Avrami-Erofeev rate equation

Incorporation of a diffusion term in nucleation and growth reaction models has been proposed by Hulbert [68]. Interface advance is assiuned to fit the parabolic law and is proportional to but the nucleation step is uninhibited. The overall rate expressions have the same form as the Avrami-Erofeev equation ... 

The kinetics of many solid state reactions have been reported as being satisfactorily represented by the first-order rate equation [70] (which is also one form of the Avrami-Erofeev equation (n = 1)). Such kinetic behaviour may be expected in decompositions of fine powders if particle nucleation occurs on a random basis and growth does not advance beyond the individual crystallite nucleated. 

Care should be taken in defining the procedure for calculating values of k fi om the experimental data. There is always the possibihty that the apparent is a compound term containing several individual rate coefficients for separable processes (such as nucleation and growth). It is important that the dimensions of k (and hence of A) should be (time). For example, the power law (Table 3.3.) should be written as = kt and not as ar = k t. Similarly the Avrami-Erofeev equation (An) is [-ln(l - a)Y = kt. The use of k in place of A in the Arrhenius equation will produce an apparent activation energy /i, which is n times the conventional activation energy obtained using k. 

For those rate equations containing exponents, e.g the Avrami-Erofeev equations (An), it would appear that plots of ln[- ln(l - a)] against In (t - Q would provide the most direct method for the determination of the value of the exponent, n. Such plots are, however, notoriously insensitive and errors in t, together with contributions from initial rate processes, can influence the apparent value of n. Non-integer values of n have been reported [12],... 

Ammonium copper(II) sulfate hexahydrate [64] loses 4H2O between 339 and 393 K with no influence of water vapour pressure on reaction rate below 172 N m . The Avrami-Erofeev equation n = 2 fitted the data below 350 K with E, = 170 kJ mol. Above this temperature the contracting area equation applied and E, = 82 kJ mol. ... 

A recent kinetic study [108] of the overall dehydration rates of KA1(S04)2.12H20 and KCr(S04)2.12H20 showed that measured ur-time data were well described by the Avrami-Erofeev equation with = 2. This was not consistent with expectation for the growth of three dimensional nuclei for which A = 3 and = 0 or 1, respectively. In accordance with the reaction models described above, there must be limited water losses from all surfaces together with an overall reaction controlled by product recrystallization that confers the apparent topotacticity on the overall... 

The thermal dehyi-ation of Na2C03.H20 between 336 and 400 K fits the Avrami-Erofeev equation with = 2 (E, = 71.5 kJ mol and., 4 = 2.2 x 10 s [110]). The apparent reduction in rate resulting from an increase of /r(H20) is ascribed to competition from the rehydration reaction. Electron micrographs confirm the nucleation and growth mechanism indicated by the kinetic behaviour, nucleation develops from circular defects that may be occluded solution. 

The final stages of the above reaction overlapped with the onset of the nucleation and growth process that continued to complete the dehydration. Growth of three dimensional nuclei was confirmed microscopically. This second rate process was well described by the Avrami-Erofeev equation with = 2 and E, for crystals was 175 30 kJ mol (with a considerable scatter of data) below 460 K and a more reproducible reaction rate, with E, = 153 10 kJ mol, for powder. Above about 450 K there were some indications of intracrystalline melting of single crystals and the value of , increased markedly to 350 50 kJ mol (again with significant scatter of data). 

Galwey and Hood [160] showed that NajCOj.l.SHjOj decomposed in vacuum (360 to 410 K) to produce Na COj + l.SHjO + O.TSOj. ar-time curves were sigmoidal and the kinetics could be described by the Avrami-Erofeev equation with = 2 or 3. The activation energy was 112 8 kJ mol. The reaction rate between 313 and 343 K was significantly increased by the presence of small amounts of liquid water. This deceleratory reaction was fitted by the first-order equation (E, = 80 10 kJ mol ) and it was concluded that breakdown of hydrogen peroxide proceeded in the liquid water, possibly with trace amounts of impurity transition-metal ions acting as catalysts. 

Although the rates of decomposition of different preparations of lithium azide differed markedly [34], reproducible behaviour was observed for salt which had been crushed and pelleted. Such pretreatment was believed to produce a uniform concentration of defects within the reactant assemblages. The sigmoid nr-time curves fitted the Avrami-Erofeev equation with = 3 between 0.02 < or < 0.58 and the contracting volume expression across the wider interval 0.05 < a< 0.95, the value of was 119 kJ mol". Reaction involved the three-dimensional growth of a constant number of nuclei and it was suggested that acceleration of rate following preirradiation resulted from an increase in the number of such nuclei. 

Decomposition of CsBrOj proceeds [30] in the molten or semi-molten state of a eutectic formed with the CsBr product. Kinetics were fitted by the Prout-Tompkins and Avrami-Erofeev equations. The reaction rate (673 K) was accelerated significantly both by y-irradiation damage (which leads to rupture of Br-0 bonds) and by the presence of added Ba " ions which introduce local strain into the crystal and thereby promote Br03 ion breakdown. 

Nickel permanganate decomposed [44] (- NiMnjOj + I.5O2) between 356 and 400 K. The sigmoid ar-time curves were well expressed by the Avrami-Erofeev equation ( = 2). An initial electron transfer step was identified as rate controlling, with = 100 5 kJ mol . The rate of the first half of reaction a < 0.5) was decreased by the presence of water vapour. The rate of this autocatalytic reaction also proceeded more rapidly in the solid state than the comparable reaction in aqueous solution. 

Iron(III) oxalate decomposed between 410 and 450 K to give CO and iron(II) oxalate which retained about 10% of the iron(lll) salt [57]. Melting was not detected. The sigmoid a - time curves were identified as being due to a nucleation and growth process. The first half of the reaction was well represented by the Avrami - Erofeev equation (n = 2) and the latter half by the contracting volume equation. Values of were relatively low, 107 to 120 kJ mol , and rate control was ascribed to either electron transfer or C - C bond rupture. 

The r-time curves for the decomposition of anhydrous cobalt oxalate (570 to 590 K) were [59] sigmoid, following an initial deceleratory process to a about 0.02. The kinetic behaviour was, however, influenced by the temperature of dehydration. For salt pretreated at 420 K, the exponential acceleratory process extended to flr= 0.5 and was followed by an approximately constant reaction rate to a = 0.92, the slope of which was almost independent of temperature. In contrast, the decomposition of salt previously dehydrated at 470 K was best described by the Prout-Tompkins equation (0.24 < a< 0.97) with 7 = 165 kJ mol . This difference in behaviour was attributed to differences in reactant texture. Decomposition of the highly porous material obtained from low temperature dehydration was believed to proceed outwards from internal pores, and inwards from external surfaces in a region of highly strained lattice. This geometry results in zero-order kinetic behaviour. Dehydration at 470 K, however, yielded non-porous material in which the strain had been relieved and the decomposition behaviour was broadly comparable with that of the nickel salt. Kadlec and Danes [55] also obtained sigmoid ar-time curves which fitted the Avrami-Erofeev equation with n = 2.4 and = 184 kJ mol" . The kinetic behaviour of cobalt oxalate [60] may be influenced by the disposition of the sample in the reaction vessel. 

The isothermal kinetics of decomposition were complex, with at least two overlapping processes taking place. The shapes of the peaks indicated that both processes were initially acceleratory, and then deceleratory. The isothermal rate was assumed to be made up of weighted contributions from individual processes which could be described by the Avrami-Erofeev equation, with various values of n. 

Decompositions of chemically dissimilar substances (including NiC204, PbCjO, KNj, (NH4)2Crj07) may exhibit similar kinetic behaviour, whereas the decompositions of similar reactants (e.g. oxides) may exhibit apparently unrelated rate characteristics [60]. Decomposition data for very different reactants may be described satisfactorily by the same rate expression, for example the Avrami-Erofeev equation, with n = 2, has been reported as representing the decompositions ofFeC204 [61], KMn04 [29], silver malonate [62], nickel squarate [18], lead citrate [63] and d - LiK tartrate [64] (no common constituents in these six reactants). 

The Avrami-Erofeev equation Nuclei growing in three (or in two) dimensions, without the inhibiting effect of any diffusion limitation, give sigmoidshaped curves (Figure 5.2) that are well represented by rate equations of the form... 

FIGURE 6.9 (a) Isothermal cx-time curves for the dehydration of theophylline monohydrate, (b) corresponding fitting to the Avrami-Erofeev equation, and (c) temperature dependence of the calculated rate constant. (Reproduced from Suzuki, E. et al., Chem. Pharm. Bull., 37, 493, 1989. With permission.)... 

Kinetic runs in step b in Fig. 8c started with a very fast reduction of approximately e per molecule, after which a slow reductioh took place, yielding sigmoidal reduction curves. This, indicates that reduction of Co2+ to Co° is controlled by the formation and slow growth of reduction nuclei of metallic cobalt on. the surface of the reduced phase in step a (nucleation model). Initially, the reduction rate increases because of the growth of nuclei already formed and the appearance of new ones. At a certain point the reduction nuclei start to overlap at the inflection point, the interface of. the oxidized and reduced phases and the reduction rate both begin to decrease. Reduction of this type is described by the Avrami-Erofeev equation (118)... 

As can be seen, with the exception of the R3, D3 and D4 models, the exclusion of the In (j)m (o max term in the generalized Kissinger equation (9) has less than a 3% effect, and for the simple n order and Avrami-Erofeev models, A2, A3, less than a 1% effect. Similar figures result if one sets m - h or m = 1 in the rate equation (1). As the Arrhenius exponent E/RTjjjqx increases, there is a gradual small increase in Umax models. For all n order and diffusion controlled... 

A modified form of the Avrami-Erofeev equation, ln[—ln(l — a)] = nln(fe/n) - - nlnr, where is a nondimen-sional coefficient and A is a generalized rate constant, has been applied by Pinakov and Logvinenko to describe the decomposition of fluorinated graphite inclusion compounds of 1,2-dichloroethane and clathrates of manganese formates with dioxane. 

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