## 3 Results  
### Results of the Calculations  
For kinetic models (Table 11) and exponents \(n\) for P4 (\(n=\)1/4), P3 (\(n=\)1/3), P2 (\(n=\) 1/2), P2/3 (\(n=\) 3/2), D1 (\(n=\) 2) and R1 (0 th order, \(n=\) 1), the left-hand side of Equation (13) is as follows:  
\[n\ln\langle\alpha\rangle=-\frac{E}{RT}+C \tag{27}\]  
which reduces the listed models to 0 th order kinetics. Similarly for models F1 (\(n=\) 1), A4 (\(n=\) 1/4), A3 (\(n=\) 1/3), A2 (\(n=\frac{1}{2}\)):  
\[n\ln[-\ln(1-\langle\alpha\rangle)]=-\frac{E}{RT}+C \tag{28}\]  
For models R2 or R3 (\(n=\) 1) we obtain, respectively:  
\[\ln\biggl{(}1-\sqrt{1-\langle\alpha\rangle}\biggr{)}=-\frac{E}{RT}+C \tag{29}\]  
or:  
\[\ln\biggl{(}1-\sqrt[3]{1-\langle\alpha\rangle}\biggr{)}=-\frac{E}{RT}+C \tag{30}\]For the popular D3 model:  
\[g(\alpha)=\left[1-\sqrt[3]{1-\langle\alpha\rangle}\right]^{2} \tag{31}\]  
after logarithmizing Equation (31) it transforms into R3 (Equation (30)). For the control test, the reference model Equation (27) for \(n=1\) and for the assumed conditions \((E,\ A,\ q)\), Figure 1. Table 1 shows the calculated activation energies, integration constants (intercept of linear equations), temperatures and the corresponding conversion degrees as \([T_{0};\ \alpha_{0}]\). The numerical data summarized in Table 1 were verified using Equation (25). A graphical representation of the relationship obtained is presented in Figure 2. From the parameters of the obtained linear relationship:  
\[\ln\left(\frac{q}{T_{0}}\right)=-\frac{23400}{T_{0}}+32.144\quad\left(r^{2}=1. 0000\right) \tag{32}\]  
The activation energy \(E=199.53\) kJ\(\cdot\)mol\({}^{-1}\) and the pre-exponential factor \(A=10^{13.966}\) s\({}^{-1}\) were determined. As one can easily see, both determined parameters have slightly lower values than those used for calculations (\(A=10^{14}\) s\({}^{-1}\), \(f(a_{0})=1\)). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Heating Rate** & \(E\)**, k j mol\({}^{-1}\)** & **Intercept C** & \(T_{0}\)**K** & \(\alpha_{0}\) **from** & **In\(A\)** \\ \(q\)**, K s\({}^{-1}\)** & **acc. Equation (27)** & **acc. Equation (27)** & **from Equation (23)** & **Equations (22) or (24)** & **(A in s\({}^{-1}\))** \\ \hline
0.01 & 207.03 & 40.890 & 557.2 & 0.02238 & 33.762 \\
0.03 & 207.13 & 39.822 & 571.5 & 0.02294 & 33.738 \\
0.1 & 207.23 & 38.651 & 587.9 & 0.02359 & 33.718 \\
0.3 & 207.33 & 37.583 & 603.7 & 0.02422 & 33.686 \\
0.5 & 207.38 & 37.087 & 611.4 & 0.02450 & 33.688 \\ average \(\pm\) standard deviation & 207.22 \(\pm\) 0.14 & 38.807 \(\pm\) 1.569 & 586.3 \(\pm\) 22.4 & 0.02353 \(\pm\) 0.00088 & 33.718 \(\pm\) 0.029 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of calculated parameters of Equation (27) for five assumed heating rates and \(n=1\big{(}r^{2}=1\big{)}\). Figure 1: Kinetic approach Equation (27) for \(n=1\) for the assumed five heating rates. The intercept value in Equation (32) is dependent on deviations of the activation energy and errors resulting from the observed variability of constants occurring in the kinetic functions. Figure 3 presents the relationship (30) function D3 (Jander) but as R3: \(\ln\left(1-\sqrt[3]{1-\left\langle\alpha\right\rangle}\right)\) (R3). Table 2 gives as an example the integration constant calculated according to Equation (14), but also comparatively Equations (16), (23) and (24) were used-all the mentioned formulas turn out to be equivalent. Figure 3: The relationship for function D3 (Jander) as Equation (30) for five different heating rates. Figure 2: Data presented in Table 1 in the Equation (25) coordinate system. Thus, if this model accepts the coordinates \([T_{0};\,\alpha_{0}]\) then by force of facts Equation (25) remains unchanged in the analytical form of Equation (32). Model D2 is very interesting because starting from infinitesimally small conversion degrees it goes from R1 through D1 to D2.