eradicating erroneous arrhenius arithmetic

Thermochimica Acta 399 (2003) 1–29

Eradicating erroneous Arrhenius arithmetic

Andrew K. Galwey18, Viewfort Park, Dunmurry, Belfast, BT17 9JY, Northern Ireland, UK

Received 25 February 2002; received in revised form 8 March 2002; accepted 19 August 2002

Abstract

A general and critical analysis is made of theories used to interpret those thermochemical rate measurements that aredirected towards investigations of the mechanisms of chemical changes that result from the heating of initially solid reactants.It is concluded that the concepts and reaction models in current usage are inadequate and that the assumptions conventionallymade in the interpretation of this type of experimental data are in urgent need of radical reappraisal. The present articlespecifically identifies several shortcomings in results obtained from the Coats–Redfern equation, which is widely used tocalculate Arrhenius parameters for reactions that are regarded as capable of representation by rate equations characteristic ofdecompositions of solids. The view that activation energy values can vary with the kinetic model used in their calculation isregarded as unacceptable for reasons that are discussed in detail.

Introduction. An appraisal is made of the limitations of the various theories available for the interpretation of rate dataobtained from thermal analytical measurements.Literature survey. From an examination of research reports, mainly publishedin the last decade, a common pattern of variation of magnitudes of Arrhenius parameters with rate expression (kinetic model),calculated using the Coats–Redfern equation, is identified as a set of Trends. This pattern of rate parameters is shown to occurwidely. Reappraisal of methods of analysis of thermochemical rate data. It is shown that the Trends identified in calculatedArrhenius parameters are mathematical artefacts that arise through identified inadequacies in the assumptions underlying theapproximate computational method.Discussion. It is concluded that the concept of the ‘variable activation energy’, whichhas recently become approved through its use in considering nonisothermal kinetic data, is unacceptable. If the theory ofthermal analysis is to remain consistent with other branches of chemistry, and if scientific order is to be developed within thethermoanalytical literature, the definitions of all terms must remain common and constant. The shortcomings, which havebeen identified here in the approximate mathematical methods of thermal rate data interpretation, introduce doubt into thesignificance of many published Arrhenius parameters. Some consequences of these uncertainties for the future developmentof the theory of the subject and our progress towards understanding the controls and mechanisms of crystolysis reactionsare examined. Meaningful kinetic data must be obtained if thermal analysis results are to be considered in the context of therecent general theoretical models developed by L’vov, which are based on a physical approach to thermal reactions. Thistheory offers insights into the controls and mechanisms of thermal reactions involving solids and introduces the possibilityof promoting scientific order in this literature, which is not currently obvious.© 2002 Elsevier Science B.V. All rights reserved.

Keywords:Activation energy; Coats–Redfern method; Kinetic analysis; Thermal analysis

1. Introduction

This critical literature review identifies importantshortcomings that have been recognized in the science

of thermoanalytical kinetics. Numerous recently pub-lished articles in this subject report the magnitudes ofcountless activation energies, and other kinetic con-clusions, based on measurements by thermal analysis

0040-6031/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved.PII: S0040-6031(02)00465-3

2 A.K. Galwey / Thermochimica Acta 399 (2003) 1–29

methods. However, critical examinations of the ex-perimental data and of the experimental methods usedin such studies show that of many reported magni-tudes of kinetic parameters, and conclusions drawntherefrom, are not supported by an adequate theo-retical justification. As emphasized throughout thissurvey, usage of the term ‘activation energy’ by ther-mal analysts differs significantly from its meaning asoriginally defined and as generally accepted through-out other branches of the literature concerned withchemical kinetics. This terminological modificationis demonstrated here to occur within an extensive andrepresentative range of published results and engen-ders doubts about the value and significance of kineticconclusions across an even wider range of rate studiesby thermal methods. In the absence of critical scrutinyof the observations and of their interpretations, kineticconclusions based on thermoanalytical investigationsshould not be accepted as possessing mechanisticsignificance and may not be capable of providing ameasure of absolute reactivity. These generalizationsrepresent a most serious criticism of practices thatare currently widely regarded as acceptable in kineticanalysis investigations of condensed phase reactionsstudied by various thermal experimental methods andinterpreted through reaction models that originatedfrom the theory of solid state decompositions (crystol-ysis reactions)[1]. This considered expression of myprofound lack of confidence in the reliability of manypublished conclusions, throughout an important sub-ject area, is not made lightly. It is forcefully expressedhere and now in the expectation of initiating an over-due debate directed towards reappraising generally theobjectives, methods and theory of thermal analysis.

The above assertions introduce a literature analysisthat identifies specific inconsistencies that are inherentin computational programs widely used to interpretkinetic measurements for decomposition reactionsproceeding in a condensed phase. It is argued belowthat conclusions reached in recent discussions haveresulted in unrecognized modifications to the subjecttheory now used throughout the thermal analysis lit-erature that is concerned with reaction rates, reactionmechanisms and reactant reactivities[1,2]. It is arguedthat the problems identified in this review must becomprehensively addressed if the methods and prac-tices routinely used in kinetic investigations of manythermal rate processes are to progress the subject as

a comprehensible and systematic science. Urgent andcritical reassessments of the significances of many re-cent published results are now overdue and a compre-hensive new approach to the mathematical analysis ofrate data obtained by thermoanalytical measurementsis essential. The theoretical understanding of chemicalbehaviour must derive from, and always include, therigorous application of chemical principles, methodsand models. The excessive reliance placed, in manyrecent thermoanalytical reports, on ever more sophis-ticated mathematical manipulations of rate measure-ment data can have only limited ability to supportproposed kinetic interpretations, even when correctlyused. The present article is directed towards identify-ing reasons for the incorrect or unreliable conclusionsthat have arisen through the use of methods of kineticanalysis that are incapable of realizing the chemicalobjectives being addressed. The consequences of sucherrors, which appear to have had unexpectedly harm-ful influences in investigating mechanisms of thermalreactions, must be addressed and corrected before thescientific foundations for kinetic interpretation of ther-moanalytical data can be restored and the subject canagain prosper. Identification of some of the problems,the extents of their implications and the remedialaction required for their removal, are discussed herein what is intended to be an optimistic appraisal of apotentially and ultimately bright future for thermoana-lytical chemistry. However, realization of this promiseis only possible if the limitations inherent in many as-pects of recent researches are identified and removed.

Historical background. Motivations for undertak-ing a rigorous survey of the field, at this time, includemy belief that thermoanalytical chemistry has effec-tively replaced the study of thermal decompositionsof solids, while selectively retaining limited aspects ofthe theory developed by the former discipline. Thermalanalysis has profitably exploited the mathematical andcomputational advantages of the advances in instru-mentation that have characterized the formative phaseof this subject. However, these developments have notbeen accompanied by the adequate provision of a suit-able theoretical framework for data interpretation orcomparable progress (or much obvious interest) in in-vestigating and recognizing the chemical and physicalcharacteristics of the reactions concerned. The con-tinuing apparent progress of the subject is maintaineddespite the insufficient foundations in theory. There

A.K. Galwey / Thermochimica Acta 399 (2003) 1–29 3

is the uncritical acceptance of important kinetic termsfor which the definitions have undergone subtle, butapparently insufficiently appreciated, or even unrecog-nized, modifications for use in discussions of thermo-analytical observations. There are also uncertaintiesin the reliability of many experimental measurementswhere the sensitivity of rate data to prevailing reactionconditions has not, apparently, been considered in theinterpretations of rate measurements. The cumulativeconsequence of these unappreciated shortcomings intheory, and inherent in some experimental practices,has been to make many reported conclusions increas-ingly empirical.

The thermoanalytical literature. Thermal analysisis now characterized by an ability to obtain and to in-terpret kinetic results rapidly and with a minimum ofeffort. This has generated an extensive but fragmentedliterature in which systematic order, together with anability to predict behaviour by induction, essentialcharacteristics of a science, are now absent or, at least(to me), far from obvious. The superficially attractiveadvantages of collecting rate data (that may be empir-ical) easily has now replaced the former appreciation[1] of the difficulties that arise in the characteriza-tion of reactivity controls and of the mechanisms ofchemical changes that occur on heating initially solidreactants. The present motivation towards contribut-ing to an ever expanding report accumulation, withinwhich order is evidently absent, continues. However,little current effort appears to be directed towardsthe recognition of trends that may be identifiable inthe literature, through provision of reviews of theextensive information that is already available. More-over, article introductions are frequently selective, byincluding relatively few citations for systems whichmight reasonably be related to (or may be inconsis-tent with) the new research being published. Thermalanalysis is a subject that contains remarkably fewcomparative discussions, either as dedicated compar-ative reviews or within the many contributed articles.One probable consequence is the absence of theorydevelopment and the recognition of few systematictrends or patterns in behaviour. Consequently, thereis a progression towards a literature composed ofisolated or individual contributions and there is noorganic, systematic growth of a coherent science.

The above criticisms, drawing attention to theuncertain significance of some recent experimental

measurements, are based on selected valuable, of-ten older, literature sources which contain important,fundamental contributions to the subject that remaininsufficiently recognized. These include reports thatemphasize the specific experimental difficulties whicharise when attempting to measure kinetic data that maybe positively used to identify a chemically controlling(rate determining) step, particularly for reactions in-volving solids. It has been conclusively demonstratedthat, for many crystolysis reactions[1], self-heating/cooling [3–6] and/or reaction reversibility[7–9] ex-ert significant, sometimes dominant, controls on thereaction rates observed. Nevertheless, kinetic in-vestigations ignoring these realities continue to bereported, and, consequently, the rate measurementsoften determined are specifically applicable (only)to some particular reaction conditions. Such dataare, therefore, empirical, but are all too frequentlyinterpreted through theory relevant to and originallyformulated to explain homogeneous rate processes,where a single, dominant controlling step operates.Recognizing the limitations of current methods andpractices, L’vov [10,11] has recently developed ageneral theoretical model, discussed further below,applicable to solid state decompositions and account-ing for the various essential characteristic features ofthese reactions. However, the useful application ofthis approach, and its further development, dependson obtaining suitable, reliable kinetic data, values ofthe activation energy in particular, that are related tothe rate of the controlling process in the reaction.

The present paper demonstrates important, but ap-parently unappreciated, limitations in some conven-tional methods of kinetic interpretation of measuredreaction rates. An appraisal is made of one of themost widely accepted techniques used for the com-putation of activation energies from nonisothermalthermoanalytical measurements. A critical reexami-nation of the assumptions and mathematical steps inthe Coats–Redfern equation[12], CR, used to deter-mine Arrhenius parameters from rising temperatureexperiments, is given below. Important shortcomingsin this well-known computation are revealed, whichhave been present, apparently undetected, in the nu-merous literature reports concerned with the uncriticalapplication of this equation. The consequences arenow discussed in the context of their probable widerimplications. A second and complementary review,


now in active preparation, considers a range of otherrelated problems besetting aspects of thermal kineticsgenerally. This includes uncertainties in the defini-tions of terms, together with the units applicable, andlimitations inherent within those theoretical modelswhich are most frequently applied in reports of ther-moanalytical work. The absence of critical appraisalof thermoanalytical results, through reviews and arti-cles, is regarded as constituting a retarding influenceon progress in the field. Together the present and theforthcoming articles pose the question, in a literaturenot noted for its ability to respond to unwelcomecontributions: is it preferable to continue the accu-mulation of unrelated largely empirical results or is ittimely (overdue) to return to the scientific method? Iunreservedly regard the latter course as preferable. Iwonder if other authors will recognize, or condescendto respond to, the criticisms of thermal analysis, meth-ods and practices, that I publicize here. I suspect thatmy unwelcome and critical analysis will be consignedto the ‘Great Repository’ of the unrecognized anduncited literature. Nonetheless, the fact remains thatthermal analysis is not fulfilling its potential becauseprogress is no longer based on chemical and physi-cal principles and (equally importantly) on adequatetheoretical scientific foundations.

Personal note. This survey is presented in directterms because I believe that kinetic studies by thermalanalysis is in a state of considerable disarray, even cri-sis, that has not been the subject of the concern that itso richly merits. I am not a thermal analyst myself, butam familiar with the literature through related interests[1,2]. Because this unsatisfactory situation continuesto remain apparently unrecognized, certainly hithertounaddressed, I consider that it is timely to present myviews (as an ‘outsider’) in the hope that the subjectoverall might eventually benefit from a forthright crit-ical comment.

1.1. The application of kinetic analysis in theinterpretation of thermoanalytical data

The concept of activation energy (E), as devel-oped, accepted and used generally throughout almostall chemical kinetics, identifies the energy with thebarrier to the bond redistribution processes for therate determining step in a single reaction[13]. It is aphysicochemical parameter determined by the magni-

tudes of the interatomic interactions that are activatedand modified during the changes occurring. For eachparticular reaction it has a characteristic and constantvalue. An exception to this generally agreed nomencla-ture is found in the field of thermal analysis, in whichmany workers hold or accept a different view: “the ac-ceptance of a variable activation energy. . . ” has beensuggested[14], though no precise definition for thisreplacement term was given. However, one possible al-ternative definition for this ‘variableE’ might be: “Forany reaction, the magnitude of ‘variableE’ is found by(uncritically) applying the Arrhenius equation to anyrate constants (k) determined from diverse kinetic ex-pressions[1] of the formg(α) = kt or f (α) = (1/k)

(dα/dt) (whereα is the fractional reaction)”. This typeof meaning appears to be the, currently fashionable,preferred choice by many, perhaps even a majority, ofresearchers who have recently published in the fieldof thermoanalytical kinetic studies. It follows thatmany E values reported, for nominally the same re-action, show wide variation and/or differentE valuesare found through the use of alternative rate equationsto analyse the same data (examples are given below).Consequently, the magnitude of this parameter mustbe regarded as having no recognizable (or intended?)mechanistic significance. Such values ofE are, there-fore, empirical and should not be cited as evidencefor the identification of a rate determining step orother mechanistic inference. Why this ‘variable’ termshould retain the descriptive label, ‘activation energy’,associated with a precise chemical concept, is not ex-plained[14]. (It also seems thatE, used in this sense,is not correctly regarded as a ‘variable’, but could bemore suitably described as a ‘multivalue’ parameter.)

Many recent articles report wide differences in ap-parent magnitudes of ‘variable’E with change of therate equation used in the calculations. This means thateither the chemical concept ofE, as originally definedand conventionally accepted, is inapplicable or themethod of calculatingE is incorrect. If the formeris believed, then the ‘activation energies’ consideredare empirical (multivalue) quantities, which, never-theless, may be capable of useful extrapolation, undersuitable circumstances, to estimate reactivities orreaction rates outside the range for which experimen-tal data are available. Alternatively, and to maintainconsistency with other branches of chemistry,E canbe regarded as a property of a controlling step and


for any reaction must have a constant value (whichmay or may not be experimentally determinable).It follows then that changes in the apparent valueof E with the kinetic model [g(α) = kt], on whichthe calculations have been based, must arise as aconsequence of unsuitable computational proceduresor inappropriate assumptions in those calculations. Iunhesitatingly express my preference for the latter ofthese alternative views. To support this opinion, andto identify the mathematical reasons for the apparentvariability of E, I have made a careful reappraisal ofthe steps involved in the calculation routine use forthe Coats–Redfern equation[12]. It is concluded thatthis method does not provide an acceptable method ofkinetic analysis and identifies a considerable uncer-tainty about the reliability of many reported values ofArrhenius parameters in numerous published articles.This outstanding concern remains to be resolved andraises important questions concerning the reliabilityof kinetic results obtained by other similar methods ofcalculation.

1.2. Objectives of kinetic studies

Kinetic studies of crystolysis[1] reactions most fre-quently appear to be motivated by an interest in re-porting magnitudes of Arrhenius parameters (E andthe frequency factor,A) together with identification ofthe rate equation, kinetic model, [g(α) = kt] that mostsatisfactorily provides the ‘best fit’ to the yield–timedata. Reasons for addressing these objectives are notalways satisfactorily explained but the reporting ofE (etc.) appears to be widely regarded as providingself-evident justification for many published researchreports. However, these achievements may often beadjudged to add relatively little of substance to ourunderstanding of the chemistry and physics of the spe-cific reactions selected as being of interest for severalreasons, some of which are outlined below. (More thanone reason may apply to the results from any particu-lar research program.)

1. Mechanism determination. Kinetic investigationsof homogeneous rate processes are often directedtowards the elucidation of the reaction mechanisms.The sequence of participating chemical changes,including the rate limiting step, through which re-actants are converted into products can often becharacterized for rate processes proceeding in so-

lution or in the gas phase. However, much less isknown about the controls and conditions that existwithin an active reaction interface during crystol-ysis reactions (solid state decompositions[1]) andpapers concerned with the kinetics of such rate pro-cesses do not elucidate the detailed mechanism ofthe chemical changes considered. Comparisons ofkinetic conclusions with similar results for relatedsystems are infrequently made. Moreover, there ap-pears to be little interest in identifying systematicorder and trends of behaviour within groups ofcomparable reactions, which is a primary objec-tive of the scientific method. The literature con-tains remarkably few reported attempts to classifythe thermal reactions of solids on the basis of thelarge numbers of kinetic results (magnitudes ofA,E and rate equation applicable) that are availablefor the thermal observations already reported fornumerous and diverse reactants. Many recent ther-moanalytical kinetic studies cannot be regarded ascontributing to the elucidation of reaction mecha-nisms.

2. Kinetic characteristics may be complex. More thana single rate-influencing factor may contributeto determining the rate characteristics of manycrystolysis reactions and the relative influencesand effects of the complementary, but differentcontrols are not always readily distinguished. Con-sequently, the calculated values ofA and ofE forthe overall reaction cannot be assumed to providea measure of the frequency of occurrence of, orthe energy barrier to, a single, dominant rate con-trolling chemical step (as may occur in many ho-mogeneous rate processes). The kinetics of manyreactions of solids are significantly influenced byrates of various diffusion or mobility (physical)controls, including the movement of heat to (orfrom) the active reaction zone (interface)[3–6]and/or removal of volatile product from the sitesof reversible dissociation steps[7–9]. Such effectsare clearly demonstrated by the well-known influ-ences on rates of many reactions that result fromchanges of the procedural variables[15,16]. Thesemodifications of conditions cause changes withinthe reaction zone which may be inhomogeneouswithin the reactant mass and undergo systematicvariations with time during progress of the chem-ical change of interest. The significance of rates


measured for such reactions must be interpretedwith quantitative consideration of all possible par-ticipating rate controls. It cannot be assumed, assometimes appears, that there is a rate determin-ing (single, chemical, dominating) step, compara-ble with the concept familiar from homogeneouskinetics.

3. Errors may arise in the calculations used in kineticinterpretation. The mathematical procedures usedto calculate Arrhenius parameters and to identifythe ‘best fit’ of measured rate data to appropriatekinetic models[1], may yield results of uncertainrelevance or reliability. The present paper is pri-marily concerned this aspect of kinetic analysis:it is concluded that calculation methods based onapproximate formulae techniques do not alwaysyield the results sought.

2. Literature survey

2.1. Variable activation energies

A principal objective of this critical survey is toexplain a characteristic pattern, discerned between rel-ative magnitudes ofE and the kinetic models,g(α) =kt, used in their calculation, that has been identified inpublished articles, for example[14]. These variationsare inconsistent with the theory of reaction kineticsand no satisfactory explanation appears to have beenprovided. Representative instances of these trends aredescribed below. This is not intended to constitute acomprehensive review or an attempt to locate and listevery traceable example of this particular pattern thatcan be found in thermal analysis data. However, theselection of examples cited here shows the generalityof this behaviour pattern which has been reportedfor studies of thermal decompositions involving nu-merous and diverse reactants by many researchesfrom geographically dispersed laboratories, includingobservations based on different experimental mea-surement methods. The constant pattern of trends andcorrelations, identified within the reported combina-tions ofE (alsoA) with [g(α) = kt], are explained (inthe subsequent section) by shortcomings and incon-sistencies inherent in the calculation methods used.The pattern identified is, therefore, a mathematicalartefact.

2.2. Identification of a representative set ofkinetic data for use in this comparativeliterature survey (StdSet)

To discuss the characteristic pattern of variations ofE with kinetic model,g(α) = kt, identified, it wasconvenient to select a suitable and representative setof results, with which the other kinetic reports foundin the literature could be compared quantitatively. Thedata set that were selected (perhaps arbitrarily), as themost appropriate and comprehensive reference avail-able for comparisons, were the kinetic parameters,E,A and [g(α) = kt] (together with the linear correla-tion coefficients,r) calculated by the Integral methodand listed in Table 4 of[17]. This is referred to be-low as the StdSet. The reaction involved was Stage1 of the thermal decomposition of the complex ofeuropiump-methylbenzoate with 2,2′-dipyridine. Re-sults reported in this paper[17] included the originalrate measurements used in the calculations, the fit ofmeasured rate data to 19 different kinetic models byboth Integral and Differential methods (38 values ofArrhenius parameters in all) andr values, correlationcoefficients, that are predominantly close to unity. Itshould be emphasized that any one of the many setsof results mentioned in the papers cited below couldequally well have been selected as the basis for thecomparisons. However, this most extensive range of‘good fit’ observations[17], with r values close tounity, was taken as a the preferred benchmark againstwhich other variations ofE with [g(α) = kt] could bemost usefully and suitably examined.

In the StdSet, only three of the 19 values ofr re-ported for the calculated Arrhenius parameters (0.661,0.932, 0.973) are less than 0.980 and six are 0.995 orgreater. (One immediate conclusion, possible from thiscomparison, is that the analytical method used exhibitsa marked inability to discriminate[18,19]between thealternative kinetic models to which the fit of data hasbeen tested.) Values of lnA andE show closely com-parable patterns of considerable variation with kineticmodel used in their calculation. These values give twostrong compensation relationships[20,21] (discussedfurther below) for the two data sets in Table 4 of[17] which represent the Integral and Differential cal-culation methods. Both lines pass through (or veryclose to) the origin of the graph (0,0). The isokinetictemperatures[20] are 510 and 481 K, respectively: as


expected, these are within the range of temperatures,443 to 511 K, of the kinetic measurements used to cal-culate the Arrhenius parameters.

2.3. Trends in magnitudes of kinetic parametersin the StdSet

The following Trends were found in the StdSet[17] which enable comparisons to be made withother published results (equations are numbered as in[17]).

Trend A1. Calculated magnitudes ofE, also lnA,decreased (irregularly) in the sequence: Eqs. (6), (4),(3), (2), (1), (5), (18), (7), (13), (12), (14), (8), (9), (15),(10), (19), (16), (11), (17). The range of variations ofE magnitudes was considerable, decreasing from 365to 26 kJ mol−1. It is noted qualitatively that the highestrelative values ofE are found for diffusion controlledreaction models[1].

Trend A2. Values of E decreased markedly andlinearly with reduction of the exponentn for boththe power law (Eqs. (1), (14)–(17)) and the Avrami–Erofeev AE (Eqs. (8)–(11)) equations[1], Fig. 1.However, both plots intercepted the axis at a smallpositive value ofn (n = 0.05–0.10, approximately)

Fig. 1. Plot of exponent,n, in power law (�) and Avrami–Erofeev (AE) (�) kinetic models[1] againstE for data calculated by theIntegral method from Table 4 in[17]. The extrapolated straight lines, through each set of points, intersect the axis at small positive valuesof n, between about 0.05–0.10.

so that magnitudes ofE/n show small progressivediminutions asE andn decrease.

Trend A3. The value ofE for Eq. (4) (diffusioncontrolled equation, D3[1]) was just more than twicethat for Eq. (13), the contracting volume equation, R3[1], and this was about 5% greater than that of Eq. (12),the contracting area rate expression, R2[1].

Trend A4. There was a pronounced compensationeffect [20], all points on a plot of lnA againstE wereclose to a straight line, that passed close to the origin(0, 0).

The relationships betweenE values obtained fromthe Integral and Differential methods (StdSet) fromTable 4 in[17], are conveniently illustrated by the un-conventional plot, comparing these quantities,Fig. 2.

The following Trends were found forE values fromthe Differential method, Table 4 of[17].

Trend B1. The sequence of decrease inEvalues was:Eqs. (6), (4), (3), (18), (2), (1), (19), (5), (7), (13), (12),(8), (9), (14), (10), (11), (15), (16), (17). The Trendis generally similar to, but not identical with, that inthe StdSet and, interestingly does not represent theclosely comparable values ofE that should have beenanticipated from application of the different forms ofthe same rate expressions to the same data,Fig. 2.


Fig. 2. Plot of values ofE calculated by Integral method (StdSet) against those calculated by Differential method from Table 4 in[17].The (upper) stronger line connects five points for the Avrami–Erofeev equation withn = 1.0, 0.667, 0.5, 0.333 and 0.25: the lower line,passing through the origin, connects the five points for the power law withn = 2.0, 1.0, 0.5, 0.333 and 0.25. The contracting geometryequation points (n = 0.5 and 0.333) are between the two lines and all the high apparent values ofE resulted from the kinetic modelscharacteristic of diffusion controlled rate processes[1].


Moreover, it might reasonably have been expected[19]that the Differential method would have provided amore discriminating method for identification of thekinetic model giving the ‘best fit’. There were someexamples of considerable differences between the al-ternatively calculated Arrhenius parameters, most no-tably Eq. (19), for whichE changed sevenfold. Therewas also a significant (50%) change for Eq. (18), to-gether with a few other, smaller variations.

Trend B2. Values from the Differential methodshowed general similarities with those described forTrend A2. It was further noticed that the differencesbetween the alternatively calculatedE values for thepower law were remarkably constant (2,908 cal mol−1)for the five values ofn. The same accuracy, but dif-ferent difference (61,100 cal mol−1), was apparentbetween each pair of values for the five AE equations,Fig. 2.

Trend B3. As Trend A3.Trend B4. A compensation effect was again found.

2.4. Comparisons of the StdSet data with resultsobtained by comparable methods

Similar sets of detailed results from the kinetic anal-ysis of rate data for Stage 3 of the thermal decomposi-tion of the same reactant are given in Table 5 of[17].Magnitudes ofE, and lnA, are generally smaller thanthose in Table 4 but a closely comparable pattern ofrelative values appears, though with some small quan-titative variations. Results from Integral methods ex-hibit almost the same sequence as in Trend A1, butvalues ofE from Eqs. (18) and (19) are relativelylarger and some minor changes in sequence are foundin Trend B1. Trend B3 was less obviously applica-ble, exceptionally Arrhenius parameters from Eq. (12)were relatively smaller. The pattern of differences inE calculated alternatively by Integral and Differentialmethods was identical with that illustrated inFig. 2(Trend A2) but absolute values were different fromthose in Table 4. Again two lines are found on thecompensation plot (Trends A4 and B4), from the twodata sets, but here the lines are almost parallel, giv-ing the isokinetic temperature[20] 824 K, within theinterval of the kinetic measurements.

These patterns of kinetic parameters[17] are read-ily and comprehensively compared with results ob-tained by almost identical kinetic studies reported

[22] for decompositions of two copper and two nickelcomplexes of Schiff bases. The presentations in botharticles are closely similar, including the same rangeof equations tested comparatively and the informa-tion reported. (The close relationships of these twocomplementary studies provided another reason forselection of the StdSet. This unusually similar corre-spondence occurs even though the investigations werecompleted in different universities with no commonauthors.) The data reported in Tables 7–10 of[22] arereadily compared with the results in Tables 4 and 5 of[17], discussed above. While the absolute magnitudesof the Arrhenius parameters vary for the several dif-ferent reactants, and manyr values are much lowerthan the high values characteristic of the StdSet,the patterns of variation are remarkably similar. Thefollowing generalizations can be made.

1. Values ofE reported for each of the data sets from[17,22] extended across a wide range of magni-tudes. While some exceptions were found, the rela-tive values ofEcalculated for the different reactantsfrequently followed virtually identical patterns ofrelative change with the kinetic model used in thecalculations: Trends A1 and B1. Negative valuesof E were reported for some equations using theDifferential method. In most tables some values ofE, alternatively calculated from Integral and Differ-ential methods, were approximately similar whileothers exhibited significant differences: reasons forthese variations were not established.

2. Every set of Arrhenius parameters recorded inall six tables of kinetic results[17,22] exhibitedmarked compensation effects, Trends A4 andB4. For all data from Integral methods, the sixlines passed through, or near, the graph origin(0, 0). Isokinetic temperatures calculated fromthe slopes were always within the temperatureranges of the rate measurements (close to 600and 660 K for both of pairs of copper and nickelsalts [22], respectively). Each of the compensa-tion [20] lines for kinetic parameters obtained bythe Differential method (Trend B4) were close to,but not coincident with, the lines obtained fromIntegral method values (Trend A4) for the samereaction.

3. E magnitudes calculated by the Integral methodvaried linearly with the exponent,n, for both power


law (Eqs. (1), (14)–(17)) and AE (Eqs. (7)–(11))equations[1]. However, extrapolation toE = 0gave a small positive intercept atn= 0.05–0.10 (ap-proximately) for all six data sets. Thus,E varieslinearly withn, but the magnitude ofE/n decreaseswith diminution of both parameters.

4. Within each group of related rate equations (powerlaw or AE [1]), the magnitude of the difference(E(Integral) − E(Differential)) was precisely con-stant for each of the two groups of five equations.However, the magnitude of this difference variedwidely and apparently randomly, from small val-ues up to about 110 kJ mol−1 within each of the sixdata sets. No pattern in the magnitudes or explana-tion for these variations could be advanced. Calcu-lated sets ofE values for all these reactants showedthe relative distribution pattern illustrated onFig. 2.

It seems probable that this characteristic, and gener-ally constant, pattern of variations in the magnitudesof calculated Arrhenius parameters arises throughcommon features, present within the mathematicalprocedure. The Trends appear, therefore, as a com-putational artefact. We first consider the generality ofthese Trends, then discuss the mathematical treatmentbecause different patterns of variation appear in thedifferent data sets, for example, the unexplained dif-ferences in point 4 of the previous paragraph. Reasonsfor the overall pattern described are considered in thesubsequent section, including a brief discussion ofsome aspects of the background theory that may be ap-plicable to enable the problems of analysis to be placedin a wider context. First, however, the above patternof Trends is compared with other relevant literaturereports. This comparison is inevitably incomplete, be-cause most of the relevant articles contain less detailedinformation.

Identification of ‘best fit’ rate equation. Recognitionof the kinetic model that most satisfactorily representsa rate process of interest, from amongst the several, ormany, possibilities included in the consideration, is of-ten based on the equation giving the relatively largestvalue of the correlation coefficient (r). However, aspointed out by Vyazovkin[23], this can be unsatisfac-tory where two or more values ofr are close and repre-sent very different magnitudes ofEand kinetic models.This might alternatively be expressed as ‘uncertainty

is introduced by the uncertainty in the uncertainty’.This approach to kinetic analysis must be regardedas unsatisfactory because it provides poor ability todiscriminate ‘best fit’ between alternative possible ki-netic models. Identifying the appropriate (‘correct’)rate equation can be difficult and aspects of the prob-lems encountered have been discussed[18,19,24,25].One proposed solution[23] to this conundrum is theuse of the ‘advanced isoconversional method’, usingmultiple heating rates to calculateE values. Temper-ature coefficients are measured for the zero order re-action rate, dα/dt, applicable over smallα intervals.This method of rate data analysis does not appear tocharacterize the kinetic model. A comprehensive ki-netic analysis should be capable of accounting for allrate characteristics. This includes distinguishing be-tween the three main types of solid state reactions[1]:sigmoid (nucleation and growth), deceleratory (con-tracting volume or area), strongly deceleratory (diffu-sion control). Where this is not achieved, as in someexamples mentioned above and also below, a prin-cipal objective of the kinetic analysis has not beenrealized.

Compensation effect. The compensation effect is alinear relationship between lnA andE: any increase inthe magnitude of one parameter is offset, or compen-sated, by an appropriate increase of the other[20,21].Instances of this pattern of behaviour are reportedwidely in the kinetic literature, representing patternsof relative rate variations with temperature. The ef-fect has been widely observed to occur within a groupof reactions related by one or more common chem-ical features or for a single reaction investigated un-der a range of different experimental conditions. It isa property of the compensation relationship[20] thatat the isokinetic temperature rate constants for all re-actions within the group exhibiting the effect havethe same value. It follows that if the same, perhapseven a single set of rate data, which necessarily applywithin the same temperature interval, are alternativelyanalysed to give a range of Arrhenius parameters,their magnitudes must exhibit compensation. Thus, al-though compensation parametersb and c ([20] fromln A = bE + c) are often reported in articles of thetype surveyed here, these terms can have no chemicalsignificance because they arise (as explained below)as an artefact inherent in the mathematical proceduresused.


2.5. Comparisons with other representativepublished studies

The other literature reports of multiple kinetic anal-yses were found to be much less comprehensive thanthose in[17,22]above. Most included only a relativelysmall number of pairs of Arrhenius parameters cal-culated for a range of kinetic models by the Integralmethod. For most of the comparable examples citedbelow, it is argued here that all of the (sometimes lim-ited) evidence available apparently presented the samerelationships, Trends, as those described above for thedata in[17,22].

Decomposition of copper ammonium chromate. Al-though the range of kinetic data reported in[26] waslimited to Arrhenius parameters calculated for nineIntegral forms of the rate equations, close parallelswere found with those of the StdSet. Reported vari-ations ofE with rate equation were relatively larger(×1.57± 0.06) than in Trend A1, but all values fol-lowed the same sequence. A plot ofE againstn, forthree AE equations, intersected the axis atn = 0.05(Trend A2) and data exhibited a compensation effect(Trend A4). Values ofr reported for the nine equationswere all 0.9920 or greater, despite the quite differentforms of the rate/time curves represented: again evi-dence of poor discrimination.

Decomposition of bis(dialkyldithiocarbamate) com-plexes of palladium. Relative magnitudes ofE andAvaried considerably between the five reactants stud-ied [27] and were approximately comparable withthe StdSet. Extrapolations from the calculated twoAE values ofE for each of the five reactants inter-cepted the axis betweenn = 0.05 and 0.10 (TrendA2). A pronounced compensation effect was found(Trend A4) for reactions that occurred within similartemperature intervals.

Ammonium dinitramide, HMX and ammonium ni-trate. Three kinetic studies, for the thermal decom-positions of ammonium dinitramide[28], of HMX[14] and of ammonium nitrate[23], have been re-ported by Vyazovkin and with Wight[28] reportingvalues ofE calculated for 12 equations by the Inte-gral method. The Trend (A1) ofE values with rateequation were as for the StdSet. From each rate equa-tion, the ratios of calculated magnitudes ofE werealmost a constant value (allE values for ammoniumdinitramide/HMX were 1.83±0.12 and for ammonium

dinitramide/ammonium nitrate were 1.11±0.11), evi-dence of a closely similar pattern of variation betweenthe calculated results for all three reactants. Trend A2was found: plots forE calculated from both powerlaw (five points each) and AE equation (four pointseach) against exponent,n, gave separate lines but allsix lines extrapolated to intercept the axis close to thesame value ofn = 0.05 (Trend A2). Trends A3 andA4 were also observed. Values ofA were not recordedfor ammonium nitrate but the slopes of the compensa-tion plots for ammonium dinitramide and HMX gaveisokinetic temperatures of 457 and 502 K, respec-tively, both within the ranges of the kinetic measure-ments. Overall these patterns of kinetic parametersshow a close resemblance to that of the StdSet.

Ortho-palladated complexes with pyridine. Activa-tion energies were reported[29] from nonisothermalkinetic studies for the thermal decompositions offive reactants analysed through 10 alternative kineticmodels. The sequences ofE magnitudes for the ninecomparable equations were as the StdSet (Trend A1).Values ofE found from four forms of the AE equation(n = 1.0, 0.667, 0.5, 0.333) extrapolated to interceptthe axis close ton = 0.05 (Trend A2). Results alsoexhibited Trend A3. Values ofA were not reported sothat the existence of a compensation effect could notbe tested. Arrhenius parameters reported for isother-mal rate measurements using the same five reactantsshowed values ofr all greater than 0.9948 for the fit ofeach data set to three different rate expressions (A1.5,A2, R2[1]): again the apparently close representationof data by the different rate equations is evidence ofpoor kinetic model discrimination[18,19].

Areneruthenium(II ) derivatives. Sanchez et al.[30]have reported kinetic (rising temperature) studiesfor the first step in the thermal decompositions ofareneruthenium(II) chloride complexes with five dif-ferent ligands. The various ligands were eliminatedbetween 424 and 487 K in air to give residual bin-uclear complexes. Rate data were analysed by threedifferent equations, conventionally used in kineticanalysis (Coats–Redfern (CR), MacCallum–Tanner(MT) and Horowitz–Metzger (HM)), to investigatethe comparative fits of data to a range of Integral rateexpressions including most of those listed in[17].For each of the 10 reactants studied, three values ofE and logA calculated by three of the expressionsgiven above are recorded, apparently selected to give


r values closest to unity. For the 90 pairs of Arrhe-nius parameters listed,r values for two are 0.99999,a further 21 are 0.99990 or above, 46 are between0.99900 and 0.99990, 15 are between 0.99800 and0.99900, the remaining six were above 0.99600.These are remarkably high correlation coefficientsand demonstrate that the approach does not satisfacto-rily discriminate between the fit of data to alternativepossible rate expressions[18,19]. Twenty-nine fromthe 30 fits found by the CR method were to diffusionor contracting geometry kinetic models, while thealternative analyses (MT and HM) of the same dataincluded some fits to the AE equations[1]. The setsof results gave evidence that the Arrhenius parametersshowed variations for the different salts studied. How-ever, within each set, the magnitudes ofE followedthe same sequence as the StdSet, Trend A1. Therewere some trends of decrease ofE with AE exponent,suggesting Trend A2, but the limited evidence avail-able was insufficient to provide confirmation. TrendA3 was found in many instances. There was a strongcompensation trend overall, Trend A4, and some ofthe slight scatter of points was attributable to linesof different slopes given by each of the ten differentreactants considered in the composite plot.

Dehydrations of sodium lanthanum sulphate mono-hydrates. Kolcu and Zumreoglu-Karan[31] havemade nonisothermal kinetic studies of the dehydra-tions of double sulphate monohydrates of sodium withfour light lanthanides. The pattern of variations ofArrhenius parameter magnitudes with kinetic modelwas again closely similar to that of the StdSet. Thesequence ofE values, for the eight Integral methodequations that could be compared, conformed withTrend A1. Extrapolations of the plots ofE againstthe three values ofn (1.0, 0.5, 0.333) for the AEequation intercepted the axis close ton = 0.1 for allfour salts (Trend A2). Values ofE from the diffusioncontrolled/interface equation were about twice thosefor the contracting volume and contracting area ex-pressions (Trend A3). There were some indicationsthat the activation parameters calculated by the Dif-ferential method showed approximate fits to TrendsB1 and B3, but the behaviour pattern showed appre-ciable variations from those described above. Againcompensation effects were found, Trends A4 and B4:points for Integral method and for Differential methodvalues were close to two parallel lines.

Other examples. Further reports that exhibit someof the behaviour patterns described above include[32,33].

Dehydration of transition metal sulphate hydrates.A further set of kinetic reports, see[34,35] and ref-erences therein, contain features similar to thosedescribed above. Magnitudes ofE, calculated usingthe AE equation for several values ofn, interceptedthe (E = 0) axis close ton = 0.05 for the four re-actions distinguished in the stepwise dehydration ofNiSO4·6H2O: however, see[36] for a discussion ofaspects of the stoichiometry of these reactions. Com-pensation behaviour was found in which, surprisingly,the Arrhenius parameters for the four dehydrationsteps distinguished as occurring in different tempera-ture intervals, between the limits 416 and 631 K, wereall close to a single line (apparent evidence of TrendA4). However, reservations must be expressed aboutincorporating these results into the present compar-isons because it appears thatα values used in thekinetic analyses of the individual and distinct consec-utive rate processes are referred to the overall reaction(–6H2O), instead of being calculated individually foreach dehydration step.

Comparative studies. In a recent comparative pro-ject [37], “Computational aspects of kinetic analysis”initiated and coordinated by ICTAC, different groupsof workers compared results of their kinetic analysesof the same eight sets of measured rate data sup-plied. A principal outcome was that there were verysignificant variations in the magnitudes of Arrheniusparameters reported by the different participants, allcalculated from the identical data. Aspects of thesevariations and the patterns of kinetic parameter dis-tributions have been discussed[38] in the context ofthe compensation trends identified. It is consideredimpossible to make a definitive comparison of theextensive tabulated data in[37] with the Trends iden-tified above, because procedures in the computationalmethods used by the several participating laborato-ries are not available in sufficient detail. However,the recognition of compensation trends in[38] sug-gests the possibility that some, at least, of the spreadof reportedE values in [37] may arise for similarreasons to those underlying the variation ofE withrate equation, [g(α) = kt], found in [17] and in theother systems listed above as exhibiting comparablebehaviour.


Similar considerations may (or may not) apply toresults obtained in the “Round robin test in the kineticevaluation of a complex solid state reaction from 13European laboratories”[39]. This study differed fromthe ICTAC project[37] in that each of the participatinglaboratories made their own kinetic measurements,using samples from the same reactant preparation,and the data obtained were kinetically analysed usinga common computer program. Again the spread ofcalculated kinetic parameters, for each of the threedistinct reactions investigated, was considerable andcompensation trends were identified. These resultsare said (Abstracts of[39]) to “allow an optimistic as-sessment for the application of kinetic procedures tosolid state reactions with well-known chemical courseinvestigated by TG (Part 1). . . by DSC (Part 2).” It isnot clear to me exactly what aspect of this work is re-garded as providing the evidence for such optimism,because the kinetic conclusions and magnitudes of thecalculated Arrhenius parameters show considerablescatter for reasons that have not been established. Itcould be that a method capable of resolving the incon-sistencies demonstrated in[39] may ultimately emergefrom the shortcomings identified here as present inthe kinetic analysis methods widely used. (It shouldalso be pointed out that the first of the reactions in thisstudy does not follow a ‘well-known chemical course’,as stated, because the dehydration of calcium oxalatemonohydrate has been shown to proceed as two con-current of overlapping reactions[40]. This mechanisticconsideration, probably influencing the kinetic inter-pretation, is not discussed in[39]. The initial dehydra-tion rate process is, therefore, complex. Consequently,the comparisons consider the kinetic analyses of threedistinct reactions, of which two may be single or sim-ple, but these certainly do not constitute a ‘complex’solid state reaction, as described in the article title.)

Additional related kinetic patterns. The pattern ofvariation of E with kinetic model,g(α) = kt, re-ported in [41], includes significant differences fromthose described above, though the specific forms ofthe rate equations used here for the data interpretationare not specified. Similarities include an approximateTrend A3 and compensation effects (Trends 4) for thethermal decompositions of the three reactants studied:cadmium, manganese and lead carbonates. However,although the magnitudes ofE diminished in the se-quence Eqs. (7), (12), (13)[17] (apparent Trend 2), the

diminutions were small and theE values reported foreach reactant might (just?) be regarded as approximat-ing to a constant value. Many of theEmagnitudes fromthe Integral and Differential equations compared weresimilar, though there were also instances of significantdifferences. Of the 60 regression coefficient valuesreported (two alternative methods of kinetic analysiscomparing fits to 10 rate equations for three reactants),10 were greater than 0.9990 (all for CdCO3), and 32were between 0.9900 and 0.9990. This is evidence ofapparently satisfactory fit of sets of measured data toalternative rate equations so that again the analysismethod gives a low level of distinguishability[18,19].

Arrhenius parameters were calculated from ratedata [42] for the first step in the decomposition of(CH3NH3)2MnCl4, by a range of different mathe-matical approximate approaches to nonisothermalkinetic analysis[1]. These results again showed con-siderable variation and also exhibited a compensationeffect. Reasons will not be discussed here, becausealternative analytical expressions were used, whereasthe present survey is mainly concerned with the CRmethod[12].

2.6. Comment

Kinetic results from[17,22] were compared andcontrasted here to exemplify the particular character-istic and systematic pattern of variations of calculatedmagnitudes of Arrhenius parameters with changes ofthe rate equation used in the analysis. While the ab-solute apparent magnitudes ofE change between dif-ferent reactants, there is an almost constant pattern ofrelative variations ofE with g(α) = kt within each ofthe series of Trends recognized above. It appears that,with minor variations and within less extensive rangesof data than those given in[17,22], the same Trendsare to be found in the many other sets of kinetic datacited above. I suggest that the most probable expla-nation for these variations is that the calculated (ap-parent) magnitude ofE for any reactant (expected tobe constant[13]) changes as a direct consequence ofalternative steps constituting the mathematical proce-dures used when applied through the different kineticmodels. In consequence, the Trends observed areidentified as computational artefacts. This means thatthe value of (the correctly defined, constant)E for anyreaction might be determined by a method other than


through the unsatisfactory[23] criterion of comparingr values. Identification of the reasons for the (ap-parently) variableE would eliminate a considerableuncertainty in the kinetic analyses of many thermo-chemical investigations. This problem is exploredbelow.

Linear correlation coefficients(r) and coefficientsof determination(r2). These statistical parameters,that provide a measure of the excellence of fit (orotherwise) of data to a kinetic expression, appear tobe regarded by researchers as being increasingly wel-come, acceptable or significant as their magnitude ap-proaches unity. Values ofr or r2 provides a measure ofthe closeness of fit of data to a perfect representationby the kinetic model considered. Such criteria are tobe regarded as valuable in kinetic analysis, as in otherscientific fields, and the sources cited above reportmany high values ofr or r2. However, when attempt-ing to identify the ‘best kinetic fit’ between alternativerate equations, the appearance of two, or more, highvalues ofr or r2 should no longer be welcomed be-cause this then becomes evidence of poor ability todiscriminate[18,19] between these alternatives. Ide-ally, in kinetic analysis, one equation should appearto be much ‘better’ (i.e.,r or r2 close to unity) thanall others and so that expression can be recognized asbeing uniquely applicable. The demonstration that al-ternative rate expressions, representing nucleation andgrowth, deceleratory or strongly deceleratoryα–time(t) relationships[1], when applied to the same data,give values ofr or r2 close to unity demonstrates thatthe method is unsuitable for identification of the ‘bestfit’. This is clearly evident in many thermoanalyticalpublications, including several discussed above.

3. Reappraisal of methods of analysis ofthermochemical rate data

The experimental techniques now exploited to ob-tain kinetic measurements, using thermoanalyticalmethods, enable large numbers of accurate data pointsto be collected rapidly and efficiently. The applicationof integrated (or associated) computer equipment,using analytical programs or spread-sheet methodsof computation, permits almost effortless data inter-pretation (partial). The results so obtained are printedin a convenient, and apparently authoritative, format

which is readily prepared for publication. The kineticconclusions arising as a consequence of this approachinclude (as in the papers cited above) multiple valuesof the Arrhenius parameters derived from single, orsometimes from multiple, data sets. (Kinetic inter-pretations reported, proposed or formulated are oftenlimited to the identification of the kinetic model giv-ing the ‘best fit’ to the data. This kinetic model is thenfrequently (and incorrectly) described as the ‘reactionmechanism’, thereby conveniently disregarding themore difficult objectives of elucidating the chemicalsteps and controls of the participating reactions. Theseare recognized as essential features of a reactionmechanismin other areas of chemistry.) The outcomeof the widespread acceptance (toleration) of theseconventions and practices is a literature consistingof individual reports, largely composed of empiricalobservations, in which the descriptive scientific terms(mechanism, activation energy) used may, or maynot, have been applied in a form intended to maintainconsistency with other branches of chemistry. What isabundantly clear is that this collection of separate anddistinct information fragments does not constitute acoherent scientific discipline within which the contri-butions are linked by unifying concepts or theoreticalmodels[13]. Consideration of these practices is essen-tial to recognize the shortcomings inherent in currentkinetic analysis methods which might then lead tomethods whereby errors in published work could becorrected. Some of the literature data might yet yieldvaluable results on critical reappraisal and reanalysis.

As in other branches of chemistry, those con-tributions to the thermoanalytical literature, whichare concerned with kinetic measurements, appear tobe motivated by the intention to elucidate reactionmechanisms. However, the published literature lackscoherence, results are without an adequate theoreticalframework and a proportion of the conclusions arebased on unsuitable experimental measurements (onlycapable of yielding empirical, conditions-sensitiverate information [3–9]). The problem, not alwaysaddressed in published articles, is how to formulatethe theoretical framework essential to introduce orderinto the extensive published material available andencourage future growth of the subject in a systematicmanner. The absence of critical reviews has undoubt-edly contributed towards the apparent absence ofinterest in theory development.


I suggest that two complementary problems mustbe recognized as existing within the accepted cus-toms and practices of thermal analysis and both nowrequire urgent recognition and reappraisal. One is thenecessity for a critical examination of the compu-tational methods that are currently employed in thekinetic analysis of thermochemical data and, in par-ticular, the necessity to ensure that all scientific termsand data units are correctly used. The present articlecommences this appraisal by critically examining themathematical procedures (apparently) used for kineticanalyses, based on the applications of the CR equationin the citations given above. This is essential becauseprogress can be expected only if the data used arereliable and the meanings, and limitations, of all con-clusions are fully understood (which, at present, is notalways obvious). The second action required is thesystematization of results, of proven reliability, whichrequires a unifying theoretical model[13], to whichthe individual systems may be related and classified.This is found in recent proposals by L’vov[10,11]which provide a rigorous and self-consistent expla-nation for the exponential dependence of the rate ofcrystolysis reactions[1] on temperature. This physicalapproach replaces earlier theoretical (chemical) treat-ments and has already been successfully applied (see[10,11] and references therein) to several groups ofthermal reactions in which systematic order for setsof rate processes has now been identified where nonehad previously been perceived. This reaction model,based on published thermochemical data, can only besuccessful in correlating reliable experimental obser-vations that are related to completely characterizedchemical changes.

The aspect of kinetic analysis addressed here is that,for a literature which includes many recommended ap-proximate calculation methods for the determinationof kinetic information (includingA, E and g(α) =kt) from nonisothermal measurements[1], the relativestrengths and weaknesses of the diverse and variousalternative expressions available[1,2] have never beenestablished nor agreed. Reports exploiting these meth-ods almost invariably choose one or other expression,or several, without explaining the reasons or providingany justification for the particular calculation methodselected. In the absence of critical and comparative re-views, the choice of approximate program for kineticanalysis becomes arbitrary. (Various approximations

were proposed in developing the alternative equations[1] which were originally formulated to enable kineticcalculations to be completed in a reasonable time.This was necessary in the era before high speed com-puting facilities became widely available. The samewell-established procedures have, however, continuedto remain as preferred methods of data analysis. Nocomparative tests, designed to identify sensitivity indistinguishability or ranges of applicability of fit todifferent kinetic models (such as[18,19]) appear tohave been reported. Even now, decades after their in-troduction, there are few suggestions that these com-putational methods should be replaced by the moresophisticated computer programs, now available, thatmust be capable of increasing the accuracy and relia-bility of calculated parameters and kinetic modelling,as pointed out by Flynn[43].)

The present paper is specifically concerned with anexamination of the properties of the widely used ap-proximate method of kinetic analysis usually referredto as the Coats–Redfern (CR) equation[12] used in[17], see also[22,23,26,29–32,37,41]. In the absenceof quantitative characterization of the properties ofthis equation, including tests of its efficacy in distin-guishing the fit of experimentally measured data toalternative kinetic models[18,19], significant, indeedfundamental, shortcomings present in this analyticalmethod appear not to have been appreciated. How-ever, in addition to their ability to interpret kineticobservations, computers are equally capable of check-ing the methods and results from such data analysis.This is initiated here. A principal consequence shouldbe the introduction of improved calculation methodsthroughout this subject area, exploiting the powerof the computer[43] to replace the obsolete, nowdemonstrably redundant, older approximate methodsof kinetic analysis. Results obtained by improvedanalysis methods may then contribute to the devel-opment of a subject ordered within the theoreticalframework recently proposed and developed[10,11].

Conclusions from my critical reappraisal of the CRequation are reported below. This analysis is based onthe results in[17] because the kinetic information isgiven in detail, is relevant to the wider literature andis in a form suitable for reconsideration by a kineticistwho now does not have access to suitable equipmentto enable me to make my own kinetic measurements.Similar investigations of the reliability of calculations


using the several other equations mentioned in[1]would certainly be worthwhile. Alvarez et al.[42],for example, have shown that apparent magnitudes ofArrhenius parameters vary with the approximate ana-lytical expression used in their calculations. The moti-vation for the present comparative survey was to showthat (so-called[14]) ‘variable activation energies’ canarise through limitations and shortcomings of the pro-grams and mathematical methods used to calculateArrhenius parameters. This is demonstrated here forthe CR method when applied to solid state type kineticmodels,g(α) = kt [1], and the conclusions are proba-bly equally applicable to other approximate methods.

3.1. Properties of the Coats–Redfern (CR)equation[12]

The CR equation is often applied in the form[17]:

ln

[g(α)

T 2

]= ln

(AR

βE

)− E

RT(1)

where the symbols have their usual meanings. Thisexpression omits the necessity to define a specific rateconstant for reaction, and, significantly, any obliga-tion to consider units. This is a fundamental, and po-tentially damaging, flaw in this approach to rate dataanalysis which is capable of accounting for featuresof the Trends described above. The isothermal kineticequation[1] is usually expressed in the formg(α) =kt where the rate constant,k, has the units (time)−1. Itwould appear that the use ofEq. (1) introduces unitsof an implied rate constant. The units of temperature,at constant heating rate, include (time)−1, whereasthe overall contribution from the temperature changethrough the (T−2) term is much smaller. Where thereare other terms, such as rate constants introduced tointerpret rate data, the Arrhenius parameters may be-come scaled accordingly, for example, the magnitudeof the conventionalE can be changed by a factorn([1], p. 121), as shown inSection 3.2.

In discussing the analysis of data below, we referto the kinetic models (from Table 3 of[17]) as FI1to FI19 for the Integral method and FD1 to FD19 forthe Differential method, and data points D1 to D17refer to the reportedα values. Representative plotsof ln[g(α)/T 2] againstT−1 are shown inFig. 3 forequations FI7 (upper steep slope), FI13 (lower steepslope) for which the values ofE are not widely dif-

ferent (161.550 and 147.661 kJ mol−1 [17]) and FI16for which E was much less (37.195 kJ mol−1). Rep-resentative repeat calculations from the data given inTables 2 and 3 of[17] confirmed and were entirely inaccordance with values reported in Table 4[17].

3.2. Power law equations

For the power law equations FI1, FI14 to FI17[17],we can writeEq. (1) in the following form:

n ln(α) − 2 lnT = ln

(AR

βE

)− E

RT(n = 2.0, 1.0,

1

2,

1

3,

1

4

)(2)

The significant feature of this expression is that thevarious forms of the power law differ only by themagnitude of the exponent,n [44]. Dividing throughby n, it is immediately apparent that the form ofEq. (2)is incapable of distinguishing between these kineticmodels and that the apparent Arrhenius parameters arescaled by the exponent.

Distinguishability. The first term inEq. (2) is sig-nificantly larger than the second. Specifically, fordata from[17] these values range from×29 to ×2.8when n decreases from 2.0 to 0.25. The range ofmagnitudes ofn ln α between points D1 and D17(i.e., �(n ln α)/T −2 = n ln D17(0.9948)/T −2 −n ln D1(0.013)/T −2) are listed inTable 1, togetherwith comparable data for the other equations fromFig. 3, similarly correlated here. For the powerlaw expressions, in the first row, these changes aremuch larger than the change in 2 lnT (which is12.471− 12.188= 0.284). Moreover, these CR func-tion increments reduce progressively from 0.0045 to0.0039 for each degree (K) rise across the particulartemperature interval studied[17]. It follows that theform of the first two terms inEq. (2) is incapable ofdistinguishing[18,19] the fit of data varying betweenthe deceleratory process (FI1), the constant rate ofreaction (FI14) and the three acceleratory processes(FI15 to 17). Consequently, this method of kineticanalysis is shown to possess almost no ability to detectany change in shape of theα–t relationships so thatdistinguishability is poor (indeed, almost nonexistent)except the effect arising from the small contributionthrough the constant systematic variation of the 2 lnTterm. This is confirmed by the small changes in the


Fig. 3. Representative plots of ln[g(α)/T2] against 1/T for three equations from data listed in[17]: FI7 (first order equation, upper steepline), FI13 (contracting volume equation, lower steep line) and FI16 (power law, cube root equation, lower slope).

Table 1Range of magnitudes of the term ln[g(α)/T2], for the different kinetic models considered inFig. 3, across the temperature interval(443.14–520.64 K) for data recorded in Tables 3 and 4 of[17], and including calculated magnitudes ofEa,b

Exponent,n 2.0 1.0 0.6667 0.5 0.3333 0.25Power law FI1, 14–17 8.392 4.054 – 1.885 1.162 0.801r2 0.9910 0.9905 – 0.9893 0.9879 0.9861E (kJ mol−1) 262.84 127.46 – 59.76 37.19 25.92Avrami–Erofeev equation FI7–11 – 5.713 3.714 2.714 1.715 1.215r2 – 0.9707 0.9694 0.9680 0.9649 0.9613E (kJ mol−1) – 161.56 105.08 76.82 48.57 34.44Contracting volume FI4, 13 10.210 4.963 – – – –r2 0.9952 0.9845 – – – –E (kJ mol−1) 303.29 147.68 – – – –

a The r2 values are the ‘coefficient of determination’, and are lower thanr, the ‘correlation coefficients’ given in[17]: overall thevalues agree well.

b The values ofE given are recorded to a greater number of significant figures than I consider appropriate to confirm the close agreementwith the results in[17].


coefficient of determination,r2 (values 0.9910–0.9861:this variation arises from the systematically changingcontribution from the 2 lnT term), listed inTable 1for the set of power law expressions considered. Ef-fectively, the comparative ‘fitting’ process is almostindependent ofn and the kinetic analysis is totallyunsatisfactory in this respect.

Arrhenius parameters. The slope of the linear plotbetween the CR exponential term and (T/K)−1 is mul-tiplied by the factorn between the different power lawexpressions and the overall influence of the secondterm (−2 lnT) remains small. Accordingly, the mag-nitudes of both apparentE and lnA values are scaledby the factorn, as found in Trend A2 (Fig. 1) and thecompensation effect (Trend A4).

3.3. Avrami–Erofeev equations

For equations FI7 to FI11 we can rewriteEq. (1)inthe form:

n ln[−ln(1 − α)] − 2 lnT = ln

(AR

βE

)− E

RT(AE equation withn = 1.0,

2

3,

1

2,

1

3,

1

4

)(3)

The factorn multiplies (or scales) the magnitude ofthe logarithm of the deceleratory first order (FI7) ex-pression which is modified only very slightly by therelatively much smaller contribution from−2 lnT.Again, the forms of these terms are incapable ofdistinguishing[18,19] between the deceleratory char-acter of Eq. (FI7) and the progressively increasinglyacceleratory character of Eqs. (FI8) through (FI11).The pattern of behaviour given by the CR analysis us-ing the AE equations is identical with that describedabove for the power law, explaining the Trends A2and 4. There is no adequate test of distinguishability.

3.4. Contracting volume equation

Only two other equations, identical in form butdiffering in exponent, FI4 and FI13, were availablefor this comparison[17] and have been included inTable 1, where the conclusions described in the twoprevious paragraphs again apply. The differencesbetween the similar rate expressions FI12 and FI13were small, Trend 3, the range of the first term, for

Eq. (FI12) in the CR equation, was 4.674,r2: 0.9980andE: 141.79 kJ mol−1.

3.5. Variation of activation energy withαrange of data

The above analysis represents selected (Fig. 3) re-sults of kinetic and statistical analyses in which thecomplete range of data points recorded in[17] wereincluded (i.e., D1–D17). In this section, the conse-quences of systematic deletion of the initial data pointsare examined. Under the rising temperature regime,used in these experiments, at the mid-point of the tem-perature range from which data were collected, onlysome 12% reaction had occurred. ApparentE valuescalculated, with selective deletion of different numbersof points from the lower end of the temperature inter-val of the kinetic measurements, are shown inTable 2.The trend of variation ofE values with range ofα in-cluded[17] in the CR calculations is illustrated for thefirst order equation inFig. 4.

The data inTable 2(again) demonstrate that appar-ent magnitudes ofE vary substantially with the rateequation used (Table 4 of[17]). Here, the calculatedE values usually, but not invariably, increase with theselective deletion of lowα values (by amounts be-tween approximately 10 to 70%, for the equationscompared). The values ofr2, which approach unity,are usually accepted as the measure of good kinetic‘fit’. However, the above variations demonstrate theexistence of a further uncertainty that requires consid-eration when basing kinetic interpretations exclusivelyon this kinetic criterion[23]. Theα range across whichthe kinetic model is applicable must be considered inany complete description of rate characteristics, but isnot always mentioned or reported.

3.6. Trends

The above discussion of the mathematical steps inanalyses of the same data by alternative rate functionsin the CR equation, shows that this approach yieldsapparentE values that vary widely with both kineticmodel (Trend A1) and range ofα considered. (Theremay well be additional controlling influences fromfactors other than those considered here, includingphysical and chemical controls such as the dependenceof rate measurements on reaction conditions, theprocedural variables[3–9,15,16].) The systematic


Table 2Apparent activation energy values calculated from selected kinetic measurements in Table 2 of[17] with selective deletion of measurementsat low α values

Rangeα Eq. FI7 FI9 FI11 FI4 FI13 FI12

E (kJ mol−1) 0.013–0.9948 161.56 76.82 34.45 303.29 147.68 141.79r2 0.9707 0.9680 0.9613 0.9852 0.9845 0.9880E (kJ mol−1) 0.0194–0.9948 177.79 84.88 38.44 328.05 160.02 152.52r2 0.9825 0.9810 0.9774 0.9949 0.9947 0.9968E (kJ mol−1) 0.0374–0.9948 192.06 91.98 41.94 345.42 168.66 158.87r2 0.9867 0.9856 0.9831 0.9983 0.9982 0.9990E (kJ mol−1) 0.0839–0.9948 206.68 99.25 45.54 357.35 174.59 161.35r2 0.9875 0.9865 0.9843 0.9993 0.9993 0.9987E (kJ mol−1) 0.195–0.9948 225.90 108.82 50.28 366.46 179.10 160.25r2 0.9875 0.9866 0.9845 0.9996 0.9996 0.9970E (kJ mol−1) 0.4339–0.9948 257.12 124.38 58.02 371.82 181.73 152.80r2 0.9904 0.9897 0.9883 0.9991 0.9991 0.9927

Fig. 4. Plots of the Coats–Redfern term withg(α) = kt as the first order equation for the StdSet data[17] with regression lines, in order ofincreasing slope, representing ‘best fit’ to data forα ranges extending from 0.013, 0.0374 and 0.195 to 0.9948, respectively (these graphsrepresent the first, third and fifth entries of the FI7 column inTable 2). There are significant changes in apparentE values with range ofα data considered; calculated values, in the same sequence, were: 162, 192 and 226 kJ mol−1.


changes ofE with n (Trends A2 and A3) are shown toarise from the form of the Integral equations (Eqs. (2)and (3)) used in the kinetic analyses. The compensa-tion effect (Trend A4) appears because the alternativerate equations are applied to the (identical) rate datameasured within the same temperature interval (isoki-netic behaviour) and bothE and lnA are scaled byn.Plots ofE againstn (Fig. 1) do not pass through theorigin but E = 0 at a small value ofn, usually be-tween 0.05 and 0.10. The magnitude of�(−2 lnT ),across the temperature interval for the data used inthis comparison, 0.284, corresponds to 0.05 to 0.10of the range of variation of the term ln[g(α)/T2] (4–6in Table 1), thus this term contributes to the intercepton then axis.

The CR equation forms used in the present analysisomit all consideration of rate constants and the unitsof time do not explicitly appear in the treatment. How-ever, the above observations suggest an interestinganalogy between the role of the rate expression expo-nent,n, here and an inconsistency that appears duringisothermal analysis using the AE equation. In the latteranalyses, the incorrect definition of rate constant unitsas (time)−n gives a value ofE which is similarly mod-ified by the factorn (see[1], p. 121). Consequently,term definitions and rigorous consistency of units mustbe regarded as essential features of kinetic analysis.This is currently being given insufficient attention.

3.7. Comparison of E values alternativelycalculated by Coats–Redfern andisoconversional methods

Vyazovkin [14,23] and Vyazovkin and Wight[28]have reported magnitudes ofE calculated alterna-

Table 3Comparison of activation energies calculated by Coats–Redfern method (“Corrected” as described in the text) and by the isoconversionalmethod from data in[14,23,28]a

Kineticorder

“Corrected” E(see text)

IsoconversionalE(approximate range of values)

IsoconversionalE(approximate mean value)

Ammonium nitrate Zero 76 85–95 92First 88.5

Ammonium dinitramide Zero 127 75–175 130First 147

HMX Zero 118 122–152 142First 134

a All activation energy valuesE (kJ mol−1) and rate equation units were (time)−1.

tively by the Coats–Redfern equation and by theisoconversional method for the thermal decomposi-tions of HMX, ammonium nitrate and ammoniumdinitramide. Attention was principally concerned withthe calculated magnitudes ofE because Vyazovkin[23] finds “. . . the preexponential factor is a depen-dent and, therefore, an inferior parameter”. (WhyAshould be considered ‘inferior’ toE in considerationof two complementary and potentially equivalent pa-rameters connected by a compensation relationshipis not explained. The essential significance ofA, asone of the factors that provides an overall measure ofabsolute reactivity, is apparently disregarded.)

A comparison of these results, based on alternativeinterpretative calculations from[14,23,28]and sum-marized inTable 3, are of interest here in examiningthe above shortcomings of the Coats–Redfern equa-tion. “Corrected”E refers to the values from Eq. (FI14)(estimated from the linear relationship betweenE andn) and FI7, to which the intercept (−E) at n = 0 hasbeen added (for both the zero and first order equa-tions the units ofk are (time)−1). No explanation forthe omission ofE values for the zero order equation isgiven, but magnitudes are easily estimated from plotsof E againstn.

It was expected that the zero order equation wouldprovide the most reliable values ofE for compari-son with those found by the isoconversional approach,which is based on consideration of successive reactionintervals of constant rate (i.e., the same kinetic model).However, the values inTable 3identify an additionaluncertainty, responsible for a difference of around 10%(compare columns three and five) between values ob-tained by the two calculation methods. The reasonsare not identified here but could be a further conse-


quence of the unreliability of the CR approximate ap-proach in estimation of the ‘temperature integral’, see[43].

3.8. Kinetic analysis of calculated model databy other rate equations

Aspects of the pattern of results of kinetic analysesidentified above were confirmed by comparative anal-ysis of simulated (calculated) rate data. Three sets ofα–T values were computed for fit to the power lawequations,α1/n = kt, with n values, 0.5, 1.0 and 2.0,representing deceleratory, constant rate and acceler-atory reactions. Rate constants for a typical reactionwere taken as log10k = 12.00–100,000/RT, duringheating at 1 K min−1 between about 320 and 400 Kand kinetic analyses were made forα in the range0.01 to virtual completion,α = 1.00. Equations testedwere from the series considered throughout this sur-vey, the power law withn = 2.0, 1.0, 1/2, 1/3, 1/4 andAE with n = 1.0, 2/3, 1/2, 1/3, 1/4. The calculatedvalues ofE andr2 are given inTable 4, ‘bold’ entriesrefer to analysis by the ‘correct’ expression, that usedto obtain the data.

Plots ofE againstn were linear for both equations,all AE values were about 15% larger than the powerlaw values (Fig. 1). All six lines extrapolated to co-incide at an apparentE = −6 to 7 kJ mol−1 whenn = 0 and the intercepts forE = 0 were betweenn = 0.03 and 0.11. An important limitation of the CRanalysis is that some magnitudes of calculatedE weresignificantly different from values used in the expres-sion from which the simulated rate data were obtained.(The divergence was even greater from similar com-putations for then = 3 power law, 93 instead of theexpected 100 kJ mol−1.)

The values ofr2 in Table 4show that the excellenceof data fit changes only marginally withn, particularlyfor tests to the power law when values are close tounity. This cannot, therefore, be used as a criterion forkinetic distinguishability. The values recorded refer tothe main reaction interval, in the range of data pointsbetween 1 and 100% completion. Tests were extendedto lower α ranges, but these were characterized bylowerr2 values and magnitudes ofE that changed withextent ofα included and diverged even further frommagnitudes associated with the highr2 values listedin Table 4. This is readily explained through consider-

ation of Eq. (1). The terms−2 lnT andE/RT changesystematically with the temperature rise, by relativelyconstant increments. The magnitude of the CR termg(α), in contrast, is determined by two parameters: themathematical form ofα and the exponential temper-ature dependency of the (usually unconsidered) rateconstant that determines the magnitude ofα and its in-cremental increases. Thus, the magnitude of the valuesof this term and the increments�(g(α)) vary relativelyto the others as reaction progresses and no linear re-lationship can be maintained between ln[g(α)/T2] andT−1. Thus the form of the CR function is most suit-able for calculation of apparentE values over limitedα intervals and the comparative data given inTable 4refer to the range that is often of greatest interest inkinetic investigations. The CR approximation may beinapplicable at lowα.

4. Discussion

The principal result from the above critical analysisof some methods of kinetic interpretation of thermalmeasurements is unambiguous. The Coats–Redfernequation, Eq. (1) [12], cannot be regarded as anacceptable method for the kinetic analyses of thermo-chemical rate data which are capable of representationthrough those kinetic models that are derived throughgeometric models applicable to crystolysis reactions[1]. The apparent magnitudes of CR-calculated Ar-rhenius parameters are scaled by the exponential term,n, present in many of the rate expressions. More-over, the methods of testing the ‘fit’ of yield–timedata to kinetic models, by the usual statistical cri-teria, are incapable of distinguishing between thealternative equations that contain different exponents.Arrhenius parameters are scaled by an undeterminedfactor, kinetic distinguishability is unsatisfactory: theCR equation is totally incapable of achieving its in-tended objectives, through its characteristic approachto kinetic analysis. While specific limitations of themethod have been mentioned in previous publications[23,44], it appears that this is the first appraisal thathas recognized the full extent, and reasons for, itsgeneral unsuitability. Reported conclusions based onthe method, and interpretations derived therefrom,must not be regarded as reliable and these now requirefundamental reconsideration.


The next obvious problem, for the subject over-all, is to establish whether the other approximateapproaches to analysis of nonisothermal kinetic data[1,2] are more reliable or suffer from similar uncon-sidered faults in the calculation methods and/or theunderlying assumptions. Critical reappraisal of theconsequences of the shortcomings now recognizedshould also be extended to reconsider all recentlypublished kinetic conclusions. This might usefullyinclude reports where only a single magnitude ofE(perhaps accompanied byA) appears as the principalresult from a kinetic study: possible limitations in thekinetic analysis method used may not be so readilyperceived. Indeed, it is now obvious that the approx-imate methods of kinetic analysis have outlived theiruse and are due for replacement. A good starting placeis the final sentence of the paper by Flynn, who con-sidered the applications of the ‘Temperature Integral’(“Its use and abuse”)[43], and who states that: “. . .

in this age of vast computational capabilities, thereis no reason not to use precise values for the temper-ature integral when calculating kinetic parameters”.Surely, by now, the practice of using the older (highly)approximate formulae in kinetic analysis might beregarded as redundant. It is unacceptable (mindfulof current, and continuing, conventional practices)that the advantages and the untapped potential of thepowerful computing facilities, that are now so widelyavailable, should remain inadequately applied in thisfield. A more fundamental approach to the kinetic in-terpretation of thermoanalytical data is now overdue.

For the future prosperity of thermochemical ratestudies, the reintroduction of the fundamental as-sumptions of chemical kinetics is essential, indeedoverdue. Methods used to interpret programmed tem-perature rate data now require radical reassessment.The potential of the computing methods that are sowidely available[43] should be applied in a construc-tive and critical manner through the use of programsthat are soundly based on and capable of being relatedto chemically acceptable reaction models[10,11,13].A new start, this proposed renaissance, must incor-porate the essential chemical assumptions and defi-nitions, rather than proceeding through mathematicalmanipulations which, in recent times, have eclipsedthe realities of the subject. It is particularly surprisingto me that the errors introduced into the CR equationduring its initial development, through its applications

to these types of reactions, were not immediatelyrecognized. It is strange that these shortcomings havebeen disregarded for so long and that this, and othercomparable, calculation methods still maintain prideof place in the forefront of modern thermokinetics.This cannot and should not be tolerated or sustainedany longer. However, only when a full appreciationof the hopelessness of the present situation has beengenerally accepted can a new generation of data in-terpretation methods be implemented. The presentdiscussion of the weaknesses exposed in the presentcriticism of the CR approach is intended to enablethese faults to be comprehensively recognized, fullyappreciated and avoided in the future. The hope nowis for the restoration of acceptable scientific foun-dations that will provide an optimistic base for theregeneration of this branch of chemical kinetics.

Nonisothermal kinetic studies of rates of reactionsin solution. Alibrandi [45] has drawn attention to theadvantages of applying nonisothemal methods to thedetermination of activation parameters for reactionsin solution, with reference to the earlier source[46].Kinetic data can be collected in a shorter time, byspectrometric methods described, and with greateraccuracy than with the usual isothermal technique.However, the approach to data analysis explained[45]bypasses the pitfalls that have substantially devaluedthe interpretation of kinetic observations in the fieldof thermal analysis. First, approximate calculationformulae are not introduced into the characterizationof the separate influences of first order rate processesand of the Arrhenius equation through the use ofcurve fitting methods. Second, the analysis is directedtowards distinguishing the relatively few homoge-neous rate expressions and the complexity arisingfrom the larger number of kinetic models associatedwith reactions of solids is avoided. For the examplesmentioned, the rate constant units are (time)−1. Third,the potentially rate-influencing effects of self-heatingand of diffusion control are inapplicable in the ho-mogeneous reaction medium. Thermal analysis couldperhaps learn from the simpler, more direct approachto rate interpretation in this treatment.

4.1. Identification of the kinetic model

Historically, the first step in data interpretation wasoften characterization of the kinetic model, though


perhaps this has become less usual in more recentwork. The form of the CR equation,Eq. (1), how-ever, makes it incapable of distinguishing[18,19], byr2 values or other criteria, the kinetic model that ‘best’describes the data from that group of rate expressionswhich constitute the set[1] most usually consideredand compared in the interpretation of rate data fromthermoanalytical measurements. A generally similarconclusion was previously discussed by House et al.[24,25]when comparing kinetic results obtained fromsuccessive identical experiments. They pointed outthat a ‘good fit’ may be found for an incorrect rate law.In most of the recent investigations, kinetic analysestake no account of the extent of reaction across whichthe fit is tested. As shown above, apparent magnitudesof E can vary with theα interval of the data includedin the CR calculations applied during the kinetic anal-ysis. Moreover, some reports do not mention rate be-haviour during the early or the late stages of reactionwhere, for many kinetic relationships, distinguishabil-ity [18,19] is most reliably tested.

4.2. Determination of the activation energy

Uncertainties in values ofE calculated by the CRmethod include the following, from which no obviousmeans of identifying the ‘correct’ value appears to beavailable.

(i) The slope of the plot of ln[g(α)/T2] againstT−1,for many rate expressions, includes the exponentn and is, therefore,−E/nR: Figs. 1 and 3andTables 1 and 4.

(ii) E magnitudes can vary significantly with therange ofα included in the calculations:Fig. 4andTable 2.

(iii) After allowance forn values, (i) above,E valueswere not reconciled with magnitudes from theisoconversional calculations:Table 3.

(iv) E values determined for standard model calcu-lated data varied with theα range included in theanalysis:Section 3.7above, alsoTable 4.

The scaling ofE by the factorn, (i) above, is an incon-sistency that parallels the variations that arise throughincorrect definition ofk, to units (time)−1, in isother-mal kinetics (see p. 121 of[1]). Even whenE is cor-rected for the exponential factor,n, to nominal units(time)−1, the CR-calculated values vary with kinetic

model and also differ from those found by the isocon-versional method, (iii) above. Reasons probably in-clude limitations inherent in the CR equation, whichdoes not suitably accommodate the kinetic model andis also based on an approximate form of ‘temperatureintegral’ [43].

4.3. Arrhenius parameter trends

The above explanations for the systematic vari-ations of E and lnA values with kinetic modelsconfirms that the Trends identified above are mathe-matical artefacts. These are implicit within the calcu-lation programs and are, therefore, without chemicalsignificance. Their appearance in the several reportsmentioned demonstrates their generality. The similarbehaviour patterns recognized in the many diversepublications cited means that the reservations ex-pressed here concerning the significance of kineticand mechanistic conclusions are, in all probability,equally applicable to all the sources mentioned.

4.4. The significance of activation energies

The convention, apparently accepted throughoutthis literature, is that virtually every parameter com-puted can be regarded as reliable and, in particular,thatE should be identified as a variable quantity[14].This appears to me to be an unreasonable and unsci-entific approach to the resolution of the uncertaintyconcerning the significance of calculated apparentEvalues. It is the computed values and not the defi-nition of E that are without meaning and, as shownhere for the CR method, this arises through short-comings in the calculation and/or the experimentalmethods. Accordingly, it is recommended that, inthermal analysis,E should retain its original definedmeaning and continue to be regarded as a property ofthe reaction chemical step[13], the model acceptedthroughout other branches of the subject. It may bedifficult, even virtually impossible, experimentally todetermineE and other kinetic parameters, includingA and g(α) = kt, but this does not justify a changeof the definition and the concept of a term that isof inestimable importance throughout reaction ratestudies. Correctly calculatedE values may be used incorrelations between different chemical changes andbe used in the formulation of reaction mechanisms.


All kinetic terms, includingE, must be meaningfullydefined before such quantities may be interpretedto provide insights into the reactivities of differentsubstances through concepts capable of providing atheoretical framework for the subject and into reactionchemistry, see L’vov[10,11]. Chemical objectivity isalways to be preferred over mathematical expediency.

The applicability of the theory of the single (identi-fiable) activation step model to reactions of solids re-mains incompletely established[47], and is regardedby many practitioners as being of doubtful value,or even impossible to confirm. We have little agree-ment concerning the factors that determine absolutereactivities of solids and the mechanisms of chem-ical changes including their controls (see, however[10,11]). The experimental measurement ofE, in itscorrectly defined meaning, is undoubtedly difficult formany reactions[3,4,7–9]and may, in practice, be im-possible for some. However, the benefits arising fromthe alternative view thatE can be usefully regarded asa function ofg(α) = kt have yet to be demonstrated.In my view this approach devalues an accepted the-oretical model without substituting a demonstrablyuseful or adequately defined replacement. An essen-tial reason for maintaining a clearly understood andsignificant meaning forE is that this parameter is po-tentially capable of being used as a criterion for theclassification of observations (which seems unlikelyfor a ‘variable’E). The recent contributions by L’vov[10,11] identify the physical significance ofE, whichis a fixed (emphatically not variable) property foreach specific chemical reaction.

4.5. One view of the future of thermal analysis

Mindful of these limitations and fundamental prob-lems inherent within current practices concerning ther-mal analytical kinetic studies, it seems to me that thereare two alternative directions in which the subject cannow progress. One view of the future is to maintainthe ‘status quo’. (The alternative view is discussed inSection 4.6. which follows.) We can continue to applythermoanalytical methods to collect even more ratedata for already well-studied thermal systems and/orto extend similar investigations to include additionalnovel reactants. This approach appears to have domi-nated the recent literature in which reports continue toappear giving further values of Arrhenius parameters

and kinetic models for reactants both familiar and un-familiar. Investigations are relatively easily completedby largely automated computer programs, that rapidlyundertake most of the labour of data interpretation.All desired significant rate parameters appear, almosteffortlessly, machine-presented in an apparently au-thoritative printed format. The recent outcome of suchresearch has been the generation of a subject litera-ture that is characterized by the accumulation of largenumbers of individual reports which remain gener-ally, or entirely, unrelated to each other and do notcontribute to the organic growth of a coherent bodyof scientific knowledge. Mechanistic and chemicalinterpretation of much of the kinetic data so obtainedhas remained incomplete for many, if not most, of thethermal decompositions studied. Such reactions areoften indicated, or implied through kinetic properties,to occur in the solid state, though usually withoutdirect or adequate confirmation. It must also be re-membered that measured rate data do not necessarily,and frequently do not, yield a magnitude ofE that ischaracteristic of any identified rate limiting step. Suchmeasured reaction rates are all too often subject tomultiple controls, due to the influences of proceduralvariables[3–9,15]. Many reported observations areempirical because, in addition to the (uncharacterized)chemical controlling step, there are significant, evendominant, contributions from self-cooling/heatingand reaction reversibility, which may be inhomoge-neous within the reactant mass and change with time.Reaction conditions must be designed to eliminate orto minimize secondary controls[3–9] if chemicallysignificant observations are to be obtained.

It has become increasingly evident, from the prac-tices and conventions that characterize the recentliterature, that the principal conclusions obtained inmost kinetic studies of crystolysis reactions (A, E andg(α) = kt) do not provide criteria whereby the prop-erties of the reactants investigated can be systematizedor correlated to establish the general features whichcontrol reactivities, mechanisms or reaction rates.This is a consequence of the absence of theoreticalmodels capable of ordering the results available andpermitting predictions, by induction, of the behaviourof hitherto untested reactants. These are principal ob-jectives of the scientific method and, in the absenceof an adequate theory, order may not be found[13].Moreover, the theoretical models inherited from the


precursor subject, thermal decompositions of solids,have found progressively fewer positive applicationsin the interpretation of thermoanalytical data. The ab-sence of principles, theories and models, capable ofordering information, reduces the scientific achieve-ments of such studies. Based on a prediction from re-cent history, the further accumulation of observations,of types that are already available in large quantities,is unlikely either to introduce scientific order or toprovide general insights capable of identifying thecontrols of reactivity, thermal properties or reactionmechanisms. The alternative view for the future is toreview radically the principles and practices of thermalanalysis with the intention of restoring scientific meth-ods based on chemical and physical concepts whichgives an optimistic view for theory development.

4.6. A more optimistic view for theory andsubject development

The recognition that current data interpretationpractices are unacceptable and should be discontin-ued represents both an advance and a challenge. Amore attractive option than continuing the unsatis-factory, even hopeless, scenario outlined above is toreappraise fundamentally and comprehensively theexisting and alternative methods that are capable ofachieving the intended objectives of the subject. Theremoval of the present impasse, arising from the ex-cessive preoccupation with unsuitable mathematicalanalysis methods, and the doubtful significances ofresults calculated therefrom, reveals an apparentlyconsiderable gap in theory. Throughout chemistry,kinetic studies are the accepted approach to the elu-cidation of reaction mechanisms, including the con-trols of reactivity and comprehensively understandingthe dynamics of chemical change. Thermochemicalmethods offer one particularly suitable approach foraddressing such problems. However, the changes ofemphasis within this subject, during its developmentand effective eclipse of studies of thermal decompo-sitions of solids, have, in practice, effectively weak-ened any motivation towards obtaining informationof chemical significance. The greatest investments ofeffort in thermal analysis have been directed towardsextending instrumental convenience and in promotingthe mathematical interpretation of data (largely usingthe kinetic models derived through solid state rate

equations). This is excellent progress but is insuffi-cient by itself to characterize all features of chemicalreactions. The limitation to the advances possible bythermal methods alone arises because of the accom-panying (but not necessarily consequent) reduced in-terest in reaction chemistry generally. Relatively lesseffort has been directed towards investigating reac-tion stoichiometries, structural features and bondingproperties of reactants, textural changes accompa-nying reactions, etc., which has inevitably restrictedthe total amount of chemical information that can bederived from such work.

4.7. Other problems in thermal analysis

Uncertainties, inconsistencies and calculation errorsare not the only problems limiting the advance of ther-mochemical kinetics. Other limitations and shortcom-ings of the subject, as now practised, that have becomeapparent and are identified as being significant, willthe subject of a forthcoming review, now in prepa-ration, including the aspects mentioned below. It isnot intended to imply, however, that each and everyfeature mentioned here is to be found in every ther-moanalytical paper, but rather that these limitationsare sufficiently frequently found in the literature tobe generally regarded as acceptable to a majority ofthe contributing researchers, and, significantly, to thejournal referees, active in the field. Overall it is ar-gued that much greater attention should be directed,through the complementary studies usually necessary,to the chemical features of the chemical reactions thatare the subject of thermochemical investigations.

1. Stoichiometry. Kinetic data must be related to anidentified chemical change or changes. Many ther-moanalytical investigations do not completely char-acterize and confirm reaction stoichiometry. EachA, E or kinetic model, rate expression, reportedmust be identified with a single, fully characterizedrate process.

2. Single reaction. Interpretation of rate data to de-termine Arrhenius parameters must be unambigu-ously identified with, and based on, data for asingle rate process, unless methods capable of re-solving complex data are used. This is not alwaysadequately demonstrated. The extent,α range offit, for each kinetic model identified as applicableshould be stated.


3. Absolute reactivities. Rate measurements obtainedto determine an absolute reaction rate, or absolutereactivity, for an interface process must show thatthe contributions from concurrent and/or over-lapping consecutive rate processes and effects ofreversibility and/or heat flow have all been consid-ered or eliminated[3–9].

4. State in which reaction proceeds. To provide asatisfactory description of any rate process, thephase in which the chemical changes of interesttake place must always be characterized. The pos-sibility of melting before or during reaction is notalways mentioned. Alternatively, an initially solidreactant may undergo precursor changes such as apolymorphic crystal transition or dehydration. Theterm ‘crystolysis’ may be used to confirm that thereaction occurs in the solid state[1].

5. All computational methods used to calculate ki-netic parameters must be fully understood, appro-priate and reliable, including term definitions andtheir units.

6. Kinetic evidence alone is almost invariably in-sufficient to characterize a reaction mechanism,particularly when solids participate, and supportfor mechanistic interpretation must be soughtthrough complementary observations. Informationof all relevant types, microscopic investigationsof textures, chemical analysis, crystallographic in-formation including topotactic[1] relationships (ifany), spectroscopic data, etc., may be expected toincrease the reliability of conclusions. The mostdependable deductions are most usually basedon investigations that include the widest range ofrelevant complementary measurements.

7. Conclusions should, where possible, be discussedin the context of related reactions and the chem-ical similarities considered. Many reports are notintegrated into the context of all relevant litera-ture, often neglecting discussions for results thatare inconsistent with the deductions being pre-sented. There are remarkably few critical reviewsor comparative surveys in this subject area.

4.8. Comment

The point of the above, highly critical and gen-eral, comments on the thermoanalytical literature isto advocate a general reappraisal of all accepted but

inappropriate practices. The realistic interpretation ofchemical measurements by chemical methods in thecontext of chemical theory should always be used, to-gether with the support of complementary observa-tions (physical and chemical). This approach might beexpected to introduce scientific order into a topic cur-rently composed of random and unrelated investiga-tions. The way forward, towards developing a coherentscientific subject, requires the recognition of two com-plementary principles. First, the introduction and ap-plication of revised methods for kinetic measurementsand for data interpretations to yield reliableE values,and all related features of kinetic analysis, that are fullyconsistent with the accepted (defined) meanings (andunits) of the terms used. The present paper demon-strates the catastrophic weaknesses of current practicesin a subject that appears to have stagnated[48], lackscritical input and apparently has ceased to be based onrecognizable scientific principles. Second, the kineticparameters, most notablyE (and not its ‘variable’ re-placement) obtained for representative systems by therevised methods can then be compared with the unify-ing theoretical concepts developed by L’vov[10,11].A revitalized attitude, intent on introducing coherenceand order based on theoretical models, should be wel-comed as offering an optimistic future. The importanceof a theoretical model, capable of providing a centralfeature in ordering results, has been stressed by Lai-dler [13]. Let us all adopt a scientific stance for theoptimistic development of a subject that is experimen-tally and theoretically difficult. L’vov[11] has stated“It is difficult to imagine how much effort, time andmoney have been spent in vain in the investigations ofkinetics of solid decompositions because of neglectingthis method” (i.e., the third-law method for the cal-culation ofE). Perhaps now, at last, we can cease tocontribute to this unnecessary and unproductive wasteby initiating a renaissance in which the principles ofthe scientific method are restored to thermal analysis.

My optimism for an impending regeneration of theinitial promise offered by thermal analysis (so manyyears ago) is based on the possibility that we mightnow replace outdated calculation programs by elimi-nating inappropriate approximate methods. This waspowerfully advocated by Flynn[43] and significantadditional faults in the currently popular methods areidentified above. Second, the formulation of a reac-tion model, by L’vov[10,11], capable of accounting


for absolute reactivities and enabling systematic com-parisons of thermochemical properties, provides theessential theoretical foundation for scientific progressthat has hitherto been missing. We should now antici-pate a much improved future for the science of thermalanalysis, a veritable renaissance.

Appendix A. Before describing a quantity as anactivation energy, make sure that it involvesactivation and that it is an energy

This heading is my response to the prefatory quo-tation by Vyazovkin in his paper “Two types ofuncertainty in the values of activation energy”[23],attributed to Daniil Kharms: “When you’re buying abird, make sure it hasn’t got teeth. If it’s got teeth, itisn’t a bird”. My dissimilar sentiment expresses mycomplete inability and unwillingness to accept thereasons for regarding activation energies as ‘variable’[14] quantities or, more usually, the explanations givenfor their apparent property of being able to adoptseveral different values for the same reaction. (Is amultivalue term correctly described as a ‘variable’?)However, this should not license the (chameleon-like)use of the term to represent any and every changeof magnitude of reaction rate with temperature underdifferent, and frequently incompletely described, re-action conditions or results obtained from the samedata by alternative, approximate calculation methods.

Why, within the current thermoanalytical kinetic lit-erature, are these quantities described as ‘activationenergies’, when they are obviously no such thing,see[13]? The activation energy is a well-establishedconcept, it has a constant value, it is a characteristicproperty within the rate limiting step of the particularchemical change being considered. It is widely andconventionally accepted as perhaps the most importantand successful concept that is applicable throughoutchemical kinetics[13]: being “. . . an energyE whichcan be related to the height of an energy barrier for thereaction.” Notwithstanding the well-founded creden-tials of this term, the convention in thermal analysisnow appears to be that any product (R×ln(temperaturecoefficient of reaction rate)) is to be regarded as an ‘ac-tivation energy’. However, there are (at least) two ob-vious disadvantages to this effective (and local) changeof terminology.

(i) It confers a spurious authority to those reportedempirical magnitudes ofE, which frequently haveno identifiable theoretical pedigree or chemicalsignificance.

(ii) It undermines and cheapens a term that, else-where throughout chemical kinetics, has ines-timable value. (Should we not use a differentterm for the quantities calculated by this method(i.e., R × ln(temperature coefficient of reactionrate)) to introduce realism into kinetic reportsand acknowledge the limitations and empiricismof the new quantity?)

The continued interests in promoting the use of thisterm (apparentE values are so easily (automatically,machine) calculated from thermoanalytical data) con-fers a veneer of respectability on a (modified) parame-ter that, to be realistic, at present lacks any theoreticalsupport and which introduces no concepts that mightbe capable of unifying the subject. The ‘variable’E isrelated to no activation process and is not identifiedwith a particular energy. However, it provides a mea-sure of the Apparent Exponential Dependence of RateOn Temperature and could, therefore, be more real-istically distinguished fromE by being referred to asthe ‘AEDROT’ Factor,G. No useful purpose is servedby including the gas constant in its calculation, so thatit can be determined as a recognized empirical termfrom any ‘rate constants’ by: lnk = ln F − G/T .

References

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eradicating erroneous arrhenius arithmetic

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