## 4 Results of Kinetic Modeling
### Calcination Kinetics Comparative Analysis
If particle sizes are too large, it has been suggested that the reaction may become limited by mass transport, as opposed to the chemical reaction. However, the influence of particle sizes has been found to be small at high CO\({}_{2}\) partial pressures [33], so intraparticle mass transport effects on the kinetic parameters can be ruled out as the cause of the overestimated kinetic parameters for the experiments under 100% CO\({}_{2}\). [7]. From the comparison of experiments under different atmospheres, it is evident that calculated activation energies can be highly variable. Although there is believed to be a single, intrinsic activation energy for the calcination reaction, values determined experimentally with model-fitting methods have been described as \"effective\" activation energies [48]. For the experiment under 100% CO\({}_{2}\), in particular, the large disparity in activation energies determined by different reaction models (310-1220 kJ mol\({}^{-1}\)) and the lower correlation coefficients suggest that model-fitting methods may be unsuitable for calcination under 100% CO\({}_{2}\). An alternative is the use of model-free methods to determine kinetic parameters, as in the previous study of [PERSON] et al., which used the isoconversional Friedman method to obtain kinetic parameters which vary over the course of the reaction [24]. For both experimental atmospheres, rate constants could not be evaluated using model-fitting methods, as the Coats-Redfern method produced unreasonably large values of _k(T)_. This is because both _k(T)_ and _f(\(\alpha\))_ vary simultaneously under non-isothermal conditions. This causes the Arrhenius parameters for non-isothermal experiments to be highly variable and exhibit a strong dependence on the reaction model (see Table 3) [62]. It has been established that the rate constant cannot be viably predicted by model-fitting methods due to the ambiguity of the kinetic triplet [62]. As suggested previously, isoconversional methods are an alternative means of determining rate constants [24]. As a means of comparison and to aid in a selection of a reaction mechanism, a master plot was produced using the experimental data (non-isothermal experiments carried out under both experimental atmospheres). Examining the _Z(\(\alpha\))_ master plot in Figure 4a for calcination under N\({}_{2}\), there is good correlation between the experimental data points and the R3 geometric model, which is most pronounced for the experiments performed at 10 and 15 K min\({}^{-1}\). The lower heating rates experiments show a poorer fit, which could be the result of heat transfer effects. This result correlates with the mechanism suggested by the Coats-Redfern method, which produces high correlation coefficients for both geometric models R2 and R3, as well as literature results (see Table 2). For calcination under CO\({}_{2}\) (Figure 4b), a good correlation exists for the experimental data points and the Avrami curves A2 and A3, as well as the first-order chemical reaction model F1. Different experimental heating rates are seen to suggest different mechanisms. For example, the 2.5 K min\({}^{-1}\) heating rate suggests the mechanism is A3, while 5 K min\({}^{-1}\) suggests A2 and 10 K min\({}^{-1}\) suggests F1. It is important to note that F1 is a special case of Avrami nucleation [63], as can be seen by the identical shape of the curves, which only differ in magnitude due to the different coefficient \(n\) in \(g(\alpha)\). Differences in the magnitude of the curves for different heating rates could be the result of heat transfer  
Figure 4: Master plot for P-CaCO3 calcination: (**a**) Under 100% N\({}_{2}\); (**b**) under 100% CO\({}_{2}\). effects. Therefore, it can be concluded that the calcination reaction under CO2 is a nucleation process. The reaction mechanism suggested by the master plot generally reflects those in the literature carried out using generalized (non-model-fitting) methods, which are summarized elsewhere [30].