# A Simple Direct Method to Obtain Kinetic Parameters for Polymer Thermal Decomposition
## 2 Approach for a New Direct Method  
Our approach uses a model-fitting method based on a direct differential method [2; 12] using the Arrhenius equation and a reaction-order model to define the kinetic process. This proposed method (PM) enables users to obtain kinetic parameters by resolution of a few simple equations that can be easily automated. The combination of Equations (2)-(4) yields:  
\[\frac{d\alpha}{dt}=A\cdot e^{(-\frac{F}{kT})}\cdot(1-\alpha)^{n} \tag{5}\]  
or  
\[\frac{d\alpha}{dt}\cdot\frac{1}{\left(1-\alpha\right)^{n}}=A\cdot e^{(-\frac {F}{kT})} \tag{6}\]  
Equation (6) can be expressed as Equation (7) using Napierian logarithms,  
\[\ln\frac{\frac{d\alpha}{dt}}{\left(1-\alpha\right)^{n}}=\ln A-\left(\frac{E}{ R\cdot T}\right) \tag{7}\]  
where the logarithmic argument on the left-hand side of the equation is represented by:  
\[F^{\prime}(\alpha)=\frac{\frac{d\alpha}{dt}}{\left(1-\alpha\right)^{n}} \tag{8}\]  
As \(n\), \(A\), \(E\), and \(R\) are considered constant during a particular decomposition reaction, the Equation (7) is a linear equation expressed in terms of the variables \(\ln F^{\prime}(\alpha)\) (y-coordinates) and \(1/T\) (x-coordinates). The activation energy is obtained from its slope, \(s\), by use of Equation (9):  
\[s=-\frac{E}{R} \tag{9}\]  
The slope is calculated by dividing the difference between the y-coordinates of two points by the difference between the x-coordinates of the same two points:  
\[s_{\alpha_{1}-\alpha_{2}}=\frac{\ln F^{\prime}(\alpha_{1})-\ln F^{\prime}( \alpha_{2})}{\frac{1}{T_{  
To evaluate this equation, we need to derivate the experimental \(\alpha\) versus the temperature. Then, by considering \(\alpha_{a}\), \(\alpha_{b}\), and \(\alpha_{c}\) equals to 0.15, 0.45, and 0.65, respectively, we can calculate the associated \(d\alpha/dT\) and \(T\) to obtain n from Equation (11). Once \(n\) is calculated, \(F^{\prime}(\alpha)\) can be evaluated with the desired experimental values of \(\alpha\) and \(d\alpha/dt\), and the calculated \(n\).  
The slope is determined from data corresponding to three intervals in the course of reaction. The points \(\alpha_{1}\) through \(\alpha_{6}\) are defined at 0.1, 0.2, 0.3, 0.6, 0.7, and 0.8, respectively, and the activation energy is calculated by Equation (12):  
\[E=-\left(\frac{\frac{\ln F^{\prime}(\alpha_{1})-\ln F^{\prime}(\alpha_{2})}{ \frac{T}{T_{\alpha_{1}}}-\frac{T}{T_{\alpha_{2}}}+\frac{\ln F^{\prime}(\alpha_ {3})-\ln F^{\prime}(\alpha_{4})}{T_{\alpha_{3}}-\frac{T}{T_{\alpha_{4}}}}+ \frac{\ln F^{\prime}(\alpha_{5})-\ln F^{\prime}(\alpha_{6})}{T_{\alpha_{5}}- \frac{T}{T_{\alpha_{6}}}}}}{3}\right).R \tag{12}\]  
Following the determination of \(E\) and \(n\), the pre-exponential factor is estimated using Equation (13), which follows from Equation (7) by considering the values of \(\alpha_{1}\) to \(\alpha_{6}\) and the average values of \(1/T_{\alpha i}\) and \(\ln(\alpha_{i})\), where \(i\) = 1-6 and \(n_{\alpha}\) = 6 are the numbers of \(\alpha\) evaluated.  
\[A=e^{\frac{\sum_{i=1}^{n_{\alpha}}\left(\frac{\alpha_{i}}{T_{\alpha_{i}}}+ \frac{1}{T_{\alpha_{i}}}+\ln F(\alpha_{i})\right)}{n_{\alpha}}} \tag{13}\]  
The methodology was applied to multiple decomposition reactions by segmenting the mass results. This may be done automatically considering the second derivative of the mass loss or by the visual association of each peak of the mass versus temperature derivative (\(dm/dT\)) curve with a reaction. The latter procedure was chosen for the cases analyzed in this study for the PVC, but the first one is described below for the poly(methaphenylene isophthalamide) (NOMEX) as it undergoes thermal decomposition in several overlapping steps.