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APPROPRIATENESS OF ARRHENIUS EQUATION FOR KINETIC ANALYSIS OF SOLID STATE REACTIONS B. Viswanathan National Centre for Catalysis Research Indian Institute of Technology, Madras, Chennai 600 036 [Presentation on 5 th February, 2008 at IGCAR].

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APPROPRIATENESS OF ARRHENIUS EQUATION FOR KINETIC ANALYSIS OF SOLID STATE REACTIONSB. ViswanathanNational Centre for Catalysis ResearchIndian Institute of Technology, Madras, Chennai 600 036[Presentation on 5th February, 2008 at IGCAR]



Thermal methods are employed for the studyDecompositionoxidationreductionsolid state reactions and so on forMaterials of qualityThermal stability andChemical processes /transformations.



It is appropriate to quote Flynn at this stage. He said “ the unfortunate fact is that, since in thermal analysis, properties of the system are measured as a function of (both) time and temperature, all thermo-analytical results are potentially kinetic data, and many people ill grounded on kinetics ( like the present author) feel obliged to perform a kinetic analysis of them”


kinetic triplet
Kinetic triplet
  • Activation energy
  • Pre-exponential factor
  • g(α) or f (α)
  • Kinetic data derived from thermal methods are always considered with some skepticism – why is this so?
  • The kinetic triplets do not have a physical meaning but can help in predicting the rate of the processes for conditions when the collection of experimental data is impossible.
  • The kinetic parameters do not have a physical meaning and can be used to help in elucidating the solid state reaction mechanisms
  • Truth is in between.


kinetic data from thermal methods
Kinetic data from thermal methods
  • Ambiguity inescapably accompanies interpretation of kinetic data obtained in thermal methods.
  • May be short comings from computation methods or experimental shortcomings.
  • Experiments are often done either isothermal or under iso-conversions or under suitable heating rates all these are unable to provide the details of all that take place under thermal methods.
  • The reactions are not normally not simple stoichiometric like dehydration decomposition – a single set of kinetic triplet can describe a simple reaction at the most or if the mechanism is independent of temperature and the progress of the reaction.
  • Finally non-isothermal kinetics is obliged to give the same results as isothermal kinetics. There are enough support for both for and against.



Table 2. Types of information obtainable from TPx techniques.

TPD, Temperature-programmed desorption

· Characterization of adsorptive properties of materials

· Characterization of surface acidity

· Temperature range of adsorbate release, temperatures of rate maxima

· Total desorbed amount, adsorption capacity, metal surface area and dispersion

· Surface energetic heterogeneity, binding states and energies of adsorbed molecules

· Mechanism and kinetics of adsorption and desorption

TPR, Temperature-programmed reduction

· Characterization of redox properties of materials, .fingerprint. of sample

· Temperature range of consumption of reducing agent, temperatures of rate maxima

· Total consumption of reducing agent, valence states of metal atoms in zeolites and metal oxides

· Interaction between metal oxide and support

· Indication of alloy formation in bimetallic catalysts

· Mechanism and kinetics of reduction

TPO, Temperature-programmed oxidation

· Characterization of redox properties of metaand metal oxides

· Characterization of coke species in deactivated catalysts

· Total coke content in deactivated catalysts

· Mechanism and kinetics of oxidation reactions



Kinetic analysis of data generated by thermal methodsVarious options(i) Model based(ii) parameter based(iii) geometry based(iv) experimental variables basedand so on………..



Why do we question the kinetic analysis of thermal methods?

In contrast to the conventional homogeneous kinetics, there is no conception of a simple reaction in the solid-state reaction kinetics. The geometric-probabilistic phenomenology currently in use is not adequate for describing the interplay between the chemical mechanism and the observed kinetic behaviour. Hence it is necessary to question the use of the mode and formulations employed for homogeneous reactions when they are applied to solid state reactions.


arrhenius equation why
Arrhenius Equation why?
  • Since temperature is the main variable and temperature and time are related by the heating rate these data are suited for kinetic analysis.
  • The basic equation in kinetics relating temperature and rate constant is the Arrhenius equation and hence we wish to seek in this presentation the appropriateness of employing this equation for the analysis of kinetic data from data obtained from thermal methods.



The applicability of Arrhenius equation for homogeneous molecular level reactions is well known and has been established beyond doubt since these systems obey the Maxwell-Boltzmann distribution. However the energy distributions in solids are not the same



However alternate functions like relating ln k with T or relating ln k with ln T in addition to ln k versus 1/T have been proposed but these equations are “theoretically sterile” since the constants of the se proposed equations do not lead to any deeper understanding of the steps of chemical reaction.



The kinetic reaction mechanism can be determined from the Arrhenius equation

K = A exp ( - Ea/RT)

Ea is the activation energy and R is the universal gas constant, A is the pre-exponential factor; T is the absolute temperature and k is the reaction rate constant.

The above equation upon log transformation can be rewritten as

Ln k = ln A - Ea/RT

The activation energy can be determined from the slope of the plot of ln k verses 1/T. and the intercept would yield the value of pre-exponential factor.


two constants of arrhenius equation

The use of Arrhenius equation gives rise to values for two important parameters namely the value of activation energy (Ea) and the pre-exponential factor (k0). Both have great significance in treating kinetic data of chemical reactions in general. They also relate to two basic thermodynamic parameters namely ΔH and ΔS but for the formation of the activated species



In the terminology of homogeneous kinetics, the activation energy Ea is usually identified as the energy barrier ( or threshold) that must be surmounted to enable the occurrence of the bond redistribution steps required to convert the reactants into products. The pre-exponential term A provides a measure of the frequency of occurrence of the reaction situation usually envisaged as incorporating the vibration frequency in the reaction coordinate.



This normally implies that the fractional reaction g (α) = kt is linear.

Polanyi-Wigner model probably assumes such a relationship for solid state decompositions as well

The equation (dx/dt) = (2νE/RT)x0 exp (-E/RT)

This equation is based on for linear rate of advance of the interface. This has been modified for geometry in deriving rate equations.

The vibration frequency ν is expected to be or the order of 1013 s-1

Experimental or derived values of A have been compared with this expected value.

Reactions are normal when A is one or two orders less than 10 13 and E is in the range of enthalpy of dissociation.

Reactions are abnormal when both A and E are different from the critical values.



The thermodynamics of the activation process ((ΔS ± and ΔH±) for an irreversible reaction is given by Eyring type of equation

The equation

(dx/dt) = (k B T/h) xo exp (ΔS ±/ R) exp (-ΔH±/RT)

The so called normal values of ν is calculated on the assumption that the entropy of activation (ΔS± ) is zero. For abnormal reactions the value of (ΔS± ) is of the order or 200 to 500 JK -1mol -1. It is possible to derive the transition state terminology using the concept of partition functions Q ± and Q. Values of Q ±/Q can be less than or equal to or greater than 1. However mechanistic differences are only postulates. The introduction of T dependence on A also has not improved the situation. Concept of discrete activated state itself is not clear. Immobility of species in solid state different from homogeneous reactions. Multi-step process electronic or phonon activation, thermal gradient and other parameters.



Fig The frequency of occurrence of Arrhenius pre-exponential values, expressed as lgA/s-1) in steps of equal increment. Note that there is no preference towards the normal expected value of A = Approx 10 13 s-1

Data from A.K.Galwey, Thermochimica Acta, 242 (1994) 259-264



Table 2. Kinetic parameters of thermal decomposition for explosives. The relevant equation is

          ln (β/Tp2) = ln (AR/T) –Ea/RTp

pentaerythritol tetranitrate (PETN),

RDX, hexanitrostilbene or 2,2′,4,4′,6,6′-hexanitrostilbene (HNS) and

HMX are investigated



Table experimental data of alpha –temperature curves for the decomposition of calcium carbonate in vaccum at different heating rates, 18,2,5,3,5,5,0,6.2,10.0 K min -1 [Data reproduced from Thermochimical Acta 355(2000)125-143



Table experimental data of alpha –time isothermal data for the decomposition of calcium carbonate in vacuum at different temperatures, 550,540,530,520,515) [Data reproduced from Thermochimical Acta 355(2000)125-143



Table experimental data of alpha –temperature curves for the decomposition of calcium carbonate in vaccum at different heating rates, 18,2,5,3,5,5,0,6.2,10.0 K min-1 [Data reproduced from Thermochimical Acta 355(2000)125-143



Table experimental data of alpha –time isothermal data for the decomposition of calcium carbonate in nitrogen at different temperatures, 773,750,740,732,719,710 and 700 0 C) [Data reproduced from Thermochimical Acta 355(2000)125-143



Graphical presentation of the Arrhenius parameters for the decomposition of CaCO3 in nitrogen [data from Thermochimica Acta 355(2000) 145-154.]



Graphical presentation of the Arrhenius parameters for the decomposition of CaCO3 in vaccum [data from thermochimica Acta 355(2000) 145-154.]



Multi step nature of solid State ReactionsInterface advance or proportional to surface area – deviation due to local reactivity local decrease of E or strain or formation of defect the interface R/P or R/inert gas or vacuum or topo-tactic reactions or R/I/P intervening phase of melt or R/R (mod)/I/PIn this context it is possible that Polanyi-Wigner treatment is applicable averaging effectEnergy in te form phonons ( vibrations internal or other wise) may be similar to the homogeneous phase reactions.



Fig. Suggested Band structure across the reaction zone

(Reproduced from A.K.Galwey abd B.E.Brown Thermochimical Acta 386(2002)91-98.

Azide excitation and activation energy arecomparable



Energy DistributionsFermi Dirac distribution when Ee is far exceeds Ef then it reduces to a Maxwell-Boltzmann distribution.If phonon energy obeys Bose Einstein statistics even then at higher energies that is hω »2πkBT then the expression is similar to M –B distribution fucntion.



Distribution of Activation Energies1.Compensation between ln(A) being linear function of E2.A single common value of A and a continuous distribution of activation energies3.A single value of E and a continuous distribution of Pre-exponential factors.


kinetics under different experimental conditions some cautions
Main reasons for studying the kinetics of thermal decompositions.

Product quality

Thermal stability

Chemical processes in the transformation

Kinetic triplet-isothermal, poly thermal atmosphere

Activation parameters should be stated with specifying the kinetic triplet. At least like under vacuum etc.

Single heating rate data should be avoided

Model filling multi heating rate data under the triplet

Iso-conversional methods (ozawa, Friedmann) are sensitive to experimental noise.

Kinetics under different experimental conditions – some cautions


some questions but not complete list
Some questions but not complete list
  • What is the significance of the values of A when they differ from the conventional expected value of 1013 s-1?
  • The value of the activation energy is not in the range of conventional bond breaking and bond formation. If the value were to differ considerably what does it signify?
  • When the values of A is far different very low or very high how are they to be accounted?



The mechanism of reaction in solid state may be different from that occurring in homogeneous systems, where movement and collision are envisaged as initiating steps of the reaction. In solids, the species are immobilized and hence this type of collisions may not be the initiator of the reaction.

The reactions ( at least some of the reactions in solid state) may be due to bond activation through electronic energy or through phonon activation.



If softening and melting were to precede the solid state reaction, then one can visualize the reaction sequence as in homogeneous medium.

The variation of E with α the extent of reaction, (smooth or abrupt) denotes the nature of consecutive steps involved in solid state reactions.

If the solid state reaction proceeds by the development and growth of a reaction interface, then the local strain, imperfections, the crystalline phases of the reactants and products all will contribute to the acceleration or deceleration of reaction rates as well as for the change of Ea with α. It is not yet clear how the various forms of the intervening phase, like a molten product, a defect crystalline phase or reorganizations in react ( like removal of water) could provide a chemical environment where the conventional Polanyi-Wigner treatment will be as much applicable as in homogeneous phase.



In terms of energy transfer, the species in the interracial zone is more ordered in homogeneous liquid medium, but less ordered as compared to the fully crystalline phase. The in-between crystalline phases (interface region) may provide additional allowed energy levein the forbidden regions of energy bands in the solid. The electronic energy levethough normally follow Fermi-Dirac statistics, can approximate to Maxwell-Boltzmann distribution under the temperature conditions employed for the solid state reaction. If this situation prevaithen one can justify the use of Arrhenius equation for evaluating Ea and pre-exponential factor.

If on the other hand phonons are the mode of activation, then one can expect the Bose-Einstein statistics will approximate to Maxwell-Boltzmann distribution for the conditions prevailing under reaction conditions and hence the use of Arrhenius equation can still be justified.

Though the summary of the arguments given justify the use of Arrhenius type of relationship to evaluate the kinetic parameters ( Ea and ln A) it has not provided any explanation for the variation of Ea and ln A. These variations are considered in terms of compensation between Ea and ln A, or by the variation of one ( Ea o



The linear relationship between Ea and ln A is termed as compensation effect in the literature. The observance of this effect is usually identified by the inherent ‘heterogeneity’ of the surface and hence the changes in reactivity of these sites. However, a simple linear variation between Ea and ln A for sites of varying reactivity is not expected unless one has to invoke additional internal reorganizations which can give rise to smooth variation in both Ea and ln A.

It must be remarked that these solid state reactions were to involve an intermediate vapourization and condensation steps, then it is probable that one can still invoke Arrhenius type of equation for evaluating Ea and ln A. However not all solid state reactions do proceed by evaporation of reactant and condensation of the product.



It is appropriate to re-quote Flynn at this stage. He said “ the unfortunate fact is that, since in thermal analysis, properties of the system are measured as a function of (both) time and temperature, all thermo-analytical results are potentially kinetic data, and many people ill grounded on kinetics ( like the present author) feel obliged to perform a kinetic analysis of them”



Even if one were to admit inexperience in treating kinetic data from thermal analysis, the physical significance of the kinetic parameters derived from the analysis is not clear yet. Secondly, it is yet to resolved, why the same experimental data can be fitted to various mode simultaneously? Do they reflect on the closeness of the models, or the inadequacy of the treatment of data based on the mode chosen. The widely practiced method of extracting Arrhenius parameters from thermal analysis experiments involves ‘force fitting’ of experimental data to simple reaction order kinetic models. The ‘force fit’ may not be suitable for the analysis of data of thermal analysis, outside the applicable range of variables and hence can be of limited utility for drawing mechanistic detail of the reaction. It should be remarked that the concept of kinetic order of the reaction has to assume a new significance in the case of solid state reactions.

Even for simple decomposition reactions, the available mode cannot appropriately take into account like sintering before decomposition, simultaneous existence of polymorphic transition.

It should be recognized that the mode so far proposed, are oversimplified and envisages one nucleation site per particle. There can be multiple and different types of nucleation simultaneously and their growth may be a complex function which cannot be treated in terms of simple geometrical considerations.

The direct observation of the texture and morphology of the substances have to be coupled with the kinetic fit of the data for developing a model and draw meaningful deductions regarding nucleation and growth.



The values of the activation energy for solid state reactions can be rationalized only in some simple restricted conditions. For example for two systems if g(α) and A were to be equal then, the magnitude of Ea can be used to postulate on the reaction kinetics. However, even in these cases , if Ea were to vary with α , the fraction of the reaction, then it denotes changes in reactivity as a result of extent of reaction and the complex nature of the reaction.

There is overwhelming tendency to compare Ea values obtained for isothermal and non-isothermal experimental conditions. Even though time and temperature are mathematically related by the heating rate (β = dT/dt), it is not clear how the species of the system will respond to the bimodal variation. Whether the changes observed will be an arithmetic sum or product of the variations observed with each of these variables. It appears that it is neither of these two mathematical functions.

In conclusion, one can state that if carefully used and complemented with other techniques, the analysis of solid state kinetics can provide indications on the reaction mechanism and may yield information on reactivity which can be exploited for synthetic strategies.



In essence, even though the thermal data is normally kinetic in nature, the interpretation of the data within the premises of kinetics requires caution. Discussing reactions on the basis of one of the kinetic triplet is not advisable. Even when the kinetic triplet is employed it is necessary to define them in the context of the experimentation.