Kinetic rate equation, Avrami-Erofeev

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. 

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. 

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. 

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. 

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). 

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)... 

In summary, the above analysis suggests that the kinetic equation of most practical use ought to be the Erofeev equation (20), —ln(l — a) = kty, where n might have any integral value from 1 to r + 3. Strictly, when k t is comparable with 1, the Avrami equation (25) ought to be used instead, but it is doubtful if experimental data of sufficient precision exist to make this a worthwhile exercise at present. In addition we note that the assumptions that the shape and the growth rates are independent of nucleus size (a, Gj, G2, G3 not functions of t) may not always be justified. 

---