## 3 Results
### Thermokinetic Analysis of Experimental Data
Equation (36) can be taken as analogous to Equation (13) for \(g(\alpha)=\alpha\), by which, using Equation (19), the initial temperature \(T_{0}\) can be determined. The obtained data were collected in Table 4, resulting in a confrontation similar to Equation (25)--Figure 6; this is a methodological comparison to Figure 2.  
Based on the data summarized in Table 4, the following form of Equation (25) was determined:  
\[\ln\left(\frac{q}{T_{0}}\right)=-\frac{23212}{T_{0}}+15.6671\quad\left(r^{2}=0.9989\right) \tag{39}\]  
from which \(E=192.99\) kJ-mol\({}^{-1}\) and \(A=10^{6.80}\)\(s^{-1}(\ln A=15.67)\) were calculated assuming \(f(a_{iso})=1\) (Figure 6). The data from Table 4[\(q\); \(T_{0}\)], as presented in Figure 6, confirm sufficient agreement with the data presented in the descriptions of Figures A1-A4, in particular, the value of both the activation energy and the frequency constant are close to the values given in the description of Figure A2. Obviously, the average values given in Table 4[\(\ln A\); \(E\)] are identical to those given in Figure A3a. Continuing the discussion, it should be stated that the experimental data analyzed with Equation (36) are interpreted twice. Originally, according to Equation (36), the intercept being a Doyle approximation allows us to determine the frequency constant \(A\) and the slope (the activation energy \(E\)). Hence, one obtains the \(KCE\) relation and further the isokinetic temperature \(T_{iso}\). On the other hand, the intercept was interpreted according to Equation (19) by iteratively determining \(T_{0}\) for each heating rate (Table 4). Analytical determination of the coordinate \([T_{0}\); \(\alpha_{0}]\) for any kinetic data is complicated. An interesting approach to the problem of determining this values \([T_{0}\); \(\alpha_{0}]\) is the use of a three-parameter equation [19; 23; 27; 37]. Equation (40) allows a different approach than that proposed in the paper [38]:  
\[\ln\alpha=A_{0}-\frac{A_{1}}{T}-A_{2}\ln T \tag{40}\]  
after differentiation of Equation (40) relative to temperature, for \(T=T_{0}\) and introducing Equation (10) we obtain:  
\[\frac{1}{T_{0}}=\alpha_{0}\left(\frac{A_{1}}{T_{0}^{2}}-\frac{A_{2}}{T_{0}}\right) \tag{41}\]  
Figure 6: Experimental data of the thermal decomposition of CaCO\({}_{3}\) in nitrogen for seven heating rates in the coordinate system of Equation (25). and after multiplying both sides by \(T_{0}^{2}\) and after ordering, we get:  
\[T_{0}=\frac{\alpha_{0}\cdot A_{1}}{1+\alpha_{0}\cdot A_{2}} \tag{42}\]  
The approximate methods can be used to determine the coordinates \([T_{0};\ \alpha_{0}]\) from the system of Equations (40) and (42). After introducing Equations (42)-(40) for \(T=T_{0}\) we obtain:  
\[\frac{1}{\alpha_{0}}+\ln\alpha_{0}+A_{2}\mathrm{ln}\bigg{(}\frac{\alpha_{0}A_ {1}}{1+\alpha_{0}A_{2}}\bigg{)}=A_{0}-A_{2},\ \ \ A_{2}\
eq 0 \tag{43}\]  
This way we can skip the tedious analytical determination of higher order derivatives referred to in [38] in order to calculate \(T_{0}\). Table 5 summarizes the results of the experimental data analysis of the thermal decomposition of CaCO\({}_{3}\) in nitrogen for seven heating rates using the three-parameter Equation (40). The initial temperature \(T_{0}\) was determined by an iterative method using the system of Equations (40) and (42), which omits the complicated search for this temperature discussed in works [21; 22; 38]. From the data presented in Table 5 it can be seen that with increasing heating rate the value of isokinetic temperature increases. According to the notation of Equations (10) and (42), it can be concluded that, with an increase in the heating rate, the reaction rate decreases monotonically. For the heating rates analyzed in this paper, the variation of the isokinetic temperature was less than 4% hence, all the data can be taken as one average value. The comparison of the initial temperatures provided in Tables 4 and 5 is surprisingly consistent, despite their determination on two independent paths.