Kinetic method. Friedman

Several different mathematical procedures have been suggested for extracting kinetic information from sets of a,T data obtained at different heating rates, p. Friedman s method [54] is to plot ln(do5 d/)i against 1/7, measured at the same value of from a,T curves at different heating rates Pi (or different isothermal reaction temperatures, T). The parallel lines obtained have slopes = -E /R and different intercepts = ln[ f( )J. A value for A is obtained by extrapolation of a plot of the intercept against to a = 0. 

To evaluate the apparent activation energy, the isoconversional methods are use as suitable analysis procedures. These methods are based on the assumption that at a constant extent of conversion degree (a), the decomposition rate da/dt is a function only of the temperature. In methods developed by Friedman and Flynn-Wall-Ozawa, linear functions are obtained from which slopes the apparent activation energy at constant conversion a is achieved. In the free kinetic method set by Kissinger is calculated from the slope of the linear function takes into consideration the relationship between the heating rate and peak temperature of the first-derivative thermogravimetric curve [97]. 

Thus even approximate analytical solutions are often more instructive than the more accurate numerical solutions. However considerable caution must be used in this approach, since some of the approximations, employed to make the equations tractable, can lead to erroneous answers. A number of approximate solution for the hot spot system (Eq 1) are reviewed by Merzhanov and their shortcomings are pointed out (Ref 14). More recently, Friedman (Ref 15) has developed approximate analytical solutions for a planar (semi-infinite slab) hot spot. These were discussed in Sec 4 of Heat Effects on p H39-R of this Vol. To compare Friedman s approximate solutions with the exact numerical solution of Merzhanov we computed r, the hot spot halfwidth, of a planar hot spot by both methods using the same thermal kinetic parameters in both calculations. Over a wide range of input variables, the numerical solution gives values of r which are 33 to 43% greater than the r s of the approximate solution. Thus it appears that the approximate solution, from which the effect of the process variables are much easier to discern than from the numerical solution, gives answers that differ from the exact numerical solution by a nearly constant factor... 

It may also be surprising how easily this racemization may occur. Friedman and Liardon (126) studied the racemization kinetics for various amino acid residues in alkali-treated soybean proteins. They report that the racemization of serine, when exposed to 0.1M NaOH at 75°C, is nearly complete after just 60 minutes. However, caution must be used when examining apparent racemization rates for protein-bound amino acids. Liardon et al. (127) have also reported that the classic acid hydrolysis, employed to liberate constituent amino acids, causes amino acids to racemize to various degrees. This will necessarily result in D-isomer determinations that are biased high. Widely applicable correction factors are not possible since the racemization behavior of free amino acids is different from that of amino acid residues in proteins (which can be further affected by sequence). Of course, this is not a problem for free amino acid isomer determinations since the acid hydrolysis is unnecessary. Liardon et al. also describe an isotopic labeling/mass spectrometric method for determining true racemization rates unbiased by the acid hydrolysis. For an extensive and excellent review of the nutritional implications of the racemization of amino acids in foods, the reader is directed to a review article written by Man and Bada (128). 

In order to assess the activation energy for development of a reasonable model for kinetic analysis of pristine PE and PE-n-MMT thermal degradation processes, a few evaluations by model-free methods have been done as the starting point. As an example, the results of a model-free Friedman analysis for thermal degradation of PE, where the activation energy is a function of partial mass loss change [24], are shown in Figure 5. 

Static resonance Raman spectroscopy has emerged as an important and powerful method for investigating details of structure in compounds with strong absorption bands, such as hemoglobin (see, for example Asher [15]), and these advantages have been extended to kinetic studies by Dr. T.G. Spiro of Princeton, and Dr. J.M. Friedman of Bell Laboratories. It seems unlikely at present that their methods will... 

Budrugeac et al. [123] examined the kinetics of the non-isothermal crystallization of (GeS2)o.3(Sb2S3)o.7 by employing the methods of Friedman and of invariant kinetic parameters and demonstrated that the process can be treated as a single step. A more complex kinetic situation has been encoimtered by Thomas and Simon in re-crystallization of nickel sulfide from the a- to P-form. Their analysis yielded evidence of at least two steps involved in the overall process [124]. 

Table 4.2 Kinetic parameters obtained from Friedman, Kissinger, and Ozawa methods [12].

Model-free methods evaluations were chosen as the starting points in kinetic analysis of neat PP and PP-mPP with 7% Cloisite 20A for determining the activation energy in the development of the model. Figure 6 shows a corresponding Friedman analysis, where the activation energy is a function of partial mass loss change [16]. 

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. 

Kinetic analyses of multiple nonisothermal, also referred to as multiple heating rate, runs are most commonly performed by using the methods of Friedman, Ozawa, and Flynn and Wall. However, application of the Kissinger method (Kissinger 1957) is discouraged because the method yields a single value of... 

Where a is the extent of decomposition, Z is a pre-exponential factor, E is the activation energy, R the universal gas constant (8.3154 Jmol K 0> and f(a) depends on the decomposition mechanisms. E, Z, and a) are commonly called the kinetic triplet [28]. There are different kinds of methods to obtain the kinetic triplet reported in the scientific literatme among which are those reported by Flynn-Wall-Ozawa [29], Friedman [30], Kissinger [21], or Coats-Redfem [31], with different proposed equations presented to solve Eq. (7.1). 

Friedman s isoconversional method [14] involves an Arrhenius analysis at constant levels of conversion, and we determined the apparent first-order frequency factor and activation energy at 1 % intervals using both LLNL and AKTS kinetics analysis programs. 

TGA provides general information about the overall reaction kinetics and is commonly used to estimate kinetic parameters. A model-free kinetic approach based on the isoconversional Friedmans method was used to analyze data obtained by TGA. The method gives simultaneously the activation energy and pre-exponential factor by assuming a reaction order without knowing the rate dependence on conversion. 

---