Modern Analytical Cheymistry - Chapter 13 pot

If the reaction’s rate is a function of the concentration of NO2–, then the rate also can be used to determine its concentration in the sample.2 There are many potential advantages to ki

Kinetic Methods of Analysis

A system 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 NO2–in groundwater involves the diazotization reaction shown in Figure 13.2.1The 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 NO2–in the original sample, the solution’s absorbance can

be used to determine the concentration of NO2– 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 NO2–, 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.

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

13A 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

Despite the variety of methods that had been developed, by 1960 kinetic 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 thepresence of interferents that activate or inhibit catalytic reactions Many of theselimitations, however, were overcome during the 1960s, 1970s, and 1980s with thedevelopment of improved instrumentation and data analysis methods compensat-ing for these errors.3

meth-13A.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 whilethe 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 therapid 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 lawshould 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 firstorder 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]0e –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 simplerate 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 kfis the rate constant for the forward reaction, and kbis the rate constant forthe reverse reaction If the forward and reverse reactions occur in single steps, thenthe rate law is

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

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.

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

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.4For 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]0e –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 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 funcconcentra-tion 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)

H3PO4+ 6Mo(VI) + 9H2O→12-MPA + 9H3O+ 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.5,6The 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 13.3.3Methods 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] tare known With a curve-fitting method, regression is used

to find the best fit between the data (e.g., [A]tas 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

Figure 13.3

Classification of chemical kinetic methods of

analysis.

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 datamake use of the integrated form of the rate law In the one-point fixed-time integralmethod, 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 separateexperiment using a standard solution of analyte Alternatively, the analyte’s initialconcentration can be determined using a calibration curve consisting of a plot of[A]tfor several standard solutions of known [A]0

EXAMPLE13.1

The concentration of nitromethane, CH3NO2, can be determined from thekinetics of its decomposition in basic solution In the presence of excess basethe reaction is pseudo-first-order in nitromethane For a standard solution of0.0100 M nitromethane, the concentration of nitromethane after 2.00 s wasfound to be 4.24×10–4M When a sample containing an unknown amount ofnitromethane 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 thesample?

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

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

since the stoichiometry between the analyte and product is 1:1 Substituting

equa-tion 13.10 into equaequa-tion 13.6 gives

The concentration of thiocyanate, SCN–, can be determined from the

pseudo-first-order kinetics of its reaction with excess Fe3+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

1

one-point fixed-time integral method cannot compensate for constant sources ofdeterminate error Such corrections can be made by making measurements at twopoints in time and using the difference between the measurements to determine theanalyte’s initial concentration Constant sources of error affect both measurementsequally, 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 t1and t2, we can write

[A]t2= [A]0e –k′t2Subtracting the second equation from the first equation and solving for [A]0gives

13.14

The rate constant for the reaction can be calculated from equation 13.14 by ing [A]t1and [A]t2for 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]t1– [A]t2)versus [A]0

measur-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 t1or t2, because the relevant centration terms can be replaced by the appropriate signal For example, when apseudo-first-order reaction is followed spectrophotometrically, when Beer’s law

con-(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 wemeasure the time required for a reaction to proceed by a fixed amount In this casethe 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’sresponse 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 inconcentration 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 ismeasured

One important application of the variable-time integral method is the tive analysis of catalysts, which is based on the catalyst’s ability to increase the rate

quantita-of a reaction As the initial concentration quantita-of catalyst is increased, the time needed toreach 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

Figure 13.5

Determination of reaction rate from a

tangent line at time t.

Chapter 13 Kinetic Methods of Analysis 629

where Fcatand Funcatare 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

EXAMPLE13.3

Sandell and Kolthoff9developed a quantitative method for iodide based on its

catalytic effect on the following redox reaction

As3++ 2Ce4+→As5++ 2Ce3+

Standards were prepared by adding a known amount of KI to fixed amounts of

As3+and Ce4+and measuring the time for all the Ce4+to be reduced The

following results were obtained:

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

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

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]0e –kt

If the rate is measured at a fixed time, then both k and e –ktare constant, and a bration curve of (rate)tversus [A]0can be used for the quantitative analysis of theanalyte

cali-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 ismeasured 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).

EXAMPLE13.4

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-orderconditions.10 The rate of reaction is monitored by the fluorescence of themetal–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+ ] ( µ M) 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 of0.313 emission intensity/s What is the concentration of aluminum in theserum sample?

SOLUTION

A calibration curve of emission intensity per second versus the concentration of

Al3+(Figure 13.6) is a straight line, where

(rate)t = 0= 1.69×[Al3+(µM)] – 0.246Substituting the sample’s initial rate into the calibration equation gives analuminum concentration of 0.331 µM

0

5 4 3 2 1

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 relaequa-tionship 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 funcconcentra-tion of time, can be determined by fitting a

re-arranged form of equation 13.12

[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 relicap-able results Although curve-fitting methods are

computationally more demanding, the calculations are easily handled by computer

EXAMPLE 13.5

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

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] = 0.0986 – 0.00677t

0 2 4 6 8 10 12 0.00

0.10 0.08 0.06 0.04 0.02

be controlled with proper instrumentation Other variables, such as interferents inthe sample matrix, are more difficult to control and may lead to significant errors.Although not discussed in this text, direct-computation and curve-fitting methodshave been developed that compensate for these sources of error.3

Representative Method Although each chemical kinetic method has its ownunique considerations, the determination of creatinine in urine based on the ki-netics of its reaction with picrate provides an instructive example of a typicalprocedure

Description of Method. Creatine is an organic acid found in muscle tissue that supplies energy for muscle contractions One of its metabolic products is creatinine, which is excreted in urine Because the concentration of creatinine in urine and serum is an important indication of renal function, rapid methods for its analysis are clinically important In this method the rate of reaction between creatinine and picrate in an alkaline medium is used to determine the concentration of creatinine in urine Under the conditions of the analysis, the reaction is first-order in picrate, creatinine, and hydroxide.

Rate = k[picrate][creatinine][OH– ] The rate of reaction is monitored using a picrate ion-selective electrode.

Procedure. Prepare a set of external standards containing 0.5 g/L to 3.0 g/L creatinine (in 5 mM H2SO4) using a stock solution of 10.00 g/L creatinine in 5 mM

H 2 SO 4 In addition, prepare a solution of 1.00 × 10 –2 M sodium picrate Pipet 25.00 mL

of 0.20 M NaOH, adjusted to an ionic strength of 1.00 M using Na 2 SO 4 , into a thermostated reaction cell at 25 °C Add 0.500 mL of the 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

Chapter 13 Kinetic Methods of Analysis 633

creatinine external standard, and record the potential as a function of time Repeat

the procedure using the remaining external standards Construct a calibration curve of

rate ( ∆ E/ ∆t) versus [creatinine] Samples of urine (2.00 mL) are analyzed in a similar

manner, with the concentration of creatinine determined from the calibration curve.

Questions

1 The analysis is carried out under conditions in which the reaction’s 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

We know from equation 13.6 that for a pseudo-first-order reaction, the

concentration of picrate at time t is

2 As carried out the rate of the reaction is pseudo-first-order in picrate and

pseudo-zero-order in creatinine and OH – Explain why it is possible to prepare a calibration curve of rate versus [creatinine].

Since the reaction is carried out under conditions in which it is order in creatinine and OH –, the rate constant, k′ , is

pseudo-zero-k′= k[creatinine][OH– ]

where k is the reaction’s true rate constant The rate, therefore, is

where c is a constant.

3 Why is it necessary to use a thermostat in the reaction cell?

The rate of a reaction is temperature-dependent To avoid a determinate error resulting from a systematic change in temperature or to minimize

indeterminate errors due to fluctuations in temperature, the reaction cell must have a thermostat to maintain a constant temperature.

4 Why is it necessary to prepare the NaOH solution so that it has an ionic

strength of 1.00 M?

The potential of the ion-selective electrode actually responds to the activity of picrate in solution By adjusting the NaOH solution to a high ionic strength, we maintain a constant ionic strength in all standards and samples Because the relationship between activity and concentration is a function of ionic strength (see Chapter 6), the use of a constant ionic strength allows us to treat the potential as though it were a function of the concentration of picrate.

Rate = RTkcreatinine][OH− = [creatinine]

13A.2 Instrumentation

Quantitative information about a chemical reaction can be made using any ofthe techniques described in the preceding chapters For reactions that are kineti-cally slow, an analysis may be performed without worrying about the possibilitythat significant changes in concentration occur while measuring the signal.When the reaction’s rate is too fast, which is usually the case, significant errorsmay be introduced if changes in concentration are ignored One solution to this

problem is to stop, or quench, the reaction by suitably adjusting experimental

conditions For example, many reactions involving enzymes show a strong pHdependency and may be quenched by adding a strong acid or strong base Oncethe reaction is stopped, the concentration of the desired species can be deter-mined at the analyst’s convenience Another approach is to use a visual indicatorthat changes color after the reaction occurs to a fixed extent You may recall thatthis variable-time method is the basis of the so-called “clock reactions” com-monly used to demonstrate kinetics in the general chemistry classroom and labo-ratory Finally, reactions with fast kinetics may be monitored continuously usingthe same types of spectroscopic and electrochemical detectors found in chro-matographic instrumentation

Two additional problems for chemical kinetic methods of analysis are theneed to control the mixing of the sample and reagents in a rapid and repro-ducible fashion and the need to control the acquisition and analysis of the signal.Many kinetic determinations are made early in the reaction when pseudo-zero-order or pseudo-first-order conditions are in effect Depending on the rate of re-action, measurements are typically made within a span of a few milliseconds orseconds This is both an advantage and a disadvantage The disadvantage is thattransferring the sample and reagent to a reaction vessel and their subsequentmixing must be automated if a reaction with rapid kinetics is to be practical.This usually requires a dedicated instrument, thereby adding an additional ex-pense to the analysis The advantage is that a rapid, automated analysis allows for

a high throughput of samples For example, an instrument for the automated netic analysis of phosphate, based on reaction 13.9, has achieved sampling rates

ki-of 3000 determinations per hour.6

A variety of designs have been developed to automate kinetic analyses.6The

stopped-flow apparatus, which is shown schematically in Figure 13.8, has found

use in kinetic determinations involving very fast reactions Sample and reagentsare loaded into separate syringes, and precisely measured volumes are dispensed

by the action of a syringe drive The two solutions are rapidly mixed in the ing chamber before flowing through an observation cell The flow of sample andreagents is stopped by applying back pressure with the stopping syringe Theback pressure completes the mixing, after which the reaction’s progress is moni-tored spectrophotometrically With a stopped-flow apparatus, it is possible tocomplete the mixing of sample and reagent and initiate the kinetic measure-ments within approximately 0.5 ms The stopped-flow apparatus shown in Fig-ure 13.8 can be modified by attaching an automatic sampler to the sample sy-ringe, thereby allowing the sequential analysis of multiple samples In this waythe stopped-flow apparatus can be used for the routine analysis of several hun-dred samples per hour

mix-Another automated approach to kinetic analyses is the centrifugal analyzer,

a partial cross section of which is shown in Figure 13.9 In this technique thesample and reagents are placed in separate wells oriented radially around a circu-lar transfer disk attached to the rotor of a centrifuge As the centrifuge spins, the

quench

To stop a reaction by suddenly changing

the reaction conditions.

stopped flow

A kinetic method of analysis designed to

rapidly mix samples and reagents when

using reactions with very fast kinetics.

Mixing chamber Syringe drive

Detector

Light source Monochromator

Figure 13.9

Schematic diagram of a centrifugal analyzer showing (a) the wells for holding the sample and reagent; (b) mixing of the sample and reagent; and (c) the configuration of the spectrophotometric detector.

sample and reagents are pulled by the centrifugal force to the cuvette, wheremixing occurs A single optical source and detector, located above and below thetransfer disk’s outer edge, allows the absorbance of the reaction mixture to bemeasured as it passes through the optical beam The centrifugal analyzer allows anumber of samples to be analyzed simultaneously For example, if a transferplate contains 30 cuvettes and rotates with a speed of 600 rpm, it is possible tocollect 10 data points per sample for each second of rotation.

The ability to collect kinetic data for several hundred samples per hour is oflittle consequence if the analysis of the data must be accomplished manually Be-sides time, the manual analysis of kinetic data is limited by noise in the detector’ssignal and the accuracy with which the analyst can determine reaction rates fromtangents drawn to differential rate curves Not surprisingly, the development ofautomated kinetic analyzers was paralleled by the development of analog anddigital circuitry, as well as computer software for the smoothing, on-line integra-tion and differentiation, and analysis of kinetic signals.12

13A.3 Quantitative Applications

Chemical kinetic methods of analysis continue to find use for the analysis of a ety of analytes, most notably in clinical laboratories, where automated methods aid

vari-in handlvari-ing a large volume of samples In this section several general quantitativeapplications are considered

Enzyme-Catalyzed Reactions Enzymes are highly specific catalysts for cal reactions, with each enzyme showing a selectivity for a single reactant, or sub- strate For example, acetylcholinesterase is an enzyme that catalyzes the decomposi-

biochemi-tion of the neurotransmitter acetylcholine to choline and acetic acid Manyenzyme–substrate reactions follow a simple mechanism consisting of the initial for-mation of an enzyme–substrate complex, ES, which subsequently decomposes toform product, releasing the enzyme to react again

13.15

If measurements are made early in the reaction, the product’s concentration is

neg-ligible, and the step described by the rate constant k–2can be ignored Under theseconditions the rate of the reaction is

13.16

To be analytically useful equation 13.16 needs to be written in terms of theconcentrations of enzyme and substrate This is accomplished by applying the

“steady-state” approximation, in which we assume that the concentration of ES is

essentially constant After an initial period in which the enzyme–substrate complexfirst forms, the rate of formation of ES

and its rate of disappearance

The specific molecule for which an

enzyme serves as a catalyst.

steady-state approximation

In a kinetic process, the assumption that

a compound formed during the reaction

reaches a concentration that remains

constant until the reaction is nearly

complete.

Figure 13.10

Plot of equation 13.18 showing limits for which a chemical kinetic method of analysis can be used to determine the concentration

of a catalyst or a substrate.

are equal Combining these equations gives

k1([E]0– [ES])[S] = k–1[ES] + k2[ES]

which is solved for the concentration of the enzyme–substrate complex

13.17

where Kmis called the Michaelis constant Substituting equation 13.17 into

equa-tion 13.16 leads to the final rate equaequa-tion

13.18

A plot of equation 13.18, shown in Figure 13.10, is instructive for defining

con-ditions under which the rate of an enzymatic reaction can be used for the

quantita-tive analysis of enzymes and substrates For high substrate concentrations, where

[S] >> Km, equation 13.18 simplifies to

13.19

where Vmaxis the maximum rate for the catalyzed reaction Under these conditions

the rate of the reaction is pseudo-zero-order in substrate, and the maximum rate

can be used to calculate the enzyme’s concentration Typically, this determination is

made by a variable-time method At lower substrate concentrations, where

[S] << Km, equation 13.18 becomes

13.20

The reaction is now first-order in substrate, and the rate of the reaction can be used

to determine the substrate’s concentration by a fixed-time method

Chemical kinetic methods have been applied to the quantitative analysis of a

number of enzymes and substrates.13One example, is the determination of glucose

based on its oxidation by the enzyme glucose oxidase.6

glucose oxidase

Glucose + H2O + O2——————→gluconolactone + H2O2Conditions are controlled, such that equation 13.20 is valid The reaction is moni-

tored by following the rate of change in the concentration of dissolved O2using an

appropriate voltammetric technique

Nonenzyme-Catalyzed Reactions The variable-time method has also been used to

determine the concentration of nonenzymatic catalysts Because a trace amount of

catalyst can substantially enhance a reaction’s rate, a kinetic determination of a

cat-alyst’s concentration is capable of providing an excellent detection limit One of the

most commonly used reactions is the reduction of H2O2by reducing agents, such as

thiosulfate, iodide, and hydroquinone These reactions are catalyzed by trace levels

of selected metal ions For example the reduction of H2O2by I–

2I–+ H2O2+ 2H3O+t2H

2O + I2

is catalyzed by Mo(VI), W(VI), and Zr(IV) A variable-time analysis is conducted

by adding a small, fixed amount of ascorbic acid to each solution As I is produced,

d dt

k K

V K

d dt

k K

Concentration of substrate

Analytical region for the analysis of enzymes Maximum rate = Vmax

Analytical region for the analysis of substrates

d [P]

d t

it rapidly oxidizes the ascorbic acid and is, itself, reduced back to I– Once all theascorbic acid is consumed, the presence of excess I2provides a visual end point.

Noncatalytic Reactions Chemical kinetic methods are not as common for thequantitative analysis of analytes in noncatalytic reactions Because they lack the en-hancement of reaction rate obtained when using a catalyst, noncatalytic methodsgenerally are not used for the determination of analytes at low concentrations.4

Noncatalytic methods for analyzing inorganic analytes are usually based on a plexation reaction One example was outlined in Example 13.4, in which the con-centration of aluminum in serum was determined by the initial rate of formation of

com-its complex with 2-hydroxy-1-naphthaldehyde p-methoxybenzoyl-hydrazone.10

The greatest number of noncatalytic methods, however, are for the quantitativeanalysis of organic analytes For example, the insecticide methyl parathion has beendetermined by measuring its rate of hydrolysis in alkaline solutions.14

13A.4 Characterization Applications

Chemical kinetic methods also find use in determining rate constants and ing reaction mechanisms These applications are illustrated by two examples fromthe chemical kinetic analysis of enzymes

elucidat-Determining Vmaxand Km for Enzyme-Catalyzed Reactions The value of Vmaxand

Km for an enzymatic reaction are of significant interest in the study of cellularchemistry.15From equation 13.19 we see that Vmaxprovides a means for determin-

ing the rate constant k2 For enzymes that follow the mechanism shown in reaction

13.15, k2is equivalent to the enzyme’s turnover number, kcat The turnover number

is the maximum number of substrate molecules converted to product by a single tive site on the enzyme, per unit time Thus, the turnover number provides a directindication of the catalytic efficiency of an enzyme’s active site The Michaelis con-

ac-stant, Km, is significant because it provides an estimate of the substrate’s lar concentration.15b

intracellu-Values of Vmaxand Kmfor reactions obeying the mechanism shown in reaction13.15 can be determined using equation 13.18 by measuring the rate of reaction as afunction of the substrate’s concentration The curved nature of the relationship be-tween rate and the concentration of substrate (see Figure 13.10), however, is incon-venient for this purpose Equation 13.18 can be rewritten in a linear form by takingits reciprocal

A plot of 1/v versus 1/[S], which is called a double reciprocal, or Lineweaver–Burk plot, is a straight line with a slope of Km/Vmax, a y-intercept of 1/Vmax, and an

x-intercept of –1/Km(Figure 13.11)

Elucidating Mechanisms for the Inhibition of Enzyme Catalysis An inhibitor acts with an enzyme in a manner that decreases the enzyme’s catalytic efficiency.Examples of inhibitors include some drugs and poisons Irreversible inhibitors co-valently bind to the enzyme’s active site, producing a permanent loss in catalytic ef-ficiency even when the inhibitor’s concentration is decreased Reversible inhibitorsform noncovalent complexes with the enzyme, thereby causing a temporary de-

Figure 13.11

Lineweaver–Burk plot of equation 13.18.

crease in catalytic efficiency If the inhibitor is removed, the enzyme’s catalytic

effi-ciency 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 inhibition (Figure 13.12a), the

substrate and the inhibitor compete for the same active site on the enzyme Because

the substrate cannot bind to an enzyme that is already bound to an inhibitor, the

enzyme’s catalytic efficiency for the substrate is decreased With noncompetitive

in-hibition (Figure 13.12b) the substrate and inhibitor bind to separate active sites on

the enzyme, forming an enzyme–substrate–inhibitor, or ESI complex The presence

of the ESI complex, however, decreases catalytic efficiency since only the

enzyme–substrate complex can react to form product Finally, in uncompetitive

in-hibition (Figure 13.12c) the inhibitor cannot bind to the free enzyme, but can bind

to the enzyme–substrate complex Again, the result is the formation of an inactive

ESI complex

The three reversible mechanisms for enzyme inhibition are distinguished by

ob-serving how changing the inhibitor’s concentration affects the relationship between

the rate of reaction and the concentration of substrate As shown in Figure 13.13,

when kinetic data are displayed as a Lineweaver–Burk plot, it is possible to determine

which mechanism is in effect

13A.5 Evaluation of Chemical Kinetic Methods

Scale of Operation The detection limit for chemical kinetic methods ranges from

minor components to ultratrace components (see Figure 3.6 in Chapter 3) and is

principally determined by two factors: the rate of the reaction and the instrumental

method used for monitoring the rate Because the signal is directly proportional to

the reaction rate, faster reactions generally result in lower detection limits All other

considerations being equal, the detection limit is smaller for catalytic reactions than

for noncatalytic reactions Not surprisingly, chemical kinetic methods based on

catalysis were among the earliest techniques for trace-level analysis Ultratrace

analysis is also possible when using catalytic reactions For example, ultratrace levels

of Cu (<1 ppb) can be determined by measuring its catalytic effect on the redox

re-action between hydroquinone and H2O2 Without a catalyst, most chemical kinetic

methods for organic compounds involve reactions with relatively slow rates,

limit-ing the method to minor and higher concentration trace analytes Noncatalytic

chemical kinetic methods for inorganic compounds involving metal–ligand

com-plexation may be fast or slow, with detection limits ranging from trace to minor

lev-els of analyte

Chapter 13 Kinetic Methods of Analysis 639

1

Vmaxy-intercept =

1 V

1 [S]

Km

VmaxSlope =

–1

Kmx-intercept =

Competitive inhibition

+ I

EI

Noncompetitive inhibition

+ I

EI + S ESI

+ I

Uncompetitive inhibition

ESI

+ I

The second factor influencing detection limits is the mental method used to monitor the reaction’s progress Mostreactions are monitored spectrophotometrically or electro-chemically The scale of operation for these methods was dis-cussed in Chapters 10 and 11 and, therefore, is not discussedhere.

instru-Accuracy As noted earlier, chemical kinetic methods are tentially subject to larger errors than equilibrium methods due

po-to the effect of uncontrolled or poorly controlled variables,such as temperature and solution pH Although the direct-computation chemical kinetic methods described in thischapter can yield results with moderate accuracy (1–5%), re-action systems are encountered in which accuracy is quitepoor An improvement in accuracy may be realized by usingcurve-fitting methods In one study, for example, accuracywas improved by two orders of magnitude (500–5%) by re-placing a direct-computation analysis with a curve-fittinganalysis.16Although not discussed in this chapter, data analy-sis methods that include the ability to compensate for experi-mental error3,17–19 can lead to a significant improvement inaccuracy

Precision The precision of a chemical kinetic method is ited by the signal-to-noise ratio of the instrumental methodused to monitor the reaction’s progress With integral methods,precisions of 1–2% are routinely possible The precision for dif-ferential methods may be somewhat poorer, particularly fornoisy signals, due to the difficulty in measuring the slope of anoisy rate curve.19It may be possible to improve the precision

lim-in this case by uslim-ing a comblim-ination of signal averaglim-ing andsmoothing of the data before its analysis

Sensitivity The sensitivity for a one-point fixed-time integral method of analysis isimproved by making measurements under conditions in which the concentration ofthe monitored species is larger rather than smaller When the analyte’s concentra-tion, or the concentration of any other reactant, is monitored, measurements arebest made early in the reaction before its concentration has substantially decreased

On the other hand, when a product is used to monitor the reaction, measurementsare more appropriately made at longer times For a two-point fixed-time integral

method, sensitivity is improved by increasing the difference between times t1and t2

As discussed earlier, the sensitivity of a rate method improves when using the initialrate

Selectivity The analysis of closely related compounds, as we have seen in lier chapters, is often complicated by their tendency to interfere with one an-other To overcome this problem, the analyte and interferent must first be sepa-rated An advantage of chemical kinetic methods is that conditions can often beadjusted so that the analyte and interferent have different reaction rates If thedifference in rates is large enough, one species may react completely before theother species has a chance to react For example, many enzymes selectively cat-

ear-1 V

1 [S]

[I]

[I] = 0

1 V

1 [S]

[I]

[I] = 0

1 V

1 [S]

Effect of the concentration of inhibitor on

the Lineweaver–Burk plots for

(a) competitive inhibition, (b) noncompetitive

inhibition, and (c) uncompetitive inhibition.

The inhibitor’s concentration increases in the

direction shown by the arrows.

alyze a single substrate, allowing the quantitative analysis of that substrate in the

presence of similar substrates

The conditions necessary to ensure that a faster-reacting species can be

kineti-cally separated from a more slowly reacting species can be determined from the

ap-propriate integrated rate laws As an example, let’s consider a system consisting of

an analyte, A, and an interferent, B, both of which show first-order kinetics with a

common reagent To avoid an interference, the relative magnitudes of their rate

constants must be sufficiently different The fractions, f, of A and B remaining at

any point in time, t, are given as

13.21

13.22

For a first-order reaction we can write, from equations 13.2, 13.21, and 13.22

ln(fA)t = –kAt ln(fB)t = –kBt

Taking the ratio of these two equations gives

Thus, for example, if 99% of A is to react before 1% of B has reacted

then the rate constant for A must be 458 times larger than that for B Under these

conditions the analyte’s concentration can be determined before the interferent

be-gins to react If the analyte has the slower reaction, then it can be determined after

the interferent’s reaction is complete

The method described here is impractical when the simultaneous analysis

of both A and B is desired The difficulty in this case is that conditions favoring

the analysis of A generally do not favor the analysis of B For example, if

condi-tions are adjusted such that 99% of A reacts in 5 s, then B must reach 99%

com-pletion in either 0.01 s if it has the faster kinetics or 2300 s if it has the slower

kinetics

Several additional approaches for analyzing mixtures have been developed that

do not require such a large difference in rate constants.3,4Because both A and B

react at the same time, the integrated form of the first-order rate law becomes

Ct= [A]t+ [B]t= [A]0e –kAt+ [B]0e –kBt 13.23

where Ctis the total concentration of A and B If Ctis measured at times t1and t2,

the resulting pair of simultaneous equations can be solved to give [A]0 and [B]0

The rate constants kAand kB must be determined in separate experiments using

standard solutions of A and B Alternatively, if A and B react to form a common

product, P, equation 13.23 can be written as

P = [A](1 – e –kAt) + [B](1 – e –kBt)

k k

f f t t

Figure 13.14

Determination of the concentration of a

slowly reacting analyte, B, in the presence of

a faster reacting analyte, A.

Atoms with the same number of protons

but different numbers of neutrons are

called isotopes.

alpha particle

A positively charged subatomic particle

equivalent to a helium nucleus ( α ).

beta particle

A charged subatomic particle produced

when a neutron converts to a proton, or

a proton converts to a neutron ( β ).

negatron

The beta particle formed when a neutron

converts to a proton; equivalent to an

electron ( –1β 0 ).

Again, a pair of simultaneous equations at times t1 and t2can be solved for [A]0and [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) as a function of time consists of two regions

At short times the plot is curved since A and B are reacting simultaneously At latertimes, however, the concentration of the faster-reacting component, A, decreases to

0, and equation 13.23 simplifies to

pro-13B Radiochemical Methods of Analysis

Atoms with the same number of protons but a different number of neutrons are

called isotopes To identify an isotope we use the symbol A

Z E, where E is the ment’s atomic symbol, Z is the element’s atomic number (which is the number of protons), and A is the element’s atomic mass number (which is the sum of the

ele-number of protons and neutrons) Although isotopes of a given element have thesame chemical properties, their nuclear properties are different The most impor-tant difference between isotopes is their stability The nuclear configuration of a sta-ble isotope remains constant with time Unstable isotopes, however, spontaneouslydisintegrate, emitting radioactive particles as they transform into a more stableform

The most important types of radioactive particles are alpha particles, beta

parti-cles, gamma rays, and X-rays An alpha particle, which is symbolized as α, is alent to a helium nucleus, 4He Thus, emission of an alpha particle results in a newisotope whose atomic number and atomic mass number are, respectively, 2 and 4less than that for the unstable parent isotope

214

82Pb →214

83Bi + 0 –1β

Converting a proton to a neutron results in the emission of a positron,0β

30P →30Si + 0βEmission 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 betaparticles

Although similar to chemical kinetic methods of analysis, radiochemical

meth-ods are best classified as nuclear kinetic methmeth-ods In this section we review the

ki-netics of radioactive decay and examine several quantitative and characterization

applications

13B.1 Theory and Practice

The rate of decay, or activity, for a radioactive isotope follows first-order kinetics

13.24

where A is the activity, N is the number of radioactive atoms present in the sample

at time t, and λis the radioisotope’s decay constant Activity is given in units of

dis-integrations per unit time, which is equivalent to the number of atoms undergoing

radioactive decay per unit time

As with any first-order process, equation 13.24 can be expressed in an

inte-grated form

Substituting equation 13.25 into equation 13.24 gives

A =λN0e– λt = A0e– λt 13.26

By measuring the activity at time t, therefore, we can determine the initial activity,

A0, or the number of radioactive atoms originally present in the sample, N0

An important characteristic property of a radioactive isotope is its half-life, t1/2,

which is the amount of time required for half of the radioactive atoms to

disinte-grate For first-order kinetics the half-life is independent of concentration and is

given as

13.27

Since the half-life is independent of the number of radioactive atoms, it remains

constant throughout the decay process Thus, 50% of the radioactive atoms

disinte-grate in one half-life, 75% in two half-lives, and 87.5% in three half-lives

Kinetic information about radioactive isotopes is usually given in terms of the

half-life because it provides a more intuitive sense of the isotope’s stability

Know-ing, for example, that the decay constant for 90Sr is 0.0247 yr–1 does not give an

im-mediate sense of how fast it disintegrates On the other hand, knowing that the

half-life for 90Sr is 28.1 years makes it clear that the concentration of 90Sr in a sample

remains essentially constant over a short period of time

13B.2 Instrumentation

Alpha particles, beta particles, gamma rays, and X-rays are measured using the

par-ticle’s energy to produce an amplified pulse of electric current in a detector These

pulses are counted to give the rate of disintegration Three types of detectors

com-monly are encountered: gas-filled detectors, scintillation counters, and

semiconduc-tor detecsemiconduc-tors The gas-filled detecsemiconduc-tor consists of a tube filled with an inert gas, such

as Ar When radioactive particles enter the tube, they ionize the inert gas, producing

a large number of Ar+/e–ion pairs Movement of the electrons toward an anode and

the Ar+toward a cathode generates a measurable electric current A Geiger counter

is one example of a gas-filled detector A scintillation counter uses a fluorescent