Method of Freeman and Carroll

If A (1/T) is maintained constant, a plot of A log (AT) against A logo should give a straight line. The slope of this line is n and its intercept gives the value of E. 

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Theoretically, it is possible to calculate the energy of activation from these data by the method of Freeman and Carroll and Anderson and Freeman, who have derived the following equation ... 

Activation energy for 100-450 C deconposition, determined by method of Freeman and Carroll (1 ). 

Differential method of Achar, Brindley and Sharp (102), n = 0.5. bIniegraiing method of Coais and Redfern 18), n = 0.5. Difference-differential method of Freeman and Carroll (83). dln kcal/mole. 

By substitution of equations (2.11) and (2.9) into equation (2.8), and by differentiating the logarithmic form, an expression is obtained which is one of the differential methods the Freeman and Carroll (83) method. 

Three kinetic methods were evaluated by Sharp and Wentworth (76) using the thermal decomposition of calcium carbonate under various conditions. The physical state of the sample was as a pellet, a powder, or as 1 1 molar ratios with a-aluminum oxide or a-iron(III) oxide. The three methods used were Method 1, Freeman and Carroll Method II, Coats and Redfern and Method HI, Achar et al. (102). The kinetic data calculated by Methods II and III are presented graphically in Figure 2.41 and in Table 2.3. In every case a linear plot was obtained over a wide range of a with n = When these methods were applied with n — j, the range of x was less, especially in the case of Method III, which led to noticeable curvature at... 

A method that is widely employed to calculate rate parameters is that of Freeman and Carroll [4], where equation 5.8 is rewritten in logarithmic form as... 

These rate equations C2in be used for quite complicated reactions, but a specific method or approach is needed. Many authors have tried to devise methods for obtaining rate constants and orders of reaction for given solid state reactions. None have been wholly successful, except for Freeman and Carroll (1948). 

The Freeman and Carroll method has been shown by Fong and Chen to be the only one which gives satisfactory answers to known reactions, whether zero order, 1st order, 2nd order, or even higher. E)ven firactional orders of reaction may be determined. This method can be used with either DTA or TGA data. 

Liu, N. A. and Fan, W. C. Critical consideration on the Freeman and Carroll method for evaluating global mass loss kinetics of polymer thermal degradation. Thermochemica Acta 1999 338 85. 

Perhaps the most widely used kinetics method is that developed by Freeman and Carroll (83) in 1958. The advantage of this method is that considerably fewer experimental data are required than in the isothermal method and that the kinetics can be obtained over an entire temperature range in a continuous manner without any missing regions. In addition, where a sample undergoes considerable reaction in being raised to the temperature of interest, the results obtained by an isothermal method are often questionable. The order of reaction, nt and the activation energy of the reaction are calculated from the equation... 

Sestak (43) compared the kinetic results calculated by five different methods for a system corresponding to the dehydration of -CaS04 0.5H2O. The five methods evaluated mathematically were (1) Freeman and Carroll (83) (2) Doyle (84) (3) Coats and Redfern (85) (4) Horowitz and Metzger (88) and (5) Van Krevelen et al. (87). From these calculations it was found that the deviations of computed values oF E did not differ by more than 10%. Thus, all the methods appear to be satisfactory for the calculation of E within the limits of accuracy required. The errors of each method due to the inaccuracy of visual deduction of values from the TG curves were also calculated. These errors, % and e (errors in calculation of E or n, respectively), were as follows (1) Freeman and Carroll method, eE = 4% and e = 12% (2) Horowitz and Metzger method, ee = 2% (when the correct value of n is assumed) (3) Doyle method. eE = 4%. However, the magnitude of this error depends primarily on the position of the point on the TG curve on which the calculations are being performed. In the case of differential methods, me most accurate data are calculated from the medium-steep parts of the curve. For the approximation method, the accuracy depends on the determination of the curve inflection point temperature. 

Rajeshwar I52) determined the kinetics of the thermal decomposition of Green River oil shale kerogen by using direct Arrhenius. Freeman and Carroll, and Coats and Redfern methods. The E, A, and values are given in Table 2.7. Rajeshwar concluded that the ability to resolve multiple processes hinges on the efficacy of the particular kinetic analysis employed and is not an inherent difficulty with nonisothermal TG techniques in general. The direct Arrhenius and Coats and Redfern methods clearly indicate the presence of two reactions with distinctly different kinetic parameters. On the olher hand, the Freeman and Carroll method is handicapped at low fractional... 

In addition to gravimetric analysis, TG has also been used to elucidate the kinetics of decomposition reactions. This involves analyzing the shape of the TG curve. In general, the rate of reaction at any measured temperature is proportional to the slope of the curve, but a number of uncertainties sometimes make these analyses of questionable value. Freeman and Carroll [/. Phys. Chem., 62, 389 (1958)] describe the most popular of the kinetics-analysis methods, while Clarke et al. [Chem. Comm., 266 (1969)] present the major objections to kinetics analysis by TG. 

Determination of Kinetic Parameters by Freeman and Carroll Method [11 j... 

The kinetics of thermal degradation have generally been studied using isothermal and nonisothermal methods. In earlier literature, isothermal methods were mostly employed for the study of the kinetics of solid-state reactions. During the past three decades, however, nonisothermal methods, for example, the Doyle method [17, 18], Freeman and Carroll method [19], Coats and Redfem method [24], Ozawa method [20], Flynn and Wall method [21, 22], Friedman method [25], and Kissinger method [26], have received more attention. 

Various workers were involved in the development of the thermogravimetric technique [1-4]. In 1958, Freeman and Carroll [5] developed a widely used method for the determination of reaction kinetics using the thermobalance. During the past few years, much progress has been made in developing more suitable methods for the determination of kinetic parameters, as discussed below. 

Reaction kinetics from DSC, DTA or TGA, have been used to examine the stability of a limited number of pharmaceutical materials. Various models have been used including the Power Law, Avarami-Erofeev and Prout-Tomkins models [72]. These methods are also based on the Kissinger [73], ASTME 698 [74] or Ozawa [75] methods [8]. Most frequently, they have been applied to the dehydration of various materials such as theophylline monohydrate [76], phenobarbitone monohydrate or hemihydrate [77], phenylbutazone [78], oxazepam [23] and trazodone tetrahydrate [79]. The uses are limited for pharmaceutical systems, not least because dehydration is particle size dependent. Thermal analysis, especially DSC, DTA and TG, has been used outside the pharmaceutical area in the prediction of reaction kinetics as described elsewhere in this handbook. Methods used include those by Borchart and Daniels [80], Kissinger [73], Freeman and Carroll [81] and Flynn and Wall [82]. Although these techniques are well established and, if used properly, can give pertinent information, their use in pharmaceutical arenas is restricted to dehydration and decomposition. 

Direct application of the differential equation is perhaps the simplest method of obtaining kinetic parameters from non-isothermal observations. However, the Freeman—Carroll difference—differential method [531] has proved reasonably easy to apply and the treatment has been expanded to cover all functions f(a). The methods are discussed in a sequence similar to that used in Sect. 6.2. 

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