Avrami-Erofe ev equation

Following a common practice in the literature, significant rate expressions will be referred to by names which have become accepted through common usage these may be descriptive (power law, etc.) or recall the names of workers who contributed towards their development (the Avrami—Erofe ev equation, etc.). Examples of systems obeying each expression are restricted in the present section since the applications are exemplified more generally in the literature surveys which constitute Chaps. 4 and 5. 

The Avrami—Erofe ev equation, eqn. (6), has been successfully used in kinetic analyses of many solid phase decomposition reactions examples are given in Chaps. 4 and 5. For no substance, however, has this expression been more comprehensively applied than in the decomposition of ammonium perchlorate. The value of n for the low temperature reaction of large crystals [268] is reduced at a 0.2 from 4 to 3, corresponding to the completion of nucleation. More recently, the same rate process has been the subject of a particularly detailed and rigorous re-analysis by Jacobs and Ng [452] who used a computer to optimize curve fitting. The main reaction (0.01 < a < 1.0) was well described by the exact Avrami equation, eqn. (4), and kinetic interpretation also included an examination of the rates of development and of multiplication of nuclei during the induction period (a < 0.01). The complete kinetic expressions required to describe quantitatively the overall reaction required a total of ten parameters. 

The strongly acceleratory character of the exponential law cannot be maintained indefinitely during any real reaction. Sooner or later the consumption of reactant must result in a diminution in reaction rate. (This behaviour is analogous to the change from power law to Avrami—Erofe ev equation obedience as a consequence of overlap of compact nuclei.) To incorporate due allowance for this effect, the nucleation law may be expanded to include an initiation term (kKN0), a branching term (k N) and a termination term [ftT(a)], in which the designation is intended to emphasize that the rate of termination is a function of a, viz. 

Fig. 2. Reduced time plots for the Avrami—Erofe ev equation [eqn. (6)] with n = 2, 3 and 4 and tT = (t/ta.g) the Prout—Tompkins expression [eqn. (9)] is included as the broken line.

Magnitudes of n have been empirically established for those kinetic expressions which have found most extensive application e.g. values of n for diffusion-limited equations are usually between 0.53 and 0.58, for the contracting area and volume relations are 1.08 and 1.04, respectively and for the Avrami—Erofe ev equation [eqn. (6)] are 2.00, 3.00 etc. The most significant problem in the use of this approach is in making an accurate allowance for any error in the measured induction period since variations in t [i.e. (f + f0)] can introduce large influences upon the initial shape of the plot. Care is needed in estimating the time required for the sample to reach reaction temperature, particularly in deceleratory reactions, and in considering the influences of an induction period and/or an initial preliminary reaction. 

Fig. 6. Two reaction models which result in obedience to the power law [eqn. (2), n = 2 ] at low a, or the Avrami—Erofe ev equation [eqn. (6), n = 2 ] over a more extensive range of a. In (a), there is growth of semi-circular nuclei in a thin plate of reactant in (b), there is cylindrical growth of linear internal nuclei. In both examples, rapid nucleation (0 = 0) is followed by two-dimensional growth (X = 2).

Ng et al. [1261] report that dehydration of copper sulphate pentahydrate (- CuS04 3 H20) 320—336 K, obeys the Avrami—Erofe ev equation [eqn. (6), n = 2] with E = 104 kj mole-1. Dehydration of the trihydrate (- CuS04 H20), 343.5—359 K, obeyed the same rate expression with E = 134 kJ mole 1. Activation energies are approximately equal to reaction enthalpies. 

Analyses of rate measurements for the decomposition of a large number of basic halides of Cd, Cu and Zn did not always identify obedience to a single kinetic expression [623—625], though in many instances a satisfactory fit to the first-order equation was found. Observations for the pyrolysis of lead salts were interpreted as indications of diffusion control. More recent work [625] has been concerned with the double salts jcM(OH)2 yMeCl2 where M is Cd or Cu and Me is Ca, Cd, Co, Cu, Mg, Mn, Ni or Zn. In the M = Cd series, with the single exception of the zinc salt, reaction was dehydroxylation with concomitant metathesis and the first-order equation was obeyed. Copper (=M) salts underwent a similar change but kinetic characteristics were more diverse and examples of obedience to the first order, the phase boundary and the Avrami—Erofe ev equations [eqns. (7) and (6)] were found for salts containing the various cations (=Me). 

Pai Vemeker and Kannan [1273] observe that data for the decomposition of BaN6 single crystals fit the Avrami—Erofe ev equation [eqn. (6), n = 3] for 0.05 < a < 0.90. Arrhenius plots (393—463 K) showed a discontinuous rise in E value from 96 to 154 kJ mole-1 at a temperature that varied with type and concentration of dopant present Na+ and CO2-impurities increased the transition temperature and sensitized the rate, whereas Al3+ caused the opposite effects. It is concluded, on the basis of these and other observations, that the rate-determining step in BaN6 decomposition is diffusion of Ba2+ interstitial ions rather than a process involving electron transfer. 

Indium sulphate is slightly more stable than the aluminium salt and decomposition curves (1073—1273 K) fitted [777] the Avrami—Erofe ev equation [eqn. (6)] with values of n increasing with temperature from 1.0 to 1.6. Gallium sulphate is less stable and decomposes between 833 and 973 K[770]. 

Microscopic examination has shown [102,922] that the compact nuclei, comprised of residual material [211], grow in three dimensions and that the rate of interface advance with time is constant [922]. These observations are important in interpreting the geometric significance of the obedience to the Avrami—Erofe ev equation [eqn. (6)] [59,923]. The rate of the low temperature decomposition of AP is influenced by the particle ageing [924] and irradiation [45], the presence of gaseous products [924], ammonia [120], perchloric acid [120] and additives [59]. 

Isothermal a—time curves were sigmoid [1024] for the anhydrous Ca and Ba salts and also for Sr formate, providing that nucleation during dehydration was prevented by refluxing in 100% formic acid. From the observed obedience to the Avrami—Erofe ev equation [eqn. (6), n = 4], the values of E calculated were 199, 228 and 270 kJ mole"1 for the Ca, Sr and Ba salts, respectively. The value for calcium formate is in good agreement with that obtained [292] for the decomposition of this solid dispersed in a pressed KBr disc. Under the latter conditions, concentrations of both reactant (HCOJ) and product (CO3") were determined by infrared measurements and their variation followed first-order kinetics. 

Recently, it has been shown [1071] that CoC204 2 H20 exists in two crystalline modifications, a and 3. Taskinen et al. [1072] prepared anhydrous cobalt oxalate of different particle sizes by dehydration of the (3 (coarser grained) phase and the a/(3 mixture (finer grained). The coarser grained preparation decomposed at 590—700 K with a sigmoid a—time curve fitted by the Avrami—Erofe ev equation [eqn. (6), n = 2] and below and above 625 K, E values were 150 and 57 kJ mole-1, respectively. Reaction of the fine preparation obeyed eqn. (6) (n = 3) and below and above 665 K, values of E were 120 and 59 kJ mole-1, respectively. Catalytic properties of the products of decomposition of cobalt oxalate have been investigated [1073]. 

In an inert atmosphere, the decomposition at 573—623 K of uranyl(VI) oxalate [1101] obeys the Prout—Tompkins equation [eqn. (9)] with E = 261 4 kJ mole-1. The residual product is U02 and, under low pressure accumulatory conditions, the final CO2/CO ratio is 9. In air, the reaction proceeds in two stages. The initial process obeys the Prout—Tompkins equation and is identified as a surface reaction. Thereafter, decomposition fits the Avrami—Erofe ev equation [eqn. (6), n = 2], involving isolated disc-like grains of reactant, to yield amorphous U03 as the final product. Values of E for both stages of reaction are close to that found for reaction in the inert atmosphere ( 260 kJ mole-1) and comparable with theoretical predictions [88],... 

Sigmoid curves, attributable to nucleation and growth reactions, were observed for the decompositions of cobalt phthalate and silver mellitate these are marked in Table 16. The decomposition of nickel terephthalate [88] obeys the Avrami—Erofe ev equation [eqn. (6)], for which n is 1.0— 1.5 and E = 226 8 kJ mole-1. Decompositions of Co—Ni mixed mellitates are discussed in Sect. 7. 

An unusual variation in kinetics and mechanisms of decomposition with temperature of the compound dioxygencarbonyl chloro-bis(triphenyl-phosphine) iridium(I) has been reported by Ball [1287]. In the lowest temperature range, 379—397 K, a nucleation and growth process was described by the Avrami—Erofe ev equation [eqn. (6), n = 2]. Between 405 and 425 K, data fitted the contracting area expression [eqn. (7), n = 2], indicative of phase boundary control. At higher temperatures, 426— 443 K, diffusion control was indicated by obedience to eqn. (13). The... 

The most reliable way to find n is to use the method of Sharp and Hancock. This consists of taking logarithms of the Avrami-Erofe ev equation, to give... 

Figure 18 Relationship of weight percent of SX-II and k values computed from the Avrami-Erofe ev equation with n = 2. (From Ref. 50.)...

Figure 143. Polymorphic transition kinetics of carbamazepine from form I to form HI according to the Avrami-Erofe ev equation (Eq. 3.3). (Reproduced from Ref. 611 with permission.)...

Figure 147. Polymorphic transition kinetics of phenylbutazone according to the Avrami-Erofe ev equation (a) and by first-order kinetics (b), as a function of exposure to varying relative humidities (60°C). >, 4, ( 0% RH A,, 50% RH , , 70% RH 0, , 80% RH. (Reproduced from Ref. 612 with permission.)...

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