the fluorescence of the metal–ligand complex. Initial rates, with units of emission intensity per second, were measured for a set of standard solutions, yielding the following results [Al 3+ ] (µM) 0.300 0.500 1.00 3.00 (rate) t = 0 0.261 0.599 1.44 4.82 A serum sample treated in the same way as the standards has an initial rate of 0.313 emission intensity/s. What is the concentration of aluminum in the serum sample? SOLUTION A calibration curve of emission intensity per second versus the concentration of Al 3+ (Figure 13.6) is a straight line, where (rate) t =0 = 1.69 × [Al 3+ (µM)] – 0.246 Substituting the sample’s initial rate into the calibration equation gives an aluminum concentration of 0.331 µM. 0123 0 5 4 3 2 1 [Al 3+ ] 4 (rate) t = 0 Figure 13.6 Result of curve-fitting for the kinetic data in Example 13.4. 1400-CH13 9/8/99 4:32 PM Page 630 Chapter 13 Kinetic Methods of Analysis 631 Curve-Fitting Methods In the direct-computation methods discussed earlier, the analyte’s concentration is determined by solving the appropriate rate equa- tion at one or two discrete times. The relationship between the analyte’s concen- tration and the measured response is a function of the rate constant, which must be measured in a separate experiment. This may be accomplished using a single external standard (as in Example 13.2) or with a calibration curve (as in Example 13.4). In a curve-fitting method the concentration of a reactant or product is moni- tored continuously as a function of time, and a regression analysis is used to fit an appropriate differential or integral rate equation to the data. For example, the initial concentration of analyte for a pseudo-first-order reaction, in which the concentra- tion of a product is followed as a function of time, can be determined by fitting a re- arranged form of equation 1 3.12 [P] t = [A] 0 (1 – e –k′t ) to the kinetic data using both [A] 0 and k′ as adjustable parameters. By using data from more than one or two discrete times, curve-fitting methods are cap- able of producing more reliable results. Although curve-fitting methods are computationally more demanding, the calculations are easily handled by computer. EXAMPLE 1 3 . 5 The data shown in the following table were collected for a reaction known to follow pseudo-zero-order kinetics during the time in which the reaction was monitored. Time [A] t (s) (mM) 3 0.0731 4 0.0728 5 0.0681 6 0.0582 7 0.0511 8 0.0448 9 0.0404 10 0.0339 11 0.0217 12 0.0143 What are the rate constant and the initial concentration of analyte in the sample? SOLUTION For a pseudo-zero-order reaction a plot of [A] t versus time should be linear with a slope of –k, and a y-intercept of [A] 0 (equation 13.8). A plot of the kinetic data is shown in Figure 13.7. Linear regression gives an equa- tion of [A] t = 0.0986 – 0.00677t 1400-CH13 9/8/99 4:32 PM Page 631 [...]... Activity (A0)x = kwx Short-lived gamma-ray Emission from impurities Initial rate of gamma-ray emission for analyte Gamma-ray emission from analyte Elapsed time after irradiation Figure 13. 15 Plot of gamma-ray emission as a function of time showing how the analyte’s initial activity is determined experimentally 1400-CH13 9/8/99 4:32 PM Page 646 646 Modern Analytical Chemistry EXAMPLE 13. 7 The concentration... x + wT  13. 31 The ratio of weights in equation 13. 31 accounts for the “dilution” of the activity due to a failure to recover all the analyte Solving equation 13. 31 for wx gives wx = AT w A − wT AA 13. 32 1400-CH13 9/8/99 4:32 PM Page 647 Chapter 13 Kinetic Methods of Analysis EXAMPLE 13. 8 The concentration of insulin in a production vat is determined by the method of isotope dilution A 1.00-mg sample... kinetics are pseudo-first-order in picrate Show that under these conditions, a plot of potential as a function of time will be linear The response of the picrate ion-selective electrode is E = K − RT ln [picrate] F We know from equation 13. 6 that for a pseudo-first-order reaction, the concentration of picrate at time t is ln [picrate]t = ln [picrate]0 – k′t where k′ is the pseudo-first-order rate constant... on the number of 28 Al 13 atoms that are present This, in turn, is equal to the difference between the rate of 28 formation for 13 Al and its rate of disintegration, d( N 28 Al ) 13 27 28 = Φσ( N 13 Al ) − λN 13 Al dt 13. 28 where Φ is the neutron flux, and σ is the reaction cross-section, or probability for 27 the capture of a neutron by the 13 Al nucleus Integrating equation 13. 28 over the time of... 1.00 × 10–2 M picrate solution to the reaction cell Suspend a picrate ion-selective electrode in the solution, and monitor the potential until it stabilizes When the potential is stable, add 2.00 mL of a —Continued 1400-CH13 9/8/99 4:32 PM Page 633 Chapter 13 Kinetic Methods of Analysis creatinine external standard, and record the potential as a function of time Repeat the procedure using the remaining... a temporary de- 1400-CH13 9/8/99 4:32 PM Page 639 Chapter 13 Kinetic Methods of Analysis 1 V x-intercept = –1 Km Slope = 639 Km Vmax y-intercept = 1 Vmax 1 [S] crease in catalytic efficiency If the inhibitor is removed, the enzyme’s catalytic efficiency returns to its normal level The reversible binding of an inhibitor to an enzyme can occur through several pathways, as shown in Figure 13. 12 In competitive... common product, P, equation 13. 23 can be written as Pt = [A]0(1 – e–kAt) + [B]0(1 – e–kBt) 641 1400-CH13 9/8/99 4:32 PM Page 642 ln(Ct) 642 Modern Analytical Chemistry 0 –0.5 –1 –1.5 –2 –2.5 –3 –3.5 –4 Again, a pair of simultaneous equations at times t1 and t2 can be solved for [A]0 and [B]0 Equation 13. 23 can also be used as the basis for a curve-fitting method As shown in Figure 13. 14, a plot of ln(Ct)... of the 14N present is 7 converted to 14C by the capture of high-energy neutrons The 14C then migrates into 6 6 Table 13. 1 Common Isotopes for Use as Tracers Isotope 3H 14C 32P 35S 45Ca 55Fe 60Co 131 I Half-Life 12.5 years 5730 years 14.3 days 87.1 days 152 days 2.91 years 5.3 years 8 days 647 1400-CH13 9/8/99 4:32 PM Page 648 648 Modern Analytical Chemistry the lower atmosphere, where it is oxidized... + 0 β 14 1 Emission of an alpha or beta particle often produces an isotope in an unstable, high-energy state This excess energy is released as a gamma ray, γ, or an X-ray Gamma ray and X-ray emission may also occur without the release of alpha or beta particles 1400-CH13 9/8/99 4:32 PM Page 643 Chapter 13 Kinetic Methods of Analysis 643 Although similar to chemical kinetic methods of analysis, radiochemical... analysis of short-lived radioactive isotopes using the method outlined in Example 13. 6 is less useful since it provides only a transient measure of the isotope’s concentration The concentration of the isotope at a particular moment 1400-CH13 9/8/99 4:32 PM Page 645 Chapter 13 Kinetic Methods of Analysis 645 can be determined by measuring its activity after an elapsed time, t, and using equation 13. 26 to calculate . initial con- centration of analyte, [A] 0 , is calculated using equation 13. 2, 13. 6, or 13. 8, depend- ing on whether the reaction follows first-order, pseudo-first-order, or pseudo-zero- order kinetics 0 Figure 13. 6 Result of curve-fitting for the kinetic data in Example 13. 4. 1400-CH13 9/8/99 4:32 PM Page 630 Chapter 13 Kinetic Methods of Analysis 631 Curve-Fitting Methods In the direct-computation. Chemistry Chemical kinetic methods Direct-computation methods Curve-fitting methods Integral methods Rate methods Fixed-time One-point Two-point Variable-time One-point Two-point Initial rate Intermediate