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Anal. Chem. 1989, 67,1136-1139
Arrhenius Parameters for Solid-state Reactions from Isothermal Rate-Time Curves Michael E. Brown* Department of Chemistry and Biochemistry, Rhodes University, Grahamstown, 6140 South Africa
Andrew K. Galwey Department of Pure and Applied Chemistry, Queen’s University, Belfast B T 9 5AG, Northern Ireland
Rate (da/dt)-time curves for a reaction may readily be obtained through use of isothermal differential scanning caiorimetry (DSC)(on the assumptlon that the evolution or absorption of heat may be used to measure the extent of reaction, a) or from isothermal derivative thermogravimetry (DTG). The maln models in solid-state kinetics give rise to two general classes of rate-time curves. The main features of the curves for the sigmoid group are the maximum rate, vmnx, at some time, t,,, (>O), while the curves for the deceleratory group have Y = Y,~, at t = 0. A secondary feature, analogous to the half-life, f ,12, in conventional kinetic analysis, is the tlme, fmI2, taken for the rate to drop to half its maximum value (Le. v,/2). These quantities, ,Y,, t,,, fmI2, and tlmI2, may readily be determined, where present, from the experimental data. It Is pointed out in this paper that these quantlties are often related directly to the rate coefficients for reaction, and hence the values measured at a set of different temperatures may be used in Arrhenlus-type plots to calculate values for the apparent activation energy of reaction, without knowledge of the particular kinetic model that applies.
INTRODUCTION Kinetic studies of solid-state reactions ( 1 ) are usually based on measurement of the extent of reaction, a, as a function of time at constant temperature. The experimentally measured a, t data are then examined to identify which, if any, of a selected group of kinetic models (Table I) best describes the observed behavior. The kinetic models most often applied to solid-state reactions are frequently and conveniently expressed in the form, f ( a ) = k t , and the experimental data are then used to determine the value of the rate coefficient, k , at the temperature of the isothermal experiment. This procedure is exactly the same as that used for conventional rate studies of homogeneous reactions, in that experiments are repeated at a series of different, but constant, temperatures and the Arrhenius parameters, E, and A , are determined from the temperature dependence of the rate coefficient. There is, of course, the alternative pathway, which has been and is still being extensively explored, in which kinetic parameters are derived from analysis of experiments in which cy is measured during the controlled variation of temperature-usually with a linear increase with time. There is, as yet, no obvious unanimity in the literature concerning the reliability of Arrhenius parameters calculated from nonisothermal measurements, or which of the many approaches proposed is to be preferred for such analyses. Although there has been much discussion (2-5) on the applicability of the Arrhenius equation to heterogeneous re-
* To whom correspondence should be addressed.
actions and the significance of the parameters, E, and A, obtained, it has to be recognized that these quantities, at the very least, have practical importance as empirical coefficients that quantitatively express the variation of the rate of reaction with temperature. The equation that most accurately expresses the form of the yield-time variation during the course of an isothermal reaction gives evidence concerning a different feature of the behavior, the systematic changes in interface geometry that occur as the chemical reaction progresses. It has been noted (6-8) many times in the literature that the magnitudes of the Arrhenius parameters for a particular reaction are not very sensitive to the kinetic model used in the derivation of the rate coefficients. Most experimental techniques provide direct measurements of a, e.g. measurements of loss of mass with time during decomposition as in isothermal thermogravimetry (TG). Such isothermal TG data can of course be differentiated with respect to time to give derivative T G (DTG) curves which are directly related to rate (u = da/dt)-time curves. Other techniques such as isothermal differential scanning calorimetry (DSC) produce a record of the rate of heat evolution or absorption (dqldt) with time, which again can be related to the da/dt-time curve if the mechanism of reaction does not change with time. Ideally, both isothermal DSC and DTG results, recorded under closely similar conditions, should be compared to confirm the kinetic behavior. These relatively recent improvements in instrumental techniques now permit the collection of sufficiently accurate data for kinetic analyses to be based on rate-time or rate-a measurements with the advantages of improved discrimination. In the present paper, attention is directed toward obtaining the Arrhenius parameters from such isothermal rate-time curves. An earlier paper (9) dealt with a more detailed identification of the kinetic model from examination of experimental results obtained by these techniques. Flynn (10) has also discussed the identification of kinetic models from isothermal DSC curves, while recently Waters and Paddy (11) have systematically considered the isothermal DSC curves to be expected from reactions proceeding according to various standard kinetic models, and they have suggested procedures for extracting values of the rate coefficients, k, from such experimental data.
KINETIC ANALYSIS The selected group of kinetic models ( I ) is conveniently divided, as shown in Table I, into two groups: a-time relationships that give either (i) sigmoid or (ii) deceleratory curves. These expressions, f ( a )= k t , may be differentiated to give the expressions g(a)= u / k listed in Table I, and by suitable substitution, these may be expressed as functions of time, u = h ( t ) ,also shown in the table. In a previous paper (9),both the rate and the time scales were normalized to produce relative-rate reduced-time (RRRT) curves, which are useful in determining the kinetic model most accurately fitting each set of experimental data.
0003-2700/89/0361-1136$01.50/0 1989 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 61, NO. 10, MAY 15, 1989
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Table I. Selected Rate Expressions for Solid-state Reactions f(a) =
g(a) = ( l / k )
kt
(daldt) = u / k
u =
da/dt = h(t)
Sigmoid a-t Curves An Avrami-Erofe'ev
[- In (1 - a)]'/" In [ a / ( l - d l
B1 Prout-Tompkins
n(1 - a ) ( - In (1 a(1 - a )
nk"t"' exp(-(kt)")
a))("-l)/n
k exp(-kt)/(l
-
exp(-kt))2
Deceleratory a-t Curves F1 unimolecular decay R2 contracting area R3 contracting volume
- I n (1 - a)
D1 one-dimensldiffusion D2 two-dimensl diffusion D3 three-dimensl diffusion D4 Ginstling-Brounshtein
a' (1- a) In (1- a) [I - (1- a)1/312 (1- 201/3) - (1 -
1 - (1 - a)l/' 1 - (1 - a)1/3
I
0 tm/2
I
tmax
t'm/2
+a
. t +
Flgure 1. Rate-time curve for a sigmoid kinetic model (Avrami-Erofe'ev, An, with n = 2).
1-a 2(1 - a)1/2 3(1 -
k exp(-kt) 2k(l - kt) 3k(l - kt)'
1/2a l/[-ln (1 - CY)] 213 2[1 - (1 ;-1/3 - 11
(1/2) (kit)'/'
iy2il@1
3/2(k/t)'/2(1 - (kt)"2)2
a)1/31
for reaction and that hence quantitative measurements of the effect of temperature on these features can yield Arrhenius parameters for the reaction under study without the necessity for identification of the kinetic model. (Although it does not appear in the table of kinetic models, the occurrence of a zero-order reaction ( I I ) , u = k, would be immediately recognizable from its characteristic constant, but nonzero, rate curve.)
THE DECELERATORY SUBGROUP OF KINETIC MODELS First-Order Model (Fl). For the F1 model (Table I) u = k exp(-kt) When t is small, exp(-kt)
-
umax
1 so that =k
When t = t m j 2 u = umax/2 = k / 2 so
k / 2 = k exp(-ktm12) 0
m/2
t-+
and hence
Figure 2. Rate-time curve for a deceleratory model (contracting volume, R3).
The two subgroups of kinetic models give rise to two general shapes of ratetime curves, illustrated schematically in Figures 1 and 2. The features of the curve relevant here, in consideration of the sigmoid group (Figure l),are the maximum rate, urn,, at some time, t, (>O), while the curve for the deceleratory group (Figure 2) has u = u, a t t = 0. A secondary feature, analogous to the half-life, tllz, in conventional kinetic analysis, is the time, tmlz,taken for the rate to drop to half its maximum value (i.e. um,/2) (Figure 2). There are two points in Figure 1 where u = um,/2, and these have been defined as tmj2on the ascent and t 'mlz on the descent. Such time values are readily determined from rate-time curves or, for example, from curves where the ordinate is proportional to u. In isothermal DSC or DTG, a value proportional to u, is obtained from the corresponding maximum ordinate, corrected, for DSC, for the mass of the sample and the sensitivity of the instrument/recorder combination. For DTG, the maximum would be corrected for the overall mass loss corresponding to cy = 1.00. These rate-time features, urn,, t, tmlZ,and t LIZ,may be determined where present (within the limits of experimental error, discussed below) from the experimental data without knowledge of the particular kinetic model that applies to the data, other than the visually obvious classification into one or the other of the two main subgroups. The main purpose of this paper is then to point out that these features are directly related to the rate coefficients, k,
tm/2 = (In 2)/k = t i p and l / t m j zis directly proportional to k. Contracting-Area Model (R2). From Table I u = 2k(1 - k t ) so when t is small u = 2k
and again k may be determined from values of initial rates. When t = tmjz u = vmax/2 = k
and so k = 2 k ( l - kt,jz) and hence t,jz
= l/2k
or l / t m l zis directly proportional to k. I t is also of interest that for this linear relationship between u and t, a t the completion of reaction, when u = 0, t = tend is then equal to l l k so that measurement of a feature such as tend,although difficult to determine experimentally and usually theoretically intermediate, may, in principle, provide a measure of l / k . Contracting-Volume Model (R3). The treatment follows that for R2 (above).
u = 3k(1 - kt)2 and when t is small
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ANALYTICAL CHEMISTRY, VOL. 61, NO. 10, MAY 15, 1989
urnax = 3k
as used by Waters and Paddy ( l l ) but , only have dimensions of time to some power. Rate coefficients should thus all have dimensions of (time)-', but problems are sometimes encountered (13) when the rate equation is written in an unconventional form, say as
When t = tmI2 u = 3k/2
and
f'(a) = k'(t)" instead
so, again l / t m I 2is directly proportional to k. Also, tend= l / k . Models Dl-D4. Note that these diffusion models do not give useful relationships in this context.
THE SIGMOID SUBGROUP OF KINETIC MODELS In this group, u = u, a t t = t,, > 0. The values of u, and t, are obtained by setting the derivatives of u with respect to time, duldt = 0. The Avrami-Erofe'ev Models (An) (where n is the exponent; see Table I). From Table I, after differentiation with respect to t d u / d t = nk"tn-2 exp(-(kt)n)((n - 1) - nkntn) and since n # 0, t,,,
# 0, and
k # 0
( n - 1) - nkntrnaxn= 0 and so that l/trnaxis proportional to k. Substitution in the rate expression for u (Table I) leads to u,
= knmm exp(-rn)
where m = ( n - l ) / n . So, again, u, is proportional to k. If assumptions are made about the value of n, rate coefficients may be calculated (9)from the values oft, and either t m / 2 or t'm12. The Prout-Tompkins Model (Bl). The most direct analysis is provided by the function g ( a ) from Table I. u = k a ( 1 - CU) Hence u = u, when a = 112 and then u, = k/4. As discussed by Waters and Paddy (11) and Mata-Perez and Perez-Benito (12),autocatalytic behavior cannot apply from a = 0, and the complete rate equation must involve some other term(s), e.g.
u = k,(l - a ) n+ k,a(l - CY) with k 2 >> 12,. The time scale for isothermal reaction is thus dependent upon the time at which the autocatalytic term becomes dominant. This time is usually referred to as the induction period, tI, to reaction, and the apparent activation energy for the process taking place during this period is usually estimated by assuming n = 1 (see above) and plotting In (l/tI) against 1/ T. For the most widely used solid-state kinetic relationships, t,, tm12, or t LIZare directly then, the magnitudes of u,, related to the rate coefficients of reaction, k. Consequently, the measured values of these parameters can be used in Arrhenius-type plots to estimate values for the apparent activation energy of reaction. The other Arrhenius parameter, the preexponential factor A , is not as readily obtained since the measured quantities, x , are only proportional to the rate coefficients; e.g. x = ak, where a is a constant, so that the Arrhenius equation becomes In k = - E / R T In A In a
+
+
where a is not known unless the kinetic model is identified. In general, in solid-state kinetics, rate equations do not contain the concentration terms and hence concentration units
of
f'(a)
= (kt)"
The use of k' instead of k in the Arrhenius equation then results in In k' = In (k") = n In k = - E ' / R T
+ In A'
so that
and the apparent (') and true values are related by E ' = nE and A' = An. The absence of dimensions of concentration (or of area) in the value determined for A should be kept in mind when one is comparing characteristic A values for solid-state reactions with A values for homogeneous bulk (or diffusion) processes. Errors in Determination of Values for the Characteristic Features of Rate-Time Curves. Integral measurements have the advantage over derivative measurements in that there are no restrictions on the time scale used other than those of the stability of the base-line signal. Derivative measurements have to be distinguishable from the base line (usually stable), and hence very slow processes cannot be accurately followed by direct measurement using DSC. Self-heating or self-cooling of reactant crystallites may also impose restrictions on the highest reaction rates that may be measured. The difficulties with slow processes do not apply to isothermal DTG. For the isothermal DSC procedures discussed above, or u,,/2 are remeasurements of u, and of times to v, quired. Measurements of ,v are readily obtained for sigmoid models (i.e. bell-shaped DSC curves) even if the peak is broad. occurs at t = 0 (deceleratory models), the DSC When v, signal may be masked to some extent by the characteristic instrument responses to changes from rapid heating to isothermal conditions. It is possible to eliminate the instrument responses by repeating the heating procedure on the "dead" sample and subtracting this record from that for the actual run. Measurements of u, have to be corrected for sample mass (which must thus be accurately known) and instrument range settings (which must obviously be reliably calibrated). Measurements of time are normally very precise relative to other measurements, but their accuracy depends on establishing their relationship to the actual events occurring. For isothermal DSC, t = 0 should be regarded as being the time at which the rapidly programmed temperature reaches its isothermal value, and the ordinate should be corrected as discussed above. When the isothermal DSC peak is broad, the error in determining a value for t, will be large. The errors in determining tmlz(and t LIZ)will depend upon the precision with which urn=, and hence urn,/ 2, is known, although the errors in assigning the time will normally be smaller because of the steeper curves in these regions. Benson (14) has pointed out that the precision of measurements of activation energies depends critically on the size of the temperature interval chosen and upon the precision of temperature control. As an example, he quotes the requirements that to measure E, over a 10 K interval to 10.5%,the temperature errors should be less than 10.03 K and k values should be known to 10.3%. Errors in isothermal DSC and DTG will be larger than this, and Heberger et al. (15) have given a detailed account of the way in which errors in Arrhenius parameters should be estimated and reported.
ANALYTICAL CHEMISTRY, VOL. 61, NO. 10, MAY 15, 1989
ds
d t ...I.............
I
t /mins
10
- _- - -.. _ _ _ 15
Flgure 3. Dehydration of nickel acetate pentahydrate: (-) 380 K, 2.37 mg, sensitivity 0.5; 375 K, 3.12 mg, sensitivity 0.5; (...) 372.5 K, 3.44 mg, sensitivity 1; (--) 370 K, 1.70 mg, sensitivity 1; (---) 365 K, 2.85 mg, sensitivity 1. (-e-)
RESULTS AND DISCUSSION Waters and Paddy (11) have quoted some examples of the use of isothermal DSC for identification of kinetic models. Further examples from our laboratories are the decompositions of copper(I1) squarate (16), silver squarate (17), potassium permanganate (18),nickel acetate (19),and nickel malonate (20). Some of these data were interpreted in the context of complementary product yield-time measurements. The isothermal DSC curves obtained a t a series of different, but constant, temperatures in the range 365-380 K for the dehydration of nickel acetate pentahydrate (19) are shown in Figure 3 (with the associated sample masses and sensitivity ranges). The curves correspond to mainly deceleratory processes, and there are the problems of accurate determination of u, as discussed above. The E, values estimated from these curves are 138 f 29 kJ mol-' (urn=) and 150 f 30 kJ mol-' (1/tm,2). Curves for the dehydration of nickel malonate dihydrate (20) over the higher temperature range of 480-520 K are not very different in shape. There is a very rapid rise to maximum rate, which then remains approximately constant for a short while before decreasing in a nonlinear manner. Activation energies could be estimated more precisely for this reaction, as 118 f 3 kJ mol-' (urn=) and 134 f 4 kJ mol-' (1/tm,2). These results suggest that the activation process for dehydration is similar in the two different hydrates. The decompositions of the two carboxylates are complex processes, as confirmed by their isothermal DSC curves over higher temperature ranges, and are discussed in ref 19 and 20. As a further example of the application of the above procedures to isothermal DSC data, we have chosen the published results of Catalano and Crawford (21) for the thermal decomposition of triaminotrinitrobenzene (TATB). Their isothermal DSC curves (over the range 615-630 K) show two overlapping exotherms (ref 21, Figure 1). The authors divided
1139
the reaction into five stages and determined activation energies of 209-251 kJ mol-' for the first (autocatalytic) stage and 209-335 kJ mol-' for the remaining stages. Through use of the variations with temperature of the maximum rates, uA and uB, and the reciprocal times to the first (A) and second (B) maxima, activation energies of 244 f 6 kJ mol-' (uA), 236 f 23 kJ mol-' (l/tA), 239 f 13 kJ mol-' (LIB),and 252 f 18 kJ mol-' (l/tB) were calculated.
CONCLUSIONS Some of the readily measured points on isothermal ratetime curves are shown to be directly related to rate coefficients for reaction. Measured values of these parameters can thus be used in Arrhenius-type plots to estimate values of apparent activation energies for reaction, without the necessity of first identifying the kinetic model that quantitatively expresses the overall progress of reaction. LITERATURE CITED (1) Brown, M. E.; Dolllmore, D.; Galwey, A. K. Reactions in the Solid State ; Comprehensive Chemical Kinetics, Vol. 22;Elsevier: Amsterdam, 1980, (2) Gomes, W. Nature (London) 1961, 192, 865-866. (3) Arnold, M.: Veress. G. E.; Paulik, J.; Paullk, F. Therm. Anal., (Roc. I n t . Conf.), 6th 1980, 1 , 69-73. (4) Garn, P. D. Crit. Rev. Anal. Chem. 1972, 3 . 65-1 11. (5) Flynn, J. H.; Brown, M. E.; Sestak, J. Thermochim. Acta 1987, 110, 101-112. (6) Johnson, D. W.; Gallagher, P. K. J . f h y s . Chem. 1971, 75, 1179-1185. (7) Escuer. A. Thermochim. Acta 1986, 104, 309-319. (8) Yankwich, P. E.; Zavitsanos, P. D. Pure Appl. Chem. 1964, 8. 287-304. (9) Brown, M. E.; Galwey, A. K. Therm. Anal., (froc. I n t . Conf.), 7th 1982, 1 , 58-64. (10) Flynn, J. H. Thermochim.Acta 1985, 9 2 , 153-156. (11) Waters, D. N.; Paddy, J. L. Anal. Chem. 1988, 60, 53-57. (12) Mata-Perez, F.; Perez-Benlto, J. F. J . Chem. Educ. 1987, 6 4 , 925-927. (13) Fatemi, N.; Whitehead, R.; Price, D.; Dollimore, D. Thermochim. Acta 1986, 104, 93-100. (14) Benson, S. W. The Foundations of Chemical Kinetics; McGraw-Hill: New York, 1960;p 91. (15) Heberger, K.;Kemeny, S.; Vidoczy, T. Int. J . Chem. Kinet. 1987, 79, 171-181. (16) Brown,-M. E.; Galwey, A. K.; Beck, M. W. I s r . J . Chem. 1982, 2 2 , 2 15-2 18. (17) Brown, M. E.; Kelly, H.; Galwey, A. K.; Mohamed. M. A. Thermochim. Acta 1988, 127, 139-158. (18) Brown, M. E.; Sole, K. C.;Beck, M. W. rhermochim. Acta 1985, 8 9 , 27-37. (19) Galwey, A. K.; McKee, S. G.; Mitchell, T. R. 8.; Brown, M. E.; Bean, A. F. React. Solids 1988, 6, 173-186. (20) Galwey, A. K.; McKee, S. G.; Mitchell, T. R. B.; Mohamed, M. A,; Brown, M. E.; Bean, A. F. React. Solids 1988, 6, 187-203. (21) Catalano, E.; Crawford, P. C. Thermochim. Acta 1983, 61, 23-36.
RECEIVED for review September 23, 1988. Accepted January 24, 1989.
