Avrami-Erofeev kinetics

Figure 11.14a compares the model based on Avrami-Erofeev kinetics with the... 

Figure 12.14a compares the model based on Avrami-Erofeev kinetics with the experimental curve. The agreement is satisfactory in light of the fact that no adjustable parameter has been used to allow curve-fitting procedures. The difference at the leading edge is attributed to the first reduction step which is not included in the model. 

The Avrami-Erofeev scheme is one of the most widely used for studying solid-state kinetics it has been effectively applied to a diversity of solid-state processes including decompositions. 

Equation (3.2) is often referred to as the Avrami-Erofeev (A-E) equation, or more fittingly, on account of the substantial contributions from other workers, especially Johnson and Mehl [26] in the field of metallurgy, as the Johnson-Mehl-Avrami-Erofeev-Kholmogorov (JMAEK) equation. The values of n obtained from kinetic... 

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. 

Elder [45] has modelled several multiple reaction schemes, including mutually independent concurrent first-order reactions, competitive first-order reactions, mutually independent n-th order reactions, and mutually independent Avrami-Erofeev models with n = 2 or 3. The criteria identified for recognizing the occurrence of multiple reactions were (i) the apparent order of reaction, n, varies with the method of calculation, and (ii) the kinetic parameters, A and vary with the extent of reaction, a. 

The dehydration and rehydration reactions of calcium sulfate dihydrate (gypsiun) are of considerable technological importance and have been the subject of many studies. On heating, CaS04.2H20 may yield the hemihydrate or the anhydrous salt and both the product formed and the kinetics of the reaction are markedly dependent upon the temperature and the water vapour pressure. At low temperatures (i.e. < 383 K) the process fits the Avrami-Erofeev equation (n = 2) [75]. The apparent activation energy for nucleation varies between 250 and 140 kJ mol in 4.6 and 17.0 Torr water v our pressure, respectively. Reactions yielding the anhydrous salt (< 10 Torr) and the hemihydrate ( (HjO) >17 Torr) proceeded by an interface mechanism, for which the values of E, were 80 to 90 kJ mol. At temperatures > 383 K the reaction was controlled by diffusion with E, = 40 to 50 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 kinetics of dehydration [128] of Na2S203.5H20 were difficult to interpret because the course of the reaction was markedly influenced by the perfection of the initial reactant surface and the reaction conditions. No reliable Arrhenius parameters could be obtained. The mechanism proposed to account for behaviour was the initial formation of a thin superficial layer of the anhydrous salt which later reorganized to form dihydrate. The first step in the reaction pentahydrate - dihydrate was satisfactorily represented by the contracting area (0.08 < or, < 0.80) expression. The second reaction, giving the anhydrous salt, fitted the Avrami-Erofeev equation (n = 2) between 0.05 < 2< 0.8. The product layer offers no impedance to product water vapoiu escape and no evidence of diffusion control was obtained. The mechanistic discussions are supported by microscopic observations of the distributions and development of nuclei as reaction proceeds. 

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. 

A complete analysis of the reaction would require measurements of the variations with time of all the phases participating. The product giving unusual textures identified by Brown et al. [18] may, perhaps, be K3(Mn04)2. The fit of kinetic data to the Avrami-Erofeev equation (n = 2) [18], together with the appearance of nuclei, illustrated in Figure 14.1., can be regarded now [17] as only an incomplete representation of this more complicated reaction. 

De Waal et al. [98] used Raman spectroscopy to measure the decomposition kinetics from the isothermal time-dependence of the totally symmetric Cr - 0 vibration mode in (NH4)2Cr04 between 343 and 363 K, The results were fitted to the Avrami-Erofeev equation with n = 2. for microcrystals was 97 10 kJ mol and 49 1 kJ mol for powdered samples. 

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. 

Selcuk and Price [101] reported that the thermal decomposition of PbC204 in Nj between 611 and 707 K, studied by isothermal thermogravimetry, fitted the Avrami-Erofeev equation with n = 2, ascribed to random nucleation followed by two dimensional growth, , = 119 7 kJ mol. Incorporation of 1% Zn was found to increase E, to 137 5 kJ mol but did not change the kinetic fit. It is suggested that zinc oxalate within the host crystal restricts the formation of Smekal cracks along which decomposition is favoured. 

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. 

A kinetic study [34] of the decomposition of chromium(III) tm-N-benzoyl-N-phenylhydroxylamine between 433 and 453 K, showed that the stable products were benzanilide, chromium(III) benzoate and an unidentified chromium product, ar-time curves were sigmoid and fitted the Avrami-Erofeev equation with = 3.9 0.1, except for the lowest (and least accurate) temperature. Observation of partially reacted material identified the formation of liquid regions, characteristically above the melting point (434 K) of the major product, benzanilide. The autocatalytic process was considered to be initial decomposition of the solid, followed by more rapid breakdown of reactant dissolved in the melt. E, was about 170 kJ mol . Progressive melting is also consistent with the sigmoid ar-time curve. 

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