CHAPTER I

KINETICS OF CONSECUTIVE,

FIRST-ORDER REACTIONS :

AN OVERVIEW

CHAPTER I

KINETICS OF CONSECUTIVE, FIRST-ORDER REACTIONS

: AN OVERVIEW

1. Preliminary Considerations.

It IS an established fact that reactlons which exhibit slmple

kinetics, such as the ac~d-catalyzed hydrolysis of simple esters like ethyl acetate.

take place In a number of mechanistic stages, each involving the formation of a

translent ~ntermed~ate or transition state [I] Each of these steps 1s cons~dered to be

revers~ble In the microscopic sense. The slowest step in the reaction sequence,

compared to whlch all the other steps are very many tlmes faster, then determ~nes

the overall rate of the reaction and is referred to as the rate-determ~nlng or rate-

11n11tlng step In the above reaction, the intermed~ate formed at each stage IS of

translent existence and non-isolable It is then convenient to formulate the entlre

reaction as proceeding in a single step

it oltcli lhappens that the product 01' a chem~cal reactlon may

undergo further react~on to glve yet another product. The reactions are then sa~d to

be consecutive or successive The two reactlons may be of the same category, I e ,

~nvolve the samc reactlon on the same type of funct~onal group, as In the acrd-

catalyred hydrolys~s of dl-esters or the alkaline hydrolysis of dihalo-alkanes, or

they may be different, as In the case of the oxidation of primary alcohols first to

give the aldehyde and then the carboxylic acld, or the hydrolysis o f a P-ketoester

first to form a P-ketoac~d whlch then undergoes decarboxylat~on to give a ketone

When the rate-determining steps in each of the sequence of reactions have

comparable velocities, the kinetics of the overall reactlon becomes complicated,

even though the individual steps may exhibit simple kinetics

The in~tial susbshate of such two-step, consecutive reactions IS

usually called the (inlt~al) reactant, the product at the end of the first step is referred

to as the intermediate, and the product obtalned at the end of the second step as the

(final) product. Thus. In the acld-catalyzed hydrolys~s of dlethyl succlnatc, the

intermediate IS the mono-ester, ethyl hydrogen succinate, and the (final) product 1s

succlnlc acid What distinguishes the ~ntermed~ate In a consecutive reactlon from

the transltlon states v~sualized as formed at each mechan~stic stage of a reactlon

sequence, is that the former IS usually sola able, the reactlon conditions permitting,

and, if stable enough, 11 can be prepared and its k~netics studled separately Thus, 11

is poss~ble to prepare the intermediate of the above reaction, ethyl hydrogen

succlnate, by esterifying succinic acid wth equ~molar amount of ethyl alcohol, and

purifying 11 by removing any di-ester formed simultaneously. In the hydrolysis of

dielhyl succlnate, the first step leading to the format~on of the intermed~ate, ethyl

hydrogen succinate, IS charactenzed by a rate-coeffic~ent, denoted by kl, and the

second step, leading to the formation of the product, succin~c acld, by a second

rate-coeffic~ent, denoted by kl. In this particular case, both kl and kl happen to be

(pseudo-) first order, but In the general case, they can be of any order and also be

different In order

It is generally convenient to study consecut~ve reactlons when

both the steps are of first-order in kinetics. The mathematical formulations are then

simpler, the data can be analyzed more easily and the individual rate-coefficients,

pertalnlng to the two steps can be evaluated from the data more readily, relatively

speaking True first-order react~ons are rare Radloactlve decay processes are

amongst those which obey first-order kinetics, and there are numerous Instances of

two-step and multi-step decay processes But, radioactive decays are no1 chemical

reacllons In the usual scnse Wh~le true first-order chelnlcal rcactlons arc rare

Indeed, ~t 1s poss~ble to make reactlons of h~gher order exhiblt first-order kinetics

by surtably adjusting the reactlon condlt~ons This is usually ach~eved by

mantaming the concentrations of all the reagents much higher than the rn~tial

concentratron of the single reactant, or by maintaining a constant concentration of'

the catalyst. Thus, in the alkuline hydrolysis of diethyl succlnate, both the steps

can be reduced to first-order by maintaining the hydroxide Ion concentratron a

constant by the use of a pH-stat. The present thesis IS concerned mainly with the

klnetrcs of two-step, consecutive, first-order reaction\

The concept of consecutlve reactions can he extended to embrace

more steps, such as the three-step, the four-step and beyond Examples of three-

step, consccutlve reactlons Include the hydrolysis of' tr~cthyl mcsrtoatc and the

ox~dation of toluene to benzo~c acid via benzyl alcohol and benzaldehyde In the

prcsenr thesis, an lnvestlgatlon 1s made of the poss~blllty of the resolut~on of the

Llnelrcs of three-step, consecutive, first-order reactlons wrth rclcrcncc to the

formation of the (final) product Examples of four-step, consecutlve reactlons

include the stepwise ligand substrtution in a four co-ord~nate complex, these are not

considered In the present work. The ult~mate IS the n-step, consccutlve reactlon

(where n 1s very large), exemplified by polymerizat~on reactions. They are

excluded from the purvrew of the present work, as their complex klnetlcs are dcalt

w ~ t h by ent~rely different procedures.

2. The Integrated Rate Equations for the Reactant, Intermediate,

Product and Co-product in Two- step, and for the Product in

Three-step, Consecutive, Irreversible, First-order Reactions

Reverting to the two-step, consecutlve, first-order reactlon, the slmplest of

the s~tuations arises when both the steps are irrevers~ble:

In t h ~ s scheme, all of the reactant A Initially present is converted, through

B. Into the product C. An example of t h~s scheme is the acid-catalyzed hydrolys~s

of di-esters such as diethyl succinate

The d~fferentlal equations are

The Integrated equations were first arrived at by Esson way back In 1866

j2j The Integration, which can be carr~ed out In a few steps, 1s glven In all

textbooks and treatises on klnetlcs, Including [3]

If A alone IS Initially presen: at concentration Ao, with B and C not

present, I e , Bo = Co = 0, the concentrations of the three species at any tlme t after

thc uommenccmenl or lhc reaction, arc given by

The equations for B and C, when they too are initially present, i.e ,

Bo, Co f 0, are more complicated ([3], pp. 33-36), but the present work is restricted

in scope to the cases were A alone is initially present in the reaction system.

It is often convenient in such reactions to follow the k~netlcs by

monitor~ng the change in the concentration of the co-product, D, defined as the

common specles formed in both the steps, such as the number of free carboxyl~c

a c ~ d groups formed in the ac~d-catalyzed hydrolys~s of a di-ester. the number of

halldc lons released during the hydrolysis of a dihalidc using excess alkali, etc

[4 a1

The exact shapes of the concentration versus tlme curves of the three

species, A, B and C, depend on the relative magnitudes of the rate coeffic~ents k,

and k2 The k ~ n e t ~ c curves of the three species, and also of the co-product, D, are

illustrated In Fig. 1 for the case kl = 0 1 min'l and k2 = 0 2 mln'l.

It must be stressed that A, B, C and D in the above equations refer to the

molar concentrations of the respective species present in the reaction mixture at

tlme 1, as determined by a method such as titrimetry, gravimetry, etc If

spcclrophotomctry has been cmployed, then only the particular spccics must have

absorptlon In the wavelength at which the progress of the reaction had been

mon~tored, in order that the absorbance can be used to calculate the concentration

of the species.

In the extreme case where k2 >> kl, B is convened ~ n t o C as fast as ~t IS

formed from A, the kinetic curve of C coincides with that of A, and the forinatlon

of C reduces to a simple first-order process with the rate coefficient k l .

The three-step, consecutive, irreversible, first-order reaction may be

represented as

(Here symbol E has been employed to designate the final product, rather than the

logical D, In order to drstrnguish ~t from the co-product in two-step, consecutive,

first-order reactions, to which the symbol D has already been assigned). The

integrated equations for the four specres were first given by Rakowsky [S]. They

can be derrved by the method of operational calculus [6] or by the method of

Morrta [7] If the reactant A alone has been initially present at a concentration of

Ao, the concentratlon of E at any Instant IS glven by

Surpr~singly, there appears to be no repon on the resolut~on of the k~netlcs,

I e , the determ~nation of all the three first-order rate coefficients, of a reactlon of

t h ~ s type from the klnet~c data pena~ning to a single species, such as E Mon~torlng

the change in the concentration of the reactant A alone, would lead to an estrmate

of kl only The concentrations of the two ~ntermedrates B and C reach maxrma (at

d~iferenl moments)before declrnrng to zero, and that of thc final product E

Increases slowly at first durlng an ~nductron penod. A typ~cal reactlon would be the

n~tration of bcnzenc to sym- or 1,3,5-t11n1trobenzene, vla nltroben~enc and m-

drnitrobenzene, but the krnetlcs can be mean~ngfully investigated only if the

formatron of lsomerlc ~ntermed~ates and products (such as o- and p- d~n~trobenzene

dur~ng the second stage) is absent at every stage; for such IS the sensitivity of the

equatrons to small errors In the estimation of the concentration of the species be~ng

monitored, the format~on of even a small quantity of an isomer would be detractive

rn respect of the evaluation of the rate coefficients. A better example to study, with

l~ttle possibility of side reactions, is the acld- catalyzed hydrolysis of symmetrical

tn-esters, such as triethyl mesitoate, E in this case being rnesito~c acid formed at

the th~rd and final step.

3. Early History.

It was In 1867 that Guldberg and Waage published the law of mass action as

goveming the kinetics of chemical reactions, on the basis of their investigation of

the revers~ble formation of ethyl acetate from ethyl alcohol and acetic acid [8]

Surprisingly, the very first paper on the k~netlcs of a two-step, consecutlve. fi rst-

order, irreversible reactlon , had appeared one year earlier Harcourt and Esson in

1866 reported thelr findings "On the Laws of Comexion between the Cond~tions of

a Chem~cal Change and its Amount", based on their study of the reaction between

potasslum permanganate and oxalic acid in the presence of manganous sulphate [2]

In an appended note, Esson derived the integrated rate equations for A, B and C,

but regretted that "the number and exactness of the experimental results are not

sufficient to enable us to extract from the complicated equation trustworthy values

[of the rate coefficients]."

'l'hls admission g~vcs a hlnt of thc problcm that constantly plagucs thc

klnet~cs of consecutive reactions even in the simplest of cases, v~z . , where there arc

just two steps and both are first-order and irreversible. the Integrated rate equatlons.

goveming the concentrations of the intermediate and the product as a funct~on of

the reactlon time, are complicated in the sense that they contain the sum (or

difference) of two exponential terms Such transcendental equations cannot be

solved directly for the two rate coefficients, kl and kl; the data must haw a hlgh

degree of accuracy for obtaining meaningful estimates of the rate coefficients.

Forty years passed after the work of Harcourt and Esson before the next

publication in the kinetics of consecutive reactions appeared. Kaufler reported the

results of just three runs involving the hydrolysis of 2,7-dicyanonaphthalene in

amyl alcohol using excess aqueous alkali at 126' C. (91. He followed the p r o p s s

of the reaction, which gives naphthalene-2,7- dicarboxylate a n ~ o n as the final

product, by sweeping the ammoma, the co-product released at each step of the

reaction, w t h a cumnt of alr, absorbing it in a known excess of standard a c ~ d and

back-titrating with alkal~, a clearly cumbersome method. He had eight or fewer

readings In each of just three runs and these were of ~ndeterm~nate levels of

accuracy Nevertheless, he was able to arrive at est~mates of kl and k2 for each of

the three runs by adopting the tedious method of successive approximat~ons. he

tr~ed out different guesses of kl and k, in the integrated rate equation for the

formation of the co-product, ammonia, till he obtained the best fit of the

experimental data. Considering the amount of effort that should have gone into it,

w~thout the aid of electronic calculators or computers, as Swain later remarked

[lo], one can find little fault with the closeness of fit achieved by Kaufler Indeed,

these three runs carried out by Kaufler in 1906 remain such a milestone In the

experimental investigation of consecutive reactions, that these data were used by

workers, including Swain [IS] and Frost [4a] (pp. 170-171), many decades later,

for demonstrating the effectiveness of their own methods of evaluat~on of the rate

coeffic~ents of two-step, consecutive, first-order reactions

I t must be emphasized that the problem becomes daunt~ng only whcn ono

tr~es to evaluate the consecutive rate coefficients from the kinet~c data ~erta~nine. to

a slnele suedes, such as ammonia, the co-product D, l~berated at different reactlon

tlmes during the hydrolys~s of 2,7-dicyanonaphthalene by excess alkal~, uslng eqn

(7) The task becomes greatly s~mpllfied if e~ther of the two steps can be followed

separately Thus. Meyer [ l l ] in 1909, carr~ed out a scparate run with glycol

monoacetate, the intermediate In the acid-catalyzed hydrolysis of glycol dtacetate.

to obtain k2 directly, then us~ng this value of k2 in eqn. (7), he was able to evaluate

kl from the klnet~c data of the hydrolysis of the diacetate But such approaches have

been rarely employed Much effort has gone into the resolut~on of the kinetlcs of

consecutive reactions from data pertaining to a single species or a system property

(such as spectrophotometric absorbance) with varying degrees of success. And the

present t h e m is concerned mainly with this challenging problem.

4. Implications of the Symmetry Properties of the Equation

for Product Formation Data in Two - step, consecutive, First

order Reactions: the Occurrence of Dual Solutions.

I'erhaps the muat ~ncta~vc theorettcal treattueut ol' two-hrcp. Lonaccut]\c.

first-order rcactlons. covering also the cases where one or both the steps arc

revers~hle. was published by Rakowsky In 1907 151. He showed that. In the

lrreverslble case, the concentratlon of the reactant A, given by eqn (4) declines In a

simple cxponent~al manner with tlme, only kl , the rate coefiic~ent of the first step,

can be evaluated by follow~ng the concentratlon of A. The concentratron of the

rntermediate B butlds up lnlt~ally, at a pace dependtng on the relat~ve magn~tudes of

kl and kz, reaches a maxlmum and then declines asymptotrcally The product C IS

very slow to form In the early stages of the reactlon, Indicating an lnduct~on perlod

in rts formation; the end of the Induction period, marked by an ~nflexlon in the C

versus t curve, colncldes w ~ t h the moment when the B reaches 11s maximum The

concentratton or C then increases rapldly wlth tlme, to level off finally at Ao. Thc

ociilrrcncc of an tndu~tton period in the formation of a compound IS indeed

d~agnostlc of consecutive processes preceding 11, be they In two steps or more.

Rakowsky [5] polnted out to an Inherent symmetry in the integrated rate

equation for product formation In the two-step reaction: the two rate coefficients

can be lnterchanged In eqn (6) w~thout affecting the value of C glven by ~t at any

Instant of reaction That IS, two reactions, in whrch the two rate-coefficients are

Interchanged between the two steps, will give identical C versus t plots. For

example, the C versus t c w e for the reaction in which kl = I mtn-I and kt = 0.2

m1n.I will be ~dent~ca l with the C versus t plot for the reaction in which kl = 0 2

mln? and k2 = 1 mln", for the same initla1 concenuation, Ao, of the reactant in the

two reactaons (Fig. I):

This lmplies that a glven set of C versus t data of a two-step reactlon wrll lead to

dual palrs of solutions for kl and k2. Thus, the above data, if had been

experrmentally obtained, would lead to two pairs of solutrons for kt and k l ,

pa11 (I). kl = 1 midl and k2 = 0.2 min.' ,

and, parr ( 2 ) kl = 0 2 min.' and k2 = I m1n.I

1 . m 1 n I

T h ~ s duality of solutions, in turn, leads to an amb~gurty In the assignment of

the two rate coefficients evaluated from the data to the two steps. whether the

smaller rate coeffic~ent, k,~,,, pertains to the first step or to the second and the larger

rate coeffic~ent, klaSl, to the second step or to the first In such situat~ons, the

resolution of the amb~guity and the correct assignment of the two ratc coefficrcnts

to the two steps usually require extraneous lnformatron, such as a theoret~cal

I ) I C I ~ I ~ C ila IO w I ~ I c I ~ ~) l ' thc Iwo S ~ P S must bc hater

Such an amblgu~ty, however, does not plague the resolut~on of the ratc

coefficients from B versus t or from D versus t data, since eqns. (5) and (7) are not

syrnmetrlc wrth respect to k~ and k2: two reactions with k l and kl rnterchanged will

have d~fferent D versus t profiles, whatever be the method employed, data

pertainrng to co-product formation usually lead to unlque solutions of pairs of k l

and k2 for the reaction.

, CIA. 100686 1 0 1959 103264 104429 / 0.5418 106241 106920 / 07477 107934 / 08308

3 2

Similarly, eqn. (8) for the formation of the product E, in a three-step,

consecutive, irreversible, first-order reaction is symmetric wlth respect to the

~nterchange of k l , k2 and kl: there will be a trilemma with regard to the assignment

^ the three rate coefficients resolved from product formation data to the three

4 5 6 9 7 10 8

Individual steps, and information extraneous to the kinetics will be needed to

surmount this uncertainty

5. Time-dependence of Spectrophotometric Absorbance in

Two-step, Consecutive, First-order, Irreversible Reactions:

Duality of Solutions.

Wh~le the fbrugolng equations express the (molar) conccntratlons of

rllc varlouh spccics In thc rcaction mlxture as functions o l tlme, the classical

methods of determining the same, such as tltrlmetry, are being decreas~ngly used

nowadays Developments over the past five decades In d~fferent ~nstrun~ental

tcchn~ques have prov~dcd illtcmatlve, sophist~cated means of studylny var~ous

system properties and of following the kinetlcs of reactions. These include a

number of spectroscopic techniques, the foremost being UV-v~s~ble

spectrophotometry With the self-recording, d~gital, double-beam spectro-

photometer, capable of monltorlng to a high degree of p rec~s~on the absorbance

changes of a reaction mixture simultaneously in five different desired wavelengths

at br~ef Intervals of tlme, even just seconds apart, k~netlc techn~ques have come a

long way from the pipette and burette routine of over fifty years ago. In fact, most

of the investigations of kinetics of reactions slnce 1950 have employed

spectrophotometry

If In the reactlon system, just one of the four species, vlz., the reactant A,

the intermed~ate B, the product C or the co-product D, alone has ahsorpt~on In the

wavelength in which the progress of the reaction IS monitored, the s~tuation

becomes simple. Spectrophotometry becomes a method of determining the

concentration of the species, whlch is given by the observed absorbance divided by

the molar extinction coefficient of the species (assuming a path length of 1 cm) and

the above stipulat~ons regarding the evaluation of the rate coefficients from

concentration data apply. if A alone absorbs, only kl can be evaluated from the

data; if B or D alone absorbs, then both k~ and k2 can be determined and ass~gned

unambiguously to the respective slcps; and if the product C alone absorbs, then

dual pairs of solutions, with kl and k2 interchanged, will result. The rate

coefficients can be determined from the absorbance data even if the molar

extinction coefficient of the slngle absorbing species is not known.

But, more often than not, more than one species in the reactlon mixture

absorb in a given wavelength. The observed absorbance, Z, is then the sum of

contrlbutlons by each of the absorbing species.

where E, is the molar extinchon coefficient of the ~ t h . absorb~ng species

and c, 1s its concentration. Thus, In a two-step, consecutive first-order irreversible

reaction (scheme I), if all the three species, viz., A, B and C, are capable of

absorbing In the part~cular wavelength, assuming a path length or cell dimension

of I cm , the observed absorbance of the reaction mlxture is given by

It may be noted that 11 la assumed that the co-product, I), lhas no absorp\ion

I I I tllis uavclcngth - i i usually does not, and also tl~at thc co-rcaclant. i c . thc

reagent that 1s generally added In great excess to br~ng about the reactlon of A,

does not absorb either

Substituting the values of A, B and C given by eqns. (5) - (7),

Since, at the end of the reaction, all of A initially present would have been

convened into C. the infinite tlme absorbance of the reaction mlxture, L, would

correspond to C&, l.e., to &Ec. Then,

Equat~on (12) contalns five constants, vlz , the molar extrnctlon coeffic~ents

of the three species bes~des the two rate coefficients.

Two dlstinct sltuatlons can arlse at thts juncture: ~f the molar extlnctlon

coeftic~ents of A, B and C are all known, then only a unlque palr of solut~ons for hl

and kl arlses, whlch can be ass~gned unamb~guously to the first and the second

steps of scheme I , cxcept under certain restricted cond~tlons discussed by Jackson

rr rri [12], when only one of the two rate coeffic~ents 1s readily obtalned En and

ti can be d~rectly determlned slnce they are the molar extlnctlon coeffic~enls of the

lnlt~al reactant and the final product. If the ~ntermed~ate I3 IS stable enough to be

~soiable, ~ t s molar extlnctlon coeffic~ent, Eg, can also be determlned and used In

cqll (45). wh~ch 1s aay~nnictric w ~ l h rcspccl Lo ~ h c lntcrchangc 01 kl and 62, to

evaluatc the two rate coefficients and asslgn them unarnb~guously to the

corresponding steps. Only the correctly asslgned values of the rate coeff~c~ents can

reproduce the spectrophotometr~c data over the entire course of the reaction.

Ja~kson el a/. [I21 have demonstrated t h ~ s approach In the Instance of the two-step

hydrolysis of a chrom~um(III) complex using the known molar extinction

coeffic~ents of the three species.

An entlrely different situation arises if the intermediate is non- sola able and its

molar extlnctlon coefficient cannot be determined Equat~on (45) then contains

three unknowns, Eg besides k l and k2, to be determined. Any method of analys~s,

be I! graphical or a regression technique, will lead to two sets of solut~ons, viz., (kl,

k2, b) and ( k ~ ' , k2'. EB'), both the sets reproducing the spectrophotometr~c data

throughout. Alcock el a/. [13] encounted such a situabon In a study of the

kinetlcs of the two-step reaction of nitrous acid with hydrogen peroxlde to glve

nitric acid, via the intermediate pemxynivous ac~d , HOONO. Equation (I I) which,

as such is asymmemc with respect to the interchange of k l and k2, permits such an

Interchange when the condition is fulfilled Alcock e/ al found that a Icast squares

analysis of that kinetic data obtained by stopped-flow spectrophotometry at 0.6"C

and pH 1 57, y~elded the two sets of solut~ons,

and kl = 8.548-2 sec", k2 = 3.76 sec", wlth Eu = 2039

It was mathematically impossible to distinguish the correct set of rate

coeffic~ents of the reactlon from the wrong one on the basls of the

spectrophotometr~c data alone. Alcock el al, were able to surmount this d~fficulty

by carrying out another run at a pH of 2 26 and obtain~ng two sets of solutions for

kl, k2 and EB as before Since the molar extlnctlon coefficient of the Intermediate

was not expected to change with pH, the correct paws of the rate coefficlents at each

pH could be readily identified as those which were associated with a constant value

ol'E13, wh~ch turned out to be 58.30

The situation becomes much more complicated when the molar extinction

coefficients of all the three species are unknown or known only imprecisely rhis

requlres the evaluation of five parameters, kl and kl. besides EA, EB and Ec

W~berg [14] had such an unenviable sltuatlon in the two-step oxiJat~on of

benzaldehyde by chromyl acetate. He adopted a computer-based iterative method

to evaluate all the five parameters, starting w ~ t h their rough initla1 estimates (q v )

Since, as a rule, the results become less accurate and less precise as the

number of parameters to be determined from a given set of experimental data

Increases, one always prefers to minimlu the number of parameters to be

determined. As all the terms preceding the two exponential terms in the right-hand

aide of cqn (45) contaln only constants, rt can be written as

where P and Q are constants, since the absorbance of the reactron mlxture

may Increase or decrease wrth trrne, the term on the left-hand side of t h ~ s cquat~on IS

expressed as rts modulus

Equation (13), which represents the time-dependence of the absorbance of

a reaction mlxture pertaining to scheme 1, contains four parameters, viz., P, Q, kl

and k2, which require evaluation from the kinetic data.. Insofar as one is interested

In evaluating the two rate coefficients alone, one would prefer to determine

(optrmize) four parameters rather than five. But this reduction In the number of

parameters to be determined can be achieved only at the expense of introducing an

ambiguity in the assignment of the two determined rate coefficients to the two

reaction steps: the lesser of the two rate coefficients evaluated from the klnetic data,

k,~,,, may pertaln to the first step and the larger, kf,, , to the second, or, vrce versa,

~rrespect~ve of the method of their evaluatron, be ~t graphical or computer-based,

lterat~ve The resolutron of the ambrgu~ty would then necessitate add~tlonai

information, extraneous to the kinettc data, or the results of additional klnetic runs

carr~ed out under d~fferent cond~tlons

In lact, as has bccn polnted out by Moodre [15], many syatcnl propertle\.

hcs~des the photometr~c absorbance, of reactlon mlxtures including those In uhlch

erther of or both the steps may be reversible, obey an equatron of the type

md data on the time-dependence of such a property . Z, as the system is undergoing

oaction can be used to evaluate the apparent rate coeffic~ents p and q

It must be emphasized that, in the simplest case where both thc steps

are ~rreversible, p and q in the above equation correspond to the actual rate

coefficients of the two steps. In the more compl~cated reaction schemes, where

elther the first or the second step, or both, are revers~ble, p and q are "apparent" or

"macroscopic" rate coefficients of the reaction system that can be determ~ned from

the kinetic data by a su~table method; they are phenomenolog~cal quantltles, w~th

units of time -I, like any first-order rate coefficient, and they descr~be the evolution

of the reacting system in time. But they are not rate coefficients pertaining to any

~nd~v~dua l step of the reaction, though their magnitude is composed of the various

rate coeffic~ents of the Individual transformations whlch are called the "elemental"

or "microscopic" rate coefficients of the system. The actual relationship between

the macroscopic and the microscopic rate coeffic~ents depends on the mechan~sm

of the reaction. As there are only two experimentally observable macroscopic rate

coeffic~ents, p and q , as against a larger number of microscop~c rate coeffic~ents

( for example, four individual rate coerficients in the case of reactions where both

the steps are reversible), it IS not possible to evaluate the latter from the lnagn~tudes

0 1 tlic li)rmu~, unless udd~tional ~n lo r~na t~o r~ about tiic rclallon\h~ps hctwccn Ihcin I \

available