Equation, Arrhenius Avrami

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. 

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. 

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. 

Figure 145. Linear Arrhenius plots for the polymorphic transitions of carbamazepine (o,) and benoxaprofen ( ,) The rate constants k were obtained according to the Avrami-Erofe ev equation (time unit min). (Reproduced from Refs. 578 and 611 with permission.)...

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... 

Davidson et al. (2008) measured the rates of this transformation over the temperature range of 60 to 190°C and fit the results to the modified Avrami equation. The average value of n from their fits was 2.1(0.25). Fitting their reported rate constants to the Arrhenius equation gives an estimated value of of 32.4 kJ/mol with log A = 1.31. These values are substituted into Eq. (9.72) to create the TTT diagram shown in Figure 9.8. 

The results shown were analyzed with the Avrami empirical equation, av=l-exp(-kf"), where k is the rate constant, and w is a constant. The data on Uv obtained at the initial and intermediate stages gave a linear relationship between ln(ln(l-av) i) and ln(f) with a correlation coefficient of more than 0.99. The Avrami parameter and rate constants obtained are summarized in Table 1. Based on the Arrhenius relationship (Fig. 9), k=Aexp(-Ev/RT) with v as the activation energy and constants A and R, on those k values which increased with increasing temperature, we obtained an activation energy of kj/mol,... 

Fig. 29. Plots of the (a) Avrami and (b) Arrhenius equations for the isothermal crystallization of Ni50.54Ti49.4s thin films(Liu E>uh, 2007)...

In this chapter, we have conducted a comparative study to elucidate the role of pore microstructures on the decomposition kinetics of several MAX phases during vacuum annealing in the temperature range 1000-1800°C. The effect of pore-size on the activation energy of decomposition was evaluated using the Arrhenius equation. The kinetics of phase decomposition was modelled using a modified Avrami equation. 

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