Kinetic analysis of solid-state reactions: Precision of the activation energy calculated by integral methods

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1 Kinetic nlysis of solid-stte rections: Precision of the ctivtion energy clculted by integrl methods L.A. PÉREZ-MAQUEDA *, P.E. SÁNCHEZ-JIMÉNEZ AND J.M. CRIADO Instituto de Cienci de Mteriles de Sevill. C.S.I.C.-Universidd de Sevill. Avd. Americo Vespucio Sevill. Spin Abstrct The integrl methods re extensively used for the kinetic nlysis of solid-stte rections. The Arrhenius integrl function [p(x)] does not hve n exct nlyticl solution. Thus, different pproches, ccomplishing the condition tht ln g(α) is liner function of either 1/T or predetermined function of T, hve been proposed for this integrl to determine the ctivtion energy from liner plot of the logrithm of g(α) versus some function of T. The first pproch ws proposed by Vn Krevelen nd fter tht, number of uthors developed new pproches, very often with the scope of incresing the precision of the Arrhenius integrl s checked from the stndrd devition of the p(x ) function determined from these pproximtion with regrds to the true vlue of the p(x) function. Besides this method, those proposed by Doyle, Horowitz nd Metzger, Cots nd Redfern, McCllum nd Tnner nd Gyuly nd Greenhow re very populr for determining ctivtion energies. In fct, we hve found more thn 4500 cittions (1300 in the lst five yers) for the ppers were these methods were proposed. However, systemtic nlysis of the errors involved in the determintion of the ctivtion energy from these methods is still missing. A comprtive study of the precision of the ctivtion energy s function of x nd T computed from the different integrl methods hs been crried out. Keywords: Arrhenius integrl, integrl methods, solid stte rections, errors in ctivtion energy. 1

2 1. INTRODUCTION Thermlly stimulted solid-stte rections, such s decompositions, solid-solid rections, crystlliztions, etc, re, in generl, heterogeneous processes. The rection rte of such processes cn be kineticlly described, when it tkes plce under conditions fr from equilibrium, by the following expression: 1 d f ( T ) f ( ) (1) dt where t is the time nd α is the extent of rection rnging from 0 before the process strts to 1 when it is over. Thus, the left hnd side term in eq. (1) is the rection rte. The right hnd side term in eq. (1) consists of two terms, i.e. f(t) nd f(α), being f(t) function tht describes the dependence of the rection rte with the temperture (T). Usully, this dependence is described by the Arrhenius eqution: f ( T ) E / RT Ae (2), being A the preexponentil fctor of Arrhenius, E the ctivtion energy nd R the gs constnt. Additionlly, f(α) is term tht describes the dependence of the rection rte with the mechnism of the process. Different functions hve been proposed in literture for describing the kinetic mechnism of the solid-stte rections. These mechnisms re proposed considering different geometricl ssumptions for the shpe of the mteril prticles (sphericl, cylindricl, plnr) nd driving forces (interfce growth, diffusion, nucletion nd growth of nuclei). Some of the most common equtions proposed for these rections re included in Tble 1. The most common heting profile used for studying solid-stte rection is the liner heting progrm. Under these experimentl conditions, T chnges in wide rnge of vlues nd entire α-t curve is recorded in single experiment. For liner heting rte conditions eq. (1) cn be written 2

3 d A E / RT e f ( ) (3) dt being β the heting rte. Mny of the experimentl methods used to perform kinetic nlysis of solid-stte rections re bsed in the mesurement of the evolution of n integrl mgnitude, i.e. proportionl to the extent of rection, such s mss loss, relesed gs, mount of contrction, s function of temperture. To perform the evlution of such experimentl dt, it is necessry either to numericlly differentite the experimentl dt or to integrte eq. (3): x AE e ( ) R 2 x x AE g dx p( x) R (4), being x=e/rt. This expression cn be written in the logrithmic form: AE ln( g( )) ln ln p( x) (5), R Under liner heting rte progrm, eqs. 4 nd 5 do not hve n exct nlyticl solution to p(x) nd, therefore, the solution cnnot be expressed in closed form. 2 Although, other T-t profiles, such s prbolic or hyperbolic progrms, yield to nlyticl solutions to the Arrhenius integrl, they re very seldom used. Thus, severl pproximted equtions hve been proposed for p(x) under liner heting progrm. The pproximtions for p(x) most commonly used in the determintion of the ctivtion energy re those proposed by Cots nd Redfern, 3,4 Doyle, 5-7 Horowitz nd Metzger, 8 McCllum nd Tnner, 9,10 Gyuli nd Greenhow, 11,12 nd Vn Krevelen. 13 All these pproximtions hve been obtined either by simplifictions of the series expressions or in n empiricl wy. For given kinetic model, the resulting equtions led to liner correltion where the kinetic ctivtion energy is esy obtined from the slope. The number of publictions where these integrl methods re used for 3

4 determining ctivtion energies is vst. Thus, bout 4500 cittions cn be found in the literture for the originl ppers 3-13 where these equtions re proposed. Besides, the populrity of these integrl methods hs not decresed, s indicted by their more thn 1300 cittions just in the lst yers, i.e In these lst five yers, the pproch with more cittions hs been tht of Cots nd Redfern 3,4 with bout 590 cittions, followed by those of Horowitz nd Metzner 8 nd Doyle 5-7 with 230 nd 102 cittions, respectively (informtion on the number of cittions hve been obtined from ISI Web of Science dt bse). Nevertheless, independently of the pproximtion used, every g() leds to high liner correltion coefficient nd, therefore, it is not possible to discriminte the kinetic model from single experimentl curve. Additionlly, the resulting ctivtion energy vlues re very much dependnt on the g() function ssumed for the nlysis (these limittions re extended not only to integrl methods but lso to ny procedure tht uses single liner heting rte curve 14,15 ). Thus, in principle, the integrl methods should be only used under the two following circumstnces: (i) when the kinetic model is lredy known for obtining the ctivtion energy or (ii) when the ctivtion energy is known for determining the kinetic model. Nevertheless, new question rises bout the precision of the ctivtion energy vlues determined by these populr integrl methods becuse, s mentioned bove, they re bsed in pproximtions to the p(x) function nd their precision for the estimtion of the kinetic prmeters re still in doubt, thus some uthors hve climed tht these methods re imprecise Some studies hve estimted the errors in the pproximted p(x) functions by compring the resulting vlues with those clculted by numericl integrtion, concluding tht the errors re quite lrge. Fig. 1 shows s wy of exmple the evolution of the reltive error of the Cots nd Redfern pproximtion for the estimtion of the p(x) function versus x. This figure indictes tht the error decreses 4

5 with x, being significntly lrge for vlues of x commonly found in literture for solidstte rections. These findings hve been used s n rgument for invlidting these pproximted equtions in the estimtion of the kinetic prmeters. Nevertheless, the im of the forementioned pproximtions is the determintion of the ctivtion energy nd not the ccurte computtion of p(x). Tking into ccount tht the integrl methods re so widely extended nd tht there is some controversy in their precision, it would be of interest to estimte the precision of such methods for the determintion of the ctivtion energy. The im of the present pper is to perform comprtive study of the precision of the most extensively used pproximtions to p(x) in the determintion of the ctivtion energy. 2. ERRORS IN THE ACTIVATION ENERGY 2.1. Cots nd Redfern method. The Cots nd Redfern 3,4 pproch to the Arrhenius integrl is the following: x e 2 p( x ) 1 (6) x x the subscript stnds for pproximted. In generl, the expression more commonly used is the simplified form: p( x x e ) (7) x 2 This pproch is nmed sometimes in literture s Fisher pproch. 20 By introducing eq. (7) into eq. (4), it follows g ( A E 2 E / RT ) T e (8) R By tking nturl logrithms, eq. (8) results 5

6 2 A RT E ln( g( )) ln (9) E RT Thus, the ctivtion energy could be esily obtined from the slope of the line resulting of plotting ln(g())-2ln(t) versus 1/T. The reltive error ε of the ctivtion energy (E ) clculted by the Cots nd Redfern eqution cn be defined by the following eqution: E E E % 100 R (10) E E R By differentiting eq. (9): ln g( ) E 2T 1/ T R (11) nd by differentiting eq. (5): lng( ) ln( p( x)) E ln( p( x )) 1/ T 1/ T R x (12) Thus, from eqs (11) nd (12), it follows E p x E R ln( ( )) 2 (13) x x R tht substituting in eq. (10) leds to ln( p( x)) 2 % (14) x x This eqution indictes tht the vlues of ε% depend on x=(e/rt), nd, therefore, on the vlue of the ctivtion energy nd of the rnge of temperture of the process. The vlues of ε% hve been computed by mens of the Mthcd softwre by numericl integrtion of the p(x) function using tolernce (precision in the clculus) of The resulting ε% vlues s function of the prmeter x re included in Tble 2. The vlues 6

7 included in Tble 2 illustrte tht there is significnt influence of x in the precision of the clculted ctivtion energy vlues. Thus, ε% rnges from lmost -20% for x=2 to less thn -1% for x vlues lrger thn 20; in the limit, for x=, the error is cero. 2.2 Doyle method. The Doyle pproch to the Arrhenius integrl is the following: 5-7 log( p( x )) x (15) From eq. (15) nd eq. (5), it follows A E E log( g( )) log (16) R RT Thus, the ctivtion energy cn be obtined from the slope of the line resulting from plotting the left hnd side of eq. (16) s function of 1/T: log( g( )) 1/ T E R (17) The reltive error ε% (eq. (10)) for the ctivtion energy obtined by the Doyle method cn be obtined from eqs. (12) nd (17): ln( p( x)) % 1100 (18) x The vlues of ε% hve been computed by the sme procedure s described in the ltter section nd the resulting error vlues re included in Tble Horowitz nd Metzger method. The integer eqution fter ssuming the Horowitz nd Metzger pproch 8 to the p(x) function is the following: 7

8 A E E ln( g( )) ln 5.33 (19) R T RT s 2 s where is chrcteristic temperture such tht =T-T s, being T s n rbitrry reference temperture. From eq. (19), it is cler tht the ctivtion energy is obtined from the slope of the line resulting of plotting the left hnd side of eq. (19) versus, or versus T tht yields the sme slope: ln( g( )) T E RT 2 s (20) From eqs. (12) nd (20), the reltive error ε% (eq. (10)) in the ctivtion energy obtined by the Horowitz nd Metzger 8 results: ln( p( x)) % 1100 (21) dx Tble 2 includes the errors estimted by eq. (21) for the ctivtion energy clculted by the Horowitz nd Metzger pproch Vn Krevelen method Considering the Vn Krevelen et l pproximtion 13 to the exponentil integrl of Arrhenius, eq. (5) hs the logrithmic form: E A RT mx E ln( g( )) ln T 1 mx lnt E T (22), mx R( Tmx 1) where T mx is the temperture t the mximum thermogrvimetric rte. The ctivtion energy is determined from the slope of the line resulting from the plot of ln(g(α)) s function of lnt: ln( g( )) lnt E R( T mx 1) (23) 8

9 Thus, the Vn Krevelen et l method, 13 even though it is integer eqution, for the determintion of the ctivtion energy, it requires of the differentil experimentl curve to obtin the T mx vlue for eq. (23). As in the previous sections, the reltive error cn be clculted from eq. (12) nd (23), resulting: ln( p( x)) 1 % (24) x x The resulting vlues for the error re included in Tble McCllum nd Tnner method The deciml logrithmic form of the integer eqution (eq. (5)) using the pproch proposed by McCllum nd Tnner 9,10 to the p(x) function results: log( g( A E E )) log E R T (25) Thus, the ctivtion energy cn be clculted from the slope of the line resulting from the plot of log(g(α)) s function of 1/T: log( g( )) (1/ T ) E (26) The reltive error ε% (eq. (10)) of the ctivtion energy cn be clculted from eqs. (12) y (26): ln( p( x)) 449 % 1100 (27) 217R x 217xRT In this cse the error depends both on x nd T. Tble 3 includes the errors in the ctivtion energy s estimted by mens of eq. (27). 9

10 2.5. Gyuli nd Greenhow method The deciml logrithmic form of the integer eqution eq. for the pproch of Gyuli nd Greenhow 11,12 cn be written s follows:.9583 A E R E log( g( )) log log log E (28) R E T The ctivtion energy cn be clculted from the slope of the line resulting from the plot of log(g(α)) s function of 1/T: log( g( )) (1/ T ) E (29) As in the previous sections, the reltive error cn be clculted from eq. (12) nd (29), resulting: 1/ ln( ( )) % xt p x (30) xrt x This expression hs some similrities with tht for the errors in the ctivtion energy of McCllum y Tnner (eq. (24)). Thus, here the ε% lso depends on x nd T (Tble 4). 3. CHECKING OF THE COMPUTED ERRORS WITH THEORETICAL AND EXPERIMENTAL CURVES. To check the vlidity of the errors clculted in the previous section, set of experimentl curves hve been simulted nd nlyzed by the integrl methods using the different pproches nlyzed here. Two different curves hve been simulted ssuming two different kinetic models, kinetic prmeters, nd liner heting rte conditions. The simulted curves hve been computed by solving the system of two differentil equtions constituted by eq. (1) nd T / t by mens of the Runge- Kutt method using the mthcd softwre nd tolernce (precision in the clculus) of 10

11 10-5. The first curve (Fig. 2) hs been simulted for β= 10 K min -1, n A2 kinetic model nd the following kinetic prmeters: E=35 kj mol -1 nd A=10 min -1. The second curve (Fig. 2b) hs been computed for β= 1 K min -1, n F1 kinetic model nd the following kinetic prmeters: E=100 kj mol -1 nd A=10 8 min -1. The verge vlues of x for these two curves re 5 nd 20 for the first (Fig. 2) nd second (Fig. 2b) curves, respectively. These curves hve been nlyzed by mens of the integer method using the different pproches nlyzed bove. The resulting vlues of ctivtion energy nd the corresponding errors re given in Tble 5. The errors included in Tble 5 for the different pproches re consistent with those reported in Tbles 2-4. The smll differences between the errors in Tble 5 nd Tbles 2-4 re due to the fct tht the errors in Tbles 2-4 hve been clculted t constnt vlues of x, while those reported in Tble 5 do not correspond to single x vlue but to rnge of x vlues becuse the temperture vries in α-t curve while the ctivtion energy is constnt. For checking the precision of the method with experimentl dt, thermogrvimetric (TG) curve ws recorded for the decomposition of BCO 3 under high vcuum conditions. This is very stble compound with low equilibrium pressure ( torr) in the temperture rnge (1000 K) t which the rection tkes plce. Therefore, the decomposition conditions should be properly controlled for mintining the prtil pressure of CO 2 fr wy from the equilibrium pressure. 21 Such condition ws fulfilled by using smll mount of smple (10 mg), low heting rte (0.2 K min -1 ), nd performing the experiment under high vcuum (the blnce ws connected to vcuum device tht reduced the totl pressure to torr). Additionlly, during the TG experiment, the prtil pressure of CO 2 ws recorded with qudrupole-mssspectrometer to mke sure tht CO 2 prtil pressure ws fr wy from equilibrium pressure. In fct, the CO 2 pressure did not increse during the therml decomposition 11

12 bove torr. The thermogrvimetric curve obtined under the conditions previously described is included in Fig. 3. Tble 6 includes the ctivtion energies nd correltion fctors resulting from the nlysis of the experimentl dt (Fig. 3) by mens of the integer methods nlyzed here. It is worth noting tht the correltion fctors re very high for ll the pproximted equtions. Additionlly, the experimentl integrl curve ws numericlly differentited with the Microcl Origin softwre nd nlyzed the logrithmic form of the eqution resulting from eqs. (1) nd (2): 14,15,22,23 d ln ln( f ( )) ln( A) dt E RT (26) The ctivtion energy cn be directly obtined from the slope of the line resulting from the plot of the left hnd side of eq. (26) s function of 1/T. The resulting ctivtion energy nd correltion fctor hs been lso included in Tble 6. Using the ctivtion energy clculted from the differentil eqution (eq. 26) s ccurte vlue (becuse no pproximtion is involved in the method), we hve clculted the errors of the ctivtion energies determined by the different integer methods (Tble 6). The resulting errors re in the rnge of the expected ones (Tbles 2-4) for the vlue of x=25 corresponding to this experiment, the smll devitions observed re due to experimentl errors nd to the fct tht x, s lso mentioned before, is not constnt during the entire process. 4. SUMMARY In this pper we hve clculted the errors in the ctivtion energy determined by the extensively used integrl methods. These methods re bsed in pproximted equtions of the Arrhenius integrl tht led to liner reltion between the logrithmic form of g() nd function of the temperture from whose slope is determined the ctivtion energy. The vlues of ctivtion energy obtined by the integrl method re 12

13 subjected to some impressions becuse these methods re bsed on pproximtions. It is worth to note tht, s mentioned bove, every kinetic model led good liner correltion nd, therefore, from single liner heting rte progrm curve it is not possible to determine the ctivtion energy of the process unless the kinetic model is known. The scope of the pper is the quntifiction of the errors in the ctivtion energy determined by integrl methods when the kinetic model is known. The quntifiction of those errors hs not been directly relted to the precision of the pproximted p(x) function for evluting the temperture integrl, becuse the ppliction of these proposed pproximtions is the determintion of the ctivtion energy nd not the clcultion of the temperture integrl. The error nlyses hve shown tht for ll the pproches nlyzed here, the reltive errors very much depend on x; tht is, on E nd on the verge temperture of the process. Additionlly, this verge temperture of the process depends on the vlue of E, A nd the kinetic model followed by the rection. Thus, the error of the ctivtion energy clculted by the integrl method is influenced by the kinetic prmeters of the process. In generl, smll vlues of x due to smll vlues of E nd/or high vlues of temperture yield reltively high errors nd the integrl methods re not pproprite. On the other hnd, for high vlues of x, the errors re quite smll. It is worth noting tht the pproch tht leds to the minimum error in the ctivtion energy is tht of Cots nd Redfern. Finlly, the clculted error vlues hve been checked with those obtined of the nlysis of simulted nd experimentl curves showing n excellent greement. 13

14 REFERENCES (1) Crido, J. M.; Pérez Mqued, L. A., In Smple Controlled Therml Anlysis: Origin, Gols, Multiple Forms, Applictions nd Future; Sorensen, O. T., Rouquerol, J., Eds.; Kluwer: Dordrecht, 2003; Vol. 3, p 55. (2) Pérez-Mqued, L. A.; Crido, J. M. J. Therm. Anl. Clorim. 2000, 60, 909. (3) Cots, A. W.; Redfern, J. P. Nture 1964, 201, 68. (4) Cots, A. W.; Redfern, J. P. Journl of Polymer Science Prt B-Polymer Letters 1965, 3, 917. (5) Doyle, C. D. Anlyticl Chemistry 1961, 33, 77. (6) Doyle, C. D. Journl of Applied Polymer Science 1961, 5, 285. (7) Doyle, C. D. Nture 1965, 207, 290. (8) Horowitz, H. H.; Metzger, G. Anl. Chem. 1963, 35, (9) McCllum, J. R.; Tnner, J. Europen Polymer Journl 1970, 6, (10) McCllum, J. R.; Tnner, J. Europen Polymer Journl 1970, 6, 907. (11) Gyuli, G.; Greenhow, E. J. Thermochimic Act 1973, 6, 239. (12) Gyuli, G.; Greenhow, E. J. Journl of Therml Anlysis 1974, 6, 279. (13) vn Krevelen, D. W.; vn Heerden, C.; Huntjens, F. J. Fuel 1951, 30, 253. (14) Perez-Mqued, L. A.; Crido, J. M.; Gotor, F. J.; Mlek, J. Journl of Physicl Chemistry A 2002, 106, (15) Perez-Mqued, L. A.; Crido, J. M.; Mlek, J. Journl of Non-Crystlline Solids 2003, 320, 84. (16) Hel, G. R. Thermochimic Act 1999, 341, 69. (17) Hel, G. R. Instrumenttion Science & Technology 1999, 27,

15 (18) Flynn, J. H. Thermochimic Act 1997, 300, 83. (19) Flynn, J. H. Thermochimic Act 1992, 203, 519. (20) Fisher, P. E.; Chon, S. J.; S.S., G. Ind. Eng. Chem. Res. 1987, 26, (21) Pérez-Mqued, L. A.; Crido, J. M.; Gotor, F. I. Interntionl Journl of Chemicl Kinetics 2002, 34, 184. (22) Friedmn, H. L. J Polym Sci C 1965, 6, 183. (23) Crido, J. M.; Perez-Mqued, L. A.; Gotor, F. J.; Mlek, J.; Kog, N. Journl of Therml Anlysis nd Clorimetry 2003, 72,

16 TABLE 1. f() nd g() kinetic functions Mechnism Symbol f() g() Phse boundry controlled rection (contrcting re) R2 1 2 ) 21 1 ( 1 ( ) 12 Phse boundry controlled rection (contrcting volume) R3 ( 1 ) ( 1 ) 13 Rndom nucletion followed by n instntneous growth of nuclei. (Avrmi-Erofeev eqn. n =1) 1 F1 ( 1 ) ln( 1) Rndom nucletion nd growth of nuclei through different nucletion nd nucleus growth models. (Avrmi- Erofeev eqn.) An 11 n ln(1 ) n(1 ) ln( 1) 1/ n Two-dimensionl diffusion D2 1 ln( 1 ) ( 1)ln( 1) Three-dimensionl diffusion 3(1 ) (Jnder eqution) D3 1/ 3 Three-dimensionl diffusion (Ginstling-Brounshtein eqution) D / ( 1 ) / / ( 1) This eqution represents n Avrmi-Erofeev kinetic model with n=1 insted of first order rection. The symbol A1 would be more proper. 16

17 TABLE 2. Vlues of the reltive error (ε%) for the ctivtion energy clculted by mens of the Cots nd Redfern, Doyle, Horowitz nd Metzger, nd Vn Krevelen et l. equtions s function of the prmeter x (E/RT). x Cots nd Redfern Doyle Horowitz nd Metzger Vn Krevelen TABLE 3. Vlues of the reltive error (ε%) for the ctivtion energy clculted by mens of the McCllum nd Tnner eqution s function of the prmeter x (E/RT) nd the temperture (T). x TABLE 4. Vlues of the reltive error (ε%) for the ctivtion energy clculted by mens of the Gyuli nd Greenhow eqution s function of the prmeter x (E/RT) nd the temperture (T). x

18 TABLE 5. Vlues of the ctivtion energies (E ) nd errors (ε%) obtined of the nlysis of the simulted curves included in Figs 2 nd 2b by mens of the different integrl methods Simulted curve Fig. 2 (x5) * Simulted curve Fig. 2b (x20) * E (kj mol -1 ) ε% E (kj mol -1 ) ε% Cots nd Redfern Doyle Horowitz nd Metzger McCllum nd Tnner Gyuli nd Greenhow Vn Krevelen et l * The verge vlue of x hs been obtined from E/RT =0.5, where T =0.5 is the temperture corresponding to =0.5. TABLE 6. Vlues of the ctivtion energies (E ) nd errors (ε%) obtined of the nlysis of the experimentl results for the therml decomposition of BCO 3 obtined under high vcuum (Fig. 3) by mens of the different integrl methods. Method Correltion fctor E % * Cots nd Redfern Doyle McCllum nd Tnner Gyuli Greenhow Horowitz Metzger Vn Krevelen et l Friedmn * The errors of the ctivtion energies hve been clculted using the ctivtion energy determined by the Friedmn eqution s the ccurte vlue (becuse no pproximtion is involved in the method). 18

19 Figure Cptions Fig. 1. Evolution of the reltive error of the Cots nd Redfern pproch for the estimtion of the p(x) function versus the vlue of x. The reltive error hs been defined by the expression: (p (x)-p(x)/ p(x)) 100, being p (x) the vlue obtined by the Cots nd Redfern pproximtion nd p(x) the vlue obtined by numericl integrtion. Fig. 2. Simulted curves () β= 10 K min -1, n A2 kinetic model nd the following kinetic prmeters: E=35 kj mol -1 nd A=10 min -1 ; nd (b) β= 1 K min -1, n F1 kinetic model nd the following kinetic prmeters: E=100 kj mol -1 nd A=10 8 min -1 Fig. 3. Experimentl TG curve obtined for the BCO 3 under high vcuum t heting rte of 0.2 K min

20 b T / K Fig. 1 20

21 T / K Fig. 2 21