I. Determination of chemical reaction rate constants by numerical nonlinear analysis: differential methods

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by  Christopher G. Jesudason

I. Determination of chemical reaction rate
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1
constants by numerical nonlinear analysis:
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2
differential methods
n
a
J
ChristopherG. Jesudason∗
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2 DepartmentofChemistryandCenterforTheoreticalandComputationalPhysics

UniversityofMalaya
]
h 50603KualaLumpur,Malaysia
p
- January 27,2011
n
e
g
.
s
c Abstract
i
s
The primary emphasis of this work on kinetics is to illustrate the a posteri-
y
h oriapproachtoapplications,wherefocusondataleadstonoveloutcomes,rather
p thantheaprioritendenciesofappliedanalysiswhichimposesconstructsonthe
[ natureoftheobservable. Thesecondaryintentionisthedevelopmentofappropri-

atemethodsconsonantwithexperimentaldefinitions. Chemicalkineticequations
1
werelargelydevelopedwiththeassumptionofrateconstantinvarianceandinpar-
v
ticularrateconstantdeterminationusuallyrequiredknowledge oftheinitialcon-
0
centrations. Thesemethodscould notdeterminetheinstantaneous rateconstant.
6
0 Previousworkbasedonprecisesimulationdata[J.Math. Chem43(2008)976–
5 1023]ofabidirectionalchemicalreactionsysteminequilibriumconcludedthatthe
. rateconstantsisafunctionofspeciesconcentrationthroughthedefinedanddeter-
1
minedreactivitycoefficientsforatleastelementaryreactions. Inhomogeneitiesin
0
thereactionmediummightalsoleadtocross-couplingofforcesandfluxes,leading
1
1 toconcentration dependencies. Byfocusing ongradients, itispossibletodeter-
: mineboththeaverageandinstantaneousrateconstantsthatcanmonitorchanges
v
intherateconstantwithconcentrationchangesassuggestedbythistheory. Here,
i
X methodsaredevelopedanddiscussedutilizingnonlinearanalysiswhichdoesnot
require exact knowledge of initial concentrations. These methods are compared
r
a withthosederivedfromstandardmethodologyforknownchemicalreactionsstud-
ied by eminent kineticistsand in one case withareaction whose initial reactant
concentrationwasnotwelldetermined. Thesegradientmethodsareshowntobe
consistentwiththeonesfromstandardmethodsandcouldreadilyserveasalterna-
tivesforstudieswheretherearelimitsorunknownsintheinitialconditions,such
asintheburgeoningfieldsofastrophysicsandastrochemistry,forensics,archeol-
ogyandbiology.Allfourreactionsstudiedexhibitedsemisinusoidal-likechange
withreactantconcentrationchangewhichstandardmethodscannotdetect,which
∗Emails:[email protected][email protected]
1
seemstoconstitutetheobservationofaneweffectthatisnotpredictedbycurrent
formulations,wherethepossibilitythattheobservationsareduetoartifactsfrom
instrumentalerrorsortheoptimizationmethodisreasonedasunlikelysincetheex-
perimentswereconductedbydifferentgroupsatverydifferenttimeswithdifferent
classesofreactions. Twobroadreasonsaregivenforthisobservation,andexper-
iments aresuggested that can discriminatebetween these twoeffects. Although
firstandsecondorderreactionswereinvestigatedhereusingdatafromprominent
experimentalists, themethodappliestoarbitraryfractionalordersbypolynomial
expansion of the ratedecay curves whereclosed form integratedexpressions do
notexist.Integralmethodsfortheabovewillbeinvestigatednext.
Keywords: numericalnonlinearanalysis; orthogonalpolynomialexpansion; chemical
reactionratelaw; apriori; aposteriori
1 INTRODUCTION AND METHODS
Current trends in mathematical applications almost always indicate the creation of
mathematicalstructuresthatarethensupposedandconsideredtomirrorphysicalreal-
ityandexperimentaloutcomes.Lesscommonareapplicationsthatanalyzeexperimen-
taldatausingascloselyaspossibletheoperationaldefinitionofvariablestoelucidate
the validity or otherwise of theories. The main thrust of this sequel is to illustrate
researchthatputspriorityontheexperimentaldataasameanstoconstructingorsug-
gestingtheoreticalandmathematicalstructures.Thedatafromhighlyempiricalfieldof
chemicalkineticsisusedwithinthescopeofthedefinitionofthemeasuredvariablesto
discover/uncoverneweffects;itissuggestedherebasedontheoutcomeoftheanalysis
thatmathematicalanalysisofthedatainotherfieldswithintheoperationaldefinitionof
theempiricalvariablescanelucidatenewphenomenaandsuggesthowappropriateand
consistent theory can be constructed a posteriori, rather than the a priori tendencies
of many applied mathematicians. One aspect of this culture is the cult of prediction
andofpredictabilityinthenaturalscienceswhereresourcesareexpendedinperform-
ing experimentsto verifyand substantiate theories. The methodsdevelopedhere are
ofsecondaryimportancecomparedtotheaposteriorianalysisofthedataanditsout-
comes;thesemethodsrefertovariableswhichcomefromtheexperimentaldefinition.
Foranelementaryreaction
A +ν A ...ν A Products (1)
1 2 2 nr nr →
wedefinetherateconstantkasthefactorintheequation
d[A ]
1 =[A˙]= k[Q] (2)
dt −
where
nr
[Q]= [A ]νi =l (t)=l (t)
i A1A2...Anr Q
iY=1
with l (t) = [R](t) in general and l (t) = [A] (t) in particular, with the notation
R A 1
l (t)l (t)...l (t) = l (t). Thesquarebracketsdenotetheconcentrationofthe
A1 A2 Anr Q
2
species, and t is the time parameter. For the above, the order O, which need not be
integerisdefinedasO = nr ν . Clearly,fortheabove
i=1 i
P
d[A ]
k = 1 /Q. (3)
−(cid:26) dt (cid:27)
Wedeterminek heredirectlybyvariousmethodsofcomputingaverageandinstanta-
neousgradientsforequation(2).Intraditionalmethods,theintegratedratelawexpres-
sion isknownforonlya handfulof integerO valuesof(2) which also requireinitial
conditions;nosuchrestrictionsapplytothecurrentnumericaltechnique.Anotherclass
ofmethodisthroughaleastsquaresoptimizationofthefunctionR(k)forndatapoints
definedas
n 2
dl (t )
A i
R(k)= kl (t )) . (4)
Q i
(cid:18) dt − (cid:19)
Xi=1
Then,
n
dl (t )
′ A i
R(k)=0 kl (t ) l (t )=0
Q i Q i
⇒ (cid:18) dt − (cid:19)
Xi=1
implies
n dlA(ti).l (t )
k = i=1 dt Q i . (5)
−P n l2(t )
i=1 Q i
P
Eq.(5)doesnotrequireiterativemethodssuchasNewton-Raphson’s(NR)todetermine
the rate constant. A variant of the R(k) optimization above is found in Sec(2.0.5);
the reason being that we optimize over an intergratedexpression rather than directly
the rate equation (2) such as (18) for the first order rate constant k and 19 for the
1
second order constant k . All variants of the above methods will be discussed in
2
sequence in what follows. Most kinetic determinations use logarithmic plots with
known initial concentrations, although there have been attempts at integral methods
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and refs. therein] that dispense with the initial
concentration. (There may be ambiguitiesin e.g. [1] concerningchoice of indepen-
dentvariablesthat will be discussed elsewhere). These standardmethodsall assume
constancyoftherateconstantk,andthereforehavenotinspiredmethodologythatcan
detectthechangestotherateconstantthat,accordingtothedetailedresultsofref.[13]
shedsimportantinformationontheactivationenergyprofilechangesduetotheforce
fields acting on the reacting species. There are conceivably many other reasons for
variationwithtimeoftherateconstants;theyincludecouplingofinhomogenoustem-
peraturefield gradientswith chemicalspecies fluxes, leading to physicalvariable in-
homogeneitiesinthereactioncellthatmodifiestherateofreactionwithtime. Thisis
discussedafterthedataispresentedinwhatfollows.
There have been detailed and specialized reports and treatises of computational
techniquesoverthemanydecadesbutthesehavebeensparseandfarbetween.Wiberg
has[14, p.757]describedvariousmoreadvancedseriesexpansiontechniquesincon-
junctionwith least squaresanalysistoderivekineticdata. Hisuse of numericalinte-
grationis confinedto solvingby Runge-Kuttaintegrationa set of coupledequations,
suchasfeatureinanenzyme-catalysedreaction[14,p.771].Wiberginturndrawsupon
3
the collectiveefforts collated by D. F. Detar [15, 16]. It seems that Detar’s collation
anticipatestosomedegreemanyofthedevelopmentscitedaboveinthiswork’sbibliog-
raphy. AfirstordertreatmentofachemicalreactionisgivenintheprogramLSKIN1
[15, p.126] requires data and time intervals that are conformable to the Roseveare-
Guggenheimtimeintervalrequirement.LSKIN2[16,p.3]solvesforrateconstantand
initialconcentrationsofasecondorderreactionbasedonaseriesexpansionofthein-
tegratedrate law expression. Here, the curvaturewould introduce"errors" if a linear
expansionwereused. Forboththesemethods,theconstancyoftherateconstantkisa
basicassumption,whichisnotthecasehere.
Nonlinearanalysis(NLA),willbeattemptedhereinpreliminaryform,inorderto
compute both the instantaneousand average rate constants. We analyze 2 first order
reactionsandonesecondorderoneusingdatafromprominentkineticists. Inaddition,
weselectonefirstorderreactionwhoseinitialconcentrationindexisambiguousutiliz-
ingtheothersasareferencetogaugethelikelihoodofourresultbasedonNLA;ifour
analysisconcurswiththe3reactionsfromtheliterature, thenonemightbeconfident
that the NLA analysis of the ambiguous reaction is reasonably accurate. Important
experimentsinscienceareconductedunderuncontrolledconditions,suchasinastro-
chemicalreactionratedeterminationsandphotochemicalemissionspectraintheMars
andTitanatmospheresovertheseveraldecades[17,18,19].Asimilarsituationobtains
inforensicscienceandarcheologyandinbiologicalphysiologicalratedeterminations.
The basic methods presented here caters for both controlled and uncontrolled initial
conditions.
The3firstorderreactions(i)-(iii)andsecondorderreaction(iv)studiedareitem-
izedbelow:
(i) the tert butyl chloride hydrolysis reaction in ethanol solvent (80%v/v) at 25oC
derivedfromtheYearIIIteachinglaboratoryofthisUniversity(UM)wherethe
initial concentration,althoughdetermined,is ambiguous. Because oftime con-
straints, the inaccurate λ∞ = 2050µScm−1 for (i) was determinedby heating
the reaction vessel at the end of the monitoringto 60oC until there was no ap-
parent change in the conductivitywhen equilibratedback at 25oC. Reaction (i)
involved0.3mLofthereactantwhichwasdissolvedin50mLofethanolinitially.
Thereactionwasconductedat25oCandmonitoredovertime(minutes)bymea-
suringconductivity(µScm−1)duetothereleaseofH+ andCl− ionsasshown
belowin(6),
C H Cl+H O ka C H OH+H++Cl−. (6)
4 9 2 4 9
−→
(ii) themethanolysisofionizedphenylsalicylatederivedfromtheliterature[20,Ta-
ble 7.1,p.381]with presumablyaccuratevaluesofboththe initialconcentration
andforalldatasetsofthekineticrun. Reaction(ii)maybewritten
PS−+CH OH kb MS−+PhOH (7)
3
−→
wherefortheratelawispseudofirst-orderexpressedas
− −
rate=k [PS] =k [CH OH][PS ].
b c 3
4
Themethanolconcentrationisinexcessandiseffectivelyconstantforthereac-
tion runs [20, p.407]. The data for this reaction is given in detail in [20, Table
7.1], conducted at 30oC where several ionic species are present in the reaction
solutionfromKOH,KCl,andH Oelectrolytes.
2
(iii) the primarily SN1 substitution reaction [25, Table IX,p.2071] of tertiary butyl
bromide(ButBr) with dilute ethylalcoholicsodiumethoxidein ethanolsolvent
where there concurrently occurs an approximately 20% contribution of an E1
eliminationreaction.Reaction(iii)maybewritten
SN1
CH CBr Products (8)
3
−−−−−→
wherethesolventwasEtOHwithinitialsodiumethoxideconcentration[NaOEt]=
0.02386Nat25oC.Theproductsconsistedofapproximately81%substitutedter-
tiarybutylethoxideand19%olefinicmoleculesduetoE1elimination.Hencethe
rate constanthere refers to a composite reaction (details in [23, 2064] and [25,
p.2070].
(iv) thesecondorderE2eliminationreaction[23,p.2059-2060andTableVII,p.2064]
withreactantsisopropylbromide(PriBr)andsodiumethoxide(NaOEt).Reaction
(iv)involvingisopropylbromidePriBr (CH )CHBr(CH )maybewritten
3 3
≡
E2
(CH )CHBr(CH ) + OEt− CH CHCH +Br− +HOEt (9)
3 3 2 3
−−−−−→
wheretheisopropylbromidereactswiththeOEt− ioninEtOHsolventat25oC
toyield80.3%oftheolefinicproductwithsomeSN2substitutionwiththe(OEt)
functionalgroup[23, Table III,p.2061]accordingto the kineticists. Furtherde-
tails and data appear [23, Table VII,p.2064]. It should be mentioned that the
E2 reactionwasinferredto besecondorderfrompriorexperimentalconsidera-
tionssincethe[NaOEt]concentrationreductionfromthedataexactlycoincides
withthereductionofPriBrandwasnotindependentlydetermined.Thisperhaps
somewhatexperimentallyquestionabletechniquemaywellbethereasonforthe
instantaneousrateconstantascomputedheretobesomewhatnon-smooth,aswill
bediscussedlater(seeFig.(21)forthegraph).
“Units” in the figuresandtextpertainto the appropriatereactionvariabledimen-
sion, for instance either the absorbancefor (i) or the conductivity(µScm−1) for (ii)
below. Eitherbecauseofevaporationorthetemperaturesnotequilibratingafterheat-
ing, the measured λ∞, it would be inferred that for (i) the measured value is larger
thantheactualonedeterminedfromtheanalysis. Reaction(ii)isveryrapidcompared
to (i) and the experimental data plots show high nonlinearity. We denote by λ the
measurementparameterwhichistheconductivityµScm−1 orabsorbanceA[20,eqn
7.24-7.26]forreactions(i)and(ii)respectively;λalsoreferstotheconcentration[X]
ofspecies X for reactions(iii) and (iv). The moreaccuratelydeterminedλ∞ = A∞
for(ii)[20,Table7.1,p.381]wasatapproximately0.897. Analysisof(ii)givevalues
of A∞ = λ∞ very close to the experimentalones that suggests that our determina-
tionforreaction(i)λ∞ iscorrect. Theexperimentaldataandnumberofreadingsfor
5
thedeterminationofrateconstantsisalwaysrelatedtothemethodusedandtheorder
of accuracy required in the study; for Khan [20, Table 7.1,first A column], (reaction
(ii)), 14normalreadingsover360seconds(s)sufficedforKhan’spurposes,whereas
for the practical class (reaction (i)), 36 readings over 55 minutes (mins) were taken.
Themeager14readingsof(ii)coveredamajorportionofthenonlinearregionofthe
reaction, whereas for (i) the many readings were confined to the near-linear regime.
Linear proportionalityis assumed between λ and the extent of reaction x, where the
firstorderlaw(cbeingtheinstantaneousconcentration,kthegeneralrateconstantand
atheinitialconcentration)is ddct = −kc = −k(a−x);withλ∞ = αa,λt = αxand
λ(0)=λ =αx ,integrationyieldsforassumedconstantk
0 0
ln (λ∞−λ0) =kt (10)
(λ∞ λ(t))
−
Eqn.(10)determineskifλ0 andλ∞areknown.
Theanalysisforthelatterreactions(ii)-(iv)wouldprovideareferenceandindica-
tionofthe predictedvaluein(i)fortheinitialconcentration,apartfromcheckingfor
overallconsistencyof the methodologyin generalsituationsespecially whenthere is
doubtconcerningthevalueoftheinitialconcentration.
Themethodspresentedhereappliestoanyorderprovidedtheexpressionscan be
expandedas ann-orderpolynomialof the concentrationvariableagainstthe time in-
dependentvariable. Togetsmoothcurvesthatarestableonehadtomodifyandusea
propercurve-interpolationtechniquethatis stable whichdoesnotformsuddenkinks
orpointsofinflexionandthisfollowsnext.
1.1 Orthogonalpolynomialstabilization
ItwasdiscoveredthattheusualleastsquarespolynomialmethodusingGaussianelim-
ination [21, Sec.6.2.4,p.318]to derive the coefficients of the polynomial was highly
unstablefornpoly > 4,whichisaknowncondition[21,p.318,Sec6.2.4]. Forhigher
orders, there is in addition the tendency to form kinks and loops in an interpolated
curve for values between two known intervals. Other methods described in special-
izedtreatises[22,Ch.5,Sec.5.7-5.13],evenifrobustandstable,suchastheChebyshev
approximation required values of the proposed experimental curve at predetermined
definite points in time, which is outside the control of one using predetermineddata
andsoforthiswork,theleastsquareapproximationwasstabilizedbyorthogonalpoly-
nomials[21, Sec.6.3]modifiedfordeterminationof differentials. Itis hopedthatthe
method can also be extended to integrals in future investigations. The usual method
defines the nth order polynomial p (t) which is then expressed as a sum of square
n
termsoverthedomainofmeasurementtoyieldQin(11).
p (t) = n h tj
n j=0 j
P (11)
Q(f,p ) = N [f p (t )]2
n i=1 i− n i
P
The Q function is minimized over the polynomial coefficient space. In the orthog-
onal method adopted here, we express our polynomial expression p (t) linearly in
m
6
coefficientsa ofϕ functionsthatareorthogonalwithrespecttoaninnerproductdef-
j j
inition. Forarbitraryfunctionsf,g,theinnerproduct(f,g)isdefinedbelow,together
withpropertiesoftheϕ orthogonalpolynomials:
j
N
(f,g) = f(t ).g(t )
k=1 k k (12)
P
(ϕ ,ϕ )=0 (i=j) and (ϕ ,ϕ )=0.
i j i i
6 6
ϕi(t) = (t bi)ϕi−1(t) ciϕi−2(t)(i 1),
− − ≥
ϕ (t) = 1, and ϕ =0 for j <1,
0 j
(13)
bi = (tϕi−1,ϕi−1)/(ϕi−1,ϕi−1) (i 1),bi =0(i<1),
≥
ci = (tϕi−1,ϕi−2)/(ϕi−2,ϕi−2) (i 2),andci =0 (i<2).
≥
Wedefinethemth orderpolynomialandassociateda coefficientsasfollows:
j
m
p (t) = a ϕ (t)
m j=0 j j
P (14)
a = (f,ϕ )/(ϕ ,ϕ ),(j =0,1,...m)
j j j j
Therecursivedefinitionsforthefirstandsecondderivativesaregivenrespectively
as:
′ ′ ′
ϕi(t) = ϕi−1(t)(t−bi)+ϕi−1(t)−ciϕi−2(t)(i≥1)
(15)
′′ ′′ ′ ′′
ϕ (t) = ϕ (t)(t b )+2ϕ (t) c ϕ (t)(i 2)
i i−1 − i i−1 − i i−2 ≥
HerethecodesweredevelopedinC/C++whichprovidesforrecursivefunctionswhich
we exploitedforthe evaluationofalltheterms. The experimentaldatawere fittedto
anmth orderexpressionλ (t)definedbelow
m
m m
λ (t)= h tj =p (t)= a ϕ (t) (16)
m j m j j
Xj=0 Xj=0
The coefficients h are all computed recursively, and the derivatives determined
i
from(16)orfrom(14)and(15). Onceh ora aredetermined,thenthegradienttothe
j j
curveλ (t)iscomputedas
m
m
λ′ (t)= jh tj−1. (17)
m j
Xj=0
The l (t) function of (2) is expanded similarly as for λ (t) for order m. The or-
Q m
thogonalpolynomialmethodisstableandthemeansquareerrordecreaseswithhigher
polynomialorderingeneralmonotonically(wheren isusedtodenotethe integeror-
der),butthedifferentialsarenotsostable,becauseofthecontributionofhigherorder
coefficients in the differentialexpression as will be shown. From the form of the of
theequationthatwillbedeveloped,therateconstantisdeterminedasthegradientof
astraight-linegraphin theappropriatesegmentofthegraph. However,thecurvature
7
oftheplotwillincreasewithincreasingn,givingapoorervalueofk,whereashigher
values of n would better fit the λvst curve. Hence inspection of the plots is neces-
sarytodecideontheappropriatenvalue,wherewechoosethelowestnvalueforthe
mostlineargraphoftheexpressionunderconsiderationthatalsoprovidesagoodλ(t)
fit over a suitable time range over which the k rate constants apply. The orthogonal
polynomialstabilizationmethodprovidesgoodλfitswithincreasingn,butnotgradi-
ents,sothattheonsetofsuddenchangestothegradientwhichonphysicalgroundsis
unreasonablecanbeusedasanindicationaswhichisthebestcurvetoselect. Thereis
inpracticelittleambiguityinselectingtheappropriatepolynomials,aswillbedemon-
strated. Reactions (i) and (ii) both gauge initial concentrationsin terms of the A∞ (
λ∞)orfinalreadingofaphysicalfactorproportionaltoconcentrationandthestructure
oftheanalysisisthesameandwillthereforebediscussedsimultaneously,followedby
reactions (iii) to (iv) , where concentrationsare measured directly during the course
of the reaction, which will be discussed together because the form of the boundary
conditionsanddataareofthesameclass.
2 ANALYSIS OF REACTIONS (I) and (II)
200
150
1
−
m
S c100 expt. curve
µ n=2
λ (t) / nn==35
50 n=6
n=10
n=16
0
0 10 20 30 40 50 60
Time/mins
Figure1: Plotof(i)usingorthogonalpolynomialsforvariousordersn. Thetheleast
squaresdeviationgoesdowndramaticallywith increasingn, whichwasfoundnotto
bethecasewiththenormalnon-orthogonalpolynomialmethod.
Figure (1) are plots for the differentpolynomialorders n for reaction (i). It will
benoticedthathighernvaluesingeneralleadstobetterfitsvisually;thenormalleast
squares method leads to severe kinks and loop formation for n 4 which is not
∼ ≥
evidenthere. The reaction (ii) data coversa far greater domain with respect to half-
lifetimeswithonlyabout14points(whichisapoordatasetwithrespecttoourmethods
but which still gives quantitatively accurate values); because of the relatively more
rapidcurvaturechanges,wewouldexpectverydifferentgradientbehaviorascompared
to(i)withitsstrongerlinearity.
8
Thecorrespondingplotsforreaction(ii)areinFig. (2).
1
0.9 Expt. curve (ii)
0.8
0.7
A (t)
0.6
0.5
0.4

0 100 200 300 400
Time /s
Figure2: ExperimentalpointsomittingpointatA∞ forreaction(ii)attime= 2135s.
Thecurveisrathernon-linear.
Inviewofthenonlinearity,wechosealimitedregimetocurvefitforpolynomial
ordern = 3,4,5inFig.(2)andthegradientwascomputedforthen = 5polynomial
to determine the rate constantsas it was the only order that gave a smooth curve for
the first 12 consecutivepointsin the range; the otherordersalso gaveconsistentand
almostequalgradientsexceptattheextremeendpointsoftherangeplottedasdepicted
forexampleinFigs. (6,8,12).
1
n=3
0.9 n=4
n=5
0.8
Expt. (ii)
0.7
A (t)
0.6
0.5
0.4

0 50 100 150 200
Time /s
Figure3: Experimentalpointscurvefittedwithpolynomialsofordern = 3,4,5. The
fitforthisrangeisexcellent,despitethenonlinearnatureofthecurve
Unlikereaction(ii),theλ∞ forreaction(i)wasambiguous. Theplotof(10)was
madeforthesameexperimentalvalueswithdifferentλ∞’s,bothhigherandlowerthan
thesupposedexperimentalvalueforthisreaction.TheplotsinFig.(4)showsincreasing
9
0.4
0.2
0
λ(t)) −0.2
λλ−)/(−∞∞0−0.4 kkk210===131...167747xxx111000−−−333 λλ∞∞((01))21005500
λ−ln( −0.6 λ∞(2)3050
−0.8
−1
−1.2
0 10 20 30 40 50 60
Time /mins
Figure 4: Integrated equation(10) plot with λ∞ from experiment (0) and from two
differentarbitrary values(1,2) for λ∞, which yields two differentvalues for the rate
constantduetogradientchange.
ka fordecreasingλ∞;thechoiceλ∞ = 1050leadstoavalueofka closetotheNLA
valuesforthedifferentmethodsdiscussedwhichdoesnotrequireλ∞,butisalsoable
to determinethis value by extrapolation. The rate constantfromNLA is higherthan
thatdeterminedfromexperiment,implyingalowerλ∞ valuewhichisconsonantwith
evaporation of solvent and/or the non-equilibrationof temperature prior to measure-
menttodetermineλ∞. HenceelementaryNLAallowsonetodeducetheaccuracyof
theactualexperimentalmethodologyinthisexample.Exceptforonesection,weshall
applyNLAbasedonconstantkassumption. WealsoquotesomevaluesofKhan’sre-
sults[20,Table7.1]inTable(1),wheresomecommentisrequired.TheAabsorbance
ismonotonicallyincreasingandathighertime(t)values(see[20, Table7.1])theex-
perimentalA valueexceedsthe A∞ thatis determinedbythe processof minimizing
d2i. Hencetheminimizationof d2i withrespecttoA∞ istakenasaprotocolfor
dPeterminingthe bestk valueevenPifit contradictsexperimentalobservation. Further,
thisprotocolishighlysensitivetoA;achangeof10−3leadstoanapproximatelyten-
foldchangein k. On the other hand, if A∞ determinedfrom experimentas 0.897is
accepted,thenthencomputedrateconstantforthisvalueisk = 2.69 10−3s−1 im-
plyingthattheuncertaintyinkisoftheorderof 14 10−3. Hencew×ecanconclude
± ×
thattheKhanmethodisaprotocolthatacceptsascorrectthekvaluethatisdetermined
bytheminimizationofA∞ foracertainA∞ range( 0.8980 .8805),whichagain
≈ −
referstoanunspecifiedprotocolastothechoiceoftherange.
Thegeneral1stand2ndorderequations Westatethestandardintegratedforms
below as a reference that requires specification of initial concentrations in order to
contrastthemtothemethodsdevelopedhere.
10