Impact of Thermodynamic Constraints on the Lability of Activation Energy as a Function of Conversion Degree

Abstract

The subject concerns the determination of activation energy under dynamic conditions using two theoretical isothermal models, and subsequently experimental data, with reference to the α–T relationship matrix. In recent years, the Vyazovkin method, classified as one of the isoconversional variants, has gained the greatest recognition. Comparison was made between two isothermal models of the thermal dissociation of calcite, which in chronological terms are associated with a kinetic–nucleation reaction/process (the H-CL, as a kinetic model) and a kinetic–desorption reaction/process (the V, as a thermodynamic model). A comparison of numerical values, understood as the logarithm of the reaction/process rate with respect to temperature, shows correspondence in the temperature range up to the equilibrium temperature. The H-CL model is characterized by a strong dominance of the nucleation process relative to the chemical reaction, whereas the V model exhibits a certain type of balance resulting from the course of the chemical decomposition reaction combined with the transformation of a metastable oxide into a crystalline form. It was confirmed that both models describe the same phenomenon within the transformation process, which implies that for a constant conversion degree, the proportions of the chemical reaction and the physical process vary. Pointwise with increasing temperature, the H-CL model leads to a minimum activation energy E → 0, whereas the V model reaches a negative activation energy E < 0. In both cases, the apparent activation energy summed over the process is constant, and the assigned conversion degree, treated as isoconversional, remains fixed and corresponds to the assumed activation energy of the completed reaction/process. Several simple methods for its determination under dynamic/isoconversion conditions are used.

1. Introduction and Kinetic Formalism

In analyses of the thermal dissociation of broadly understood solid phases, or directly of pyrolysis under isothermal conditions, the starting equation is the General Rate Equation (GRE):

$${\displaystyle \frac{d\alpha }{dt}}=k·f(\alpha )$$

(1)

after expressing the Arrhenius rate constant in the usual form as

$$k=A \mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{E}{RT}}\right)$$

(2)

and taking the logarithm leads to the linear relationship:

$$\mathrm{l}\mathrm{n}{\displaystyle \frac{d\alpha /\left(dt\right)}{f\left(\alpha \right)}}=\ln A-{\displaystyle \frac{E}{RT}}$$

(3)

An extensive development of the applicability of Equation (3) to extended forms of the kinetic function $f\left(\alpha \right)$ is attributed to Koga et al. [1,2,3], which can be expressed in an expanded form of Equations (1) and (3) as

$${\displaystyle \frac{d\alpha }{dt}}=A \mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{E}{RT}}\right)·f\left(\alpha \right)·h\left({\displaystyle \frac{P_i}{P_{eq}}}\right)$$

(4)

where $h\left({\displaystyle \frac{P_i}{P_{eq}}}\right)$ denotes a function accounting for the partial pressure $P_i$ relative to the equilibrium pressure $P_{eq}$ at a given temperature; the analytical forms of this function are presented in [3].

Equation (3), known as the Friedman equation [4,5] (FR method), can be applied under dynamic conditions for a linear temperature increase q:

$$q={\displaystyle \frac{dT}{dt}}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$$

(5)

$$\mathrm{l}\mathrm{n}\left(q{\displaystyle \frac{d\alpha }{dT}}\right)=\ln A·f(\alpha )-{\displaystyle \frac{E}{RT}}$$

(6)

In the case of the non-linear method, it leads to results similar to those of the Kissinger–Akahira–Sunose (KAS) method:

$$\ln {\displaystyle \frac{q}{T^2}}=\ln {\displaystyle \frac{AR}{Eg\left(\alpha \right)}}-{\displaystyle \frac{E}{RT}}$$

(7)

and Flynn–Wall–Ozawa (FWO) [6]:

$$\log q=\log {\displaystyle \frac{AE}{Rg\left(\alpha \right)}}-2.135-0.4567{\displaystyle \frac{E}{RT}}$$

(8)

In Equations (7) and (8), classified as model-free approaches, the necessity of integration is omitted; however, as a consequence, the value of the pre-exponential factor depends on the knowledge of the kinetic model $g\left(\alpha \right)$.

When moving to integral methods, extensive compilations of numerous approaches emerge, primarily focused on the analytical solution of the non-elementary temperature integral. The most widely used solution of the temperature integral is attributed to the simplified Coats–Redfern method [6], and the starting equation is expressed in the form:

$$g\left(\alpha \right)={\int }_0^{\alpha }{\displaystyle \frac{d\alpha }{f\left(\alpha \right)}}={\displaystyle \frac{A}{q}}{\int }_0^{T_{\alpha }}\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{E}{RT}}\right)dT={\displaystyle \frac{A}{q}} I(E,T_{\alpha })$$

(9)

and, by applying the approximation, we obtain

$$I\left(E,T_{\alpha }\right)={\displaystyle \frac{R{T}_{\alpha }^2}{E}}\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{E}{R{T}_{\alpha }}}\right)$$

(10)

The general form can be expressed as

$$g\left(\alpha \right)={\displaystyle \frac{AE}{Rq}} {\int }_x^{\infty }{\displaystyle \frac{e^{-x}}{x^2}}dx={\displaystyle \frac{AE}{Rq}}p\left(x\right)$$

(11)

In Equations (7)–(9), $g\left(\alpha \right)$ denotes the weight integral, and the problem of solving/approximating the integral $I(E,T_{\alpha })$ has been the subject of studies [5,7,8,9,10,11,12,13], where $p\left(x\right)$ represents one of the known approximate solutions of the integral, in particular, the one recommended in [7]. Consequently, a comparison of Equations (9) and (11) yields

$$I\left(E,T_{\alpha }\right)={\displaystyle \frac{E}{R}}p\left(x\right)\mathrm{where} x={\displaystyle \frac{E}{R{T}_{\alpha }}}$$

(12)

In recent years, for dynamic conditions and isoconversional temperature profiles ($\alpha =$ constant), the Vyazovkin method has been widely recognized. It is described in detail in the book [14], and of particular interest is the compilation of publications under his editorship, presenting numerous studies that elucidate the influence of mechanisms accompanying thermolysis reactions through processes of a physical nature [15].

In its simplest formulation, the fundamental principle of the Vyazovkin method is the equality at individual measurement points for $\alpha =$ constant and $q=$ var expressed as $A·E/R$ = constant for i and j where i < j:

$$g\left(\alpha \right)={\displaystyle \frac{AE}{{Rq}_i}}p\left(x_i\right)={\displaystyle \frac{AE}{{Rq}_j}}p\left(x_j\right)=\cdots {\displaystyle \frac{AE}{{Rq}_n}}p\left(x_n\right), x={\displaystyle \frac{E}{R{T}_{\alpha }}}$$

(13)

The algorithm based on an optimization procedure for N-heating rate is presented in [16].

Focusing on a single element of this method and using the Coats–Redfern approach, it is possible to calculate the activation energy for two measurement points with indices in the normalized sense expressed by Equation (13) [17]:

$$\ln {\displaystyle \frac{q_i}{q_j}}+2\ln {\displaystyle \frac{T_i}{T_j}}={\displaystyle \frac{E}{R}}\left({\displaystyle \frac{1}{T_i}}-{\displaystyle \frac{1}{T_j}}\right), \alpha =\mathrm{constant}, q=\mathrm{var}$$

(14)

On the other hand, the temperature criterion, concerning small conversion degrees as presented in [18,19], allows the determination of activation energy with a tolerance of up to 20%:

$$\ln \alpha =-{\displaystyle \frac{E}{RT}}+\mathrm{constant}, 0<\alpha <\approx 0.2$$

(15)

In Equations (2)–(4), (6)–(8), (10) and (15), regardless of the method, the dimensionless activation energy ${\displaystyle \frac{E}{RT}}$ appears explicitly, which in particular cases denotes for isothermal conditions $T=$ const, and for dynamic conditions $T=$ var, whereas $T_{\alpha }$ represents the temperature corresponding to the isoconversional level ($\alpha =$ constant).

The essence of the Vyazovkin method is to demonstrate the variability of activation energy as a function of conversion degree. This has also been observed by other authors, for example Sbirrazzuoli [20], although the activation energy can remain constant over a wide range of conversion degrees ($\alpha$ ≅ 0.1–0.9) [2,3]. In particular, pronounced non-linear dependencies are evident in the pyrolysis of biomass [21] or higher alcohols [22]. If the phenomenon occurs under isoconversional dynamic conditions, it should also be apparent under isothermal conditions. Considering the potential for extensive studies on the thermal dissociation of calcite, given the wealth of experimental research presented by Maciejewski et al. [23,24,25,26,27,28] and Koga et al. [2,3,29,30,31], this problem has been revised. The topic is important due to the attractiveness of the calcium looping (CaL) method, which relies on carbonation and calcination processes [32,33,34,35,36]. Recent studies have also highlighted the issue of excessively rapid sorbent sintering [37].

2. Basis of the Analysis and Aim of the Work

From a historical perspective, studies on modeling the reaction/process rate of calcite in a kinetic–thermodynamic sense include the works of Hyatt et al. [38], Valverde [39], Valverde et al. [40,41,42], and more recently Cai and Li [43]. Owing to the specific characteristics of the subsequent analysis, two models are considered further: the H-CL model, based on [38,43] and referred to as the kinetic–nucleation model, and the V model, referred to as the kinetic–desorption model [33,34,39,40,41,42]. In both cases, admissible simplifications are introduced in the analysis.

It was decided to return to the basic kinetic models of the thermal dissociation of calcite under isothermal conditions, determining how the activation energy—often referred to as apparent—varies with conversion degree. In symbolic form, the relationship $E=E\left(\alpha \right)$ is investigated, which has been substantially developed through the application of the Vyazovkin method [14,15].

The basic models of the thermal dissociation of CaCO₃ (T = constant) account for reaction/process rate relationships as a function of individual rate constants in the atmosphere of the gaseous product, i.e., CO₂. In the two models discussed below, their characteristic feature is the mechanism involving the occurrence of metastable calcium oxide (CaO*).

As a basic assumption, Arrhenius-type kinetic parameters were adopted, namely $E=$ 191.0 kJ·mol⁻¹ and $\ln A=$ 15.4 ($A$ in s⁻¹), which simultaneously define the reference rate of the reaction/process. It was further assumed that the relevant temperature range is $T_{cr}<T<T_{eq}=$1166.13 K [44]. Here, $T_{cr}$ denotes the temperature at which the principal condition ${\alpha }_{eq}>\alpha$ is satisfied. Accordingly, in view of Equations (2)–(4), (6)–(8), (10) and (15), a functional scaling of the form $\mathrm{l}\mathrm{n}$(kinetic expression) vs. 1/$T$ is assumed to be valid for determining the activation energy. For isothermal conditions, the interpretation was carried out within the isoconversional concept, characteristic of studies under dynamic conditions. Thus, the temperature criterion was employed in the framework of the Vyazovkin repetition concept, consistent with Equation (14).

The aim of the present work is to analyze the dependence of the apparent activation energy on the conversion degree E = E(α) using two theoretical models of calcite dissociation (H-CL and V) under isothermal conditions. Subsequently, using the procedure calculated according to [17], a comparison of the activation energies determined from Equation (13) was performed for dynamic conditions.

3. Models of the Thermal Dissociation of CaCO₃ (Calcite)

3.1. Model H-CL

The reaction/process rate should be understood as mass loss over time [38]. The model hereafter denoted as H-CL is based on the equilibrium course of the reaction/process in both directions, indexed as forwards (1, n) and backwards (2, −n) [H,CL], where the index (n) refers to nucleation and (−n) to denucleation.

$${\mathrm{CaCO}}_3\stackrel{k_1}{\leftrightarrow }\underset{k_2}{\leftrightarrow }\mathrm{CaO}*+{\mathrm{CO}}_2$$

$$\mathrm{CaCO}*\stackrel{k_n}{\leftrightarrow }\underset{k_{-n}}{\leftrightarrow }\mathrm{CaO}$$

Neglecting the surface roughness conversion factor, the following equation was proposed [38]:

$$r={\displaystyle \frac{1-\frac{P_i}{P^{\varnothing }}}{\frac{k_2}{k_1{k}_n} \frac{P_i}{P^{\varnothing }}+\frac{1}{r_0}}}, {\mathrm{s}}^{-1}$$

(16)

where $r_0$ refers to decomposition in pure nitrogen, and

$${\displaystyle \frac{1}{r_0}}={\displaystyle \frac{1}{k_1}}+{\displaystyle \frac{1}{k_n}}+{\displaystyle \frac{k_{-n}}{k_1{k}_n}}$$

(17)

In paper [43], Equation (16) was reduced to the following equation:

$$r=k_1{\displaystyle \frac{1-\frac{P_i}{P_{eq}}}{1+\frac{k_1}{k_n}\left(1+\frac{P_i}{P_{eq}}\right)+\frac{k_{-n}}{k_n}}}, {\mathrm{s}}^{-1}$$

(18)

For $P_i$ = 0, Equation (18) reduces to Equation (17) in the form $r=r_0.$

In the original work [43], the ratio ${\displaystyle \frac{k_{-n}}{k_n}}$ was neglected as a quantity close to zero. In this framework, however, according to [45], this assumption becomes valid only for $T\gg T_{eq}$, whereas the equality $k_{-n}=k_n$ is observed at $T\cong$ 860 K (

Table 1. Compilation of Arrhenius constants for the reaction/process of calcite decomposition according to [45,46].

Rate Constant, s−1	E, kJmol−1	lnA,A in s−1
$k_1$	191.0	15.4
$k_n$	0 to 55.70, $k_n=A_n=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$	−6.20
$k_{-n}$	−52.41	−13.53

). Ultimately, when the reaction/process proceeds in the atmosphere of its own gaseous products, the ratio

$${\displaystyle \frac{P_i}{P_{eq}}}={\displaystyle \frac{\alpha }{{\alpha }_{eq}}}=m, T=\mathrm{constant}$$

(19)

and the proposed notation m can be regarded as the standard conversion degree when $0<\alpha <{\alpha }_{eq}$.

For calculation purposes, Equation (19) is more useful in the following form, where the equilibrium conversion degree $\left({\alpha }_{eq}\right)$ was adopted from [45]:

$$m=\alpha ·\mathrm{e}\mathrm{x}\mathrm{p}\left(-18.095+{\displaystyle \frac{21039.57}{T}}\right), \alpha =\mathrm{constant}$$

(20)

Equation (20) is valid for $T>T_{cr}$ due to the constraints $(\alpha ,{\alpha }_{eq}<1$), which follow from Equation (19) and $0<\alpha <{\alpha }_{eq},$ and thus $0<m\le 1$. In particular, when $m=1$, the system reaches thermodynamic equilibrium, and consequently $r=0.$

For the data in Table 1, Equations (18) and (20) take the following form:

$$r=(1-m){\displaystyle \frac{\mathrm{e}\mathrm{x}\mathrm{p}\left(15.4-\frac{22973.3}{T}\right)}{1+\left(1+m\right)\exp \left(21.6-\frac{22973.3}{T}\right)+\mathrm{e}\mathrm{x}\mathrm{p}\left(-7.322+\frac{6293.82}{T}\right)}}$$

(21)

The compilation of functions according to Equation (21) is shown in Figure 1a in the form $\ln r \mathrm{v}\mathrm{s}. {\displaystyle \frac{1}{T}},$ whereas Figure 1b presents an alternative use of this relationship, namely $\ln {\displaystyle \frac{r}{1-m}}\mathrm{v}\mathrm{s}. {\displaystyle \frac{1}{T}}$. The latter case corresponds to the approach presented in Equation (3) and, after logarithmization, also to Equation (4).

Figure 1a suggests that at higher temperatures, in the vicinity of the equilibrium temperature $\left(T_{eq}\right)$, the reaction/process rate approaches a constant value, as $E\to 0.$ In contrast, Figure 1b, for which temperature constraints were maintained, allows for the determination of new Arrhenius parameters.

Figure 1a provides a graphical representation of the relationship in Equation (21), aimed at showing the general variation with temperature for a constant conversion degree, noting the temperature profile constraints above T_cr. It is evident that, as the system approaches the equilibrium temperature, the region enters a statistical insignificance zone due to the lack of sufficient variability in this range. For the determination of apparent or appearing activation energy, the functional scaling was modified as shown in Figure 1b, allowing the right-hand side of Equation (21) to be expressed under the logarithm of a new reaction/process rate, here denoted as a new lnk₁. Graphically, the KCE equation (ln A vs. E) illustrated in Figure 1b defines the isokinetic temperature, T_iso = 1307.4 K, which lies outside the temperature range used in the calculations (T_iso > T_eq). The differentiation corresponding to the linearity of the KCE indicates that the activation energy varies with conversion degree.

Referring to previous considerations [45], one can return to the nucleation process. For this purpose, the equation from [46] was adopted in the following form:

$$\ln k_n=-6.20-{\displaystyle \frac{9.11·{10}^{10}}{\begin{array}{c}T(174.92·{10}^3-150T+RT\ln \alpha {)}^2\end{array}}}$$

(22)

Figure 2 shows the constant rate of nucleation vs. ${\displaystyle \frac{1}{T}}$ both according to Equation (22) (Figure 2a) and with reference to [45] (Figure 2b).

Figure 2b differs from that presented in [45] in that, for $T>T_{eq}$, Equation (22) does not cover this range. Comparable Arrhenius plots are obtained for the isoconversion levels.

3.2. Model V

The model based on work [39] and the other ones [40,41,42] treats the reaction/process course as initiated by chemical decomposition. The model, hereafter denoted as V, is based on the equilibrium course of the reaction/process in both directions, indexed as forwards (1, d) and backwards (2, a), where the index (d) refers to desorption and (a) to adsorption:

$${\mathrm{CaCO}}_3+\mathrm{L} \stackrel{k_1}{\leftrightarrow }\underset{k_2}{\leftrightarrow }\mathrm{CaO}*+\mathrm{L}({\mathrm{CO}}_2)$$

and desorption/structural transformation:

$$\mathrm{CaO}*+\mathrm{L}({\mathrm{CO}}_2) \stackrel{k_d}{\leftrightarrow }\underset{k_a}{\leftrightarrow } \mathrm{CaO}+\mathrm{L}+{\mathrm{CO}}_2$$

where L denotes the active site where calcination takes place.

Adopting the equation from [39], in which Equation (19) is directly applied, it takes the following form:

$$r=k_1{\displaystyle \frac{1-m}{1+m·\mathrm{e}\mathrm{x}\mathrm{p}\left(- \frac{{∆}_{1}G+ {∆}_{*}G}{RT}\right) }}$$

(23)

The individual Gibbs functions are based on the temperature dependence $∆G=∆H-T∆S,$ which have been determined in analytical form:

$${∆}_{1}G=208·{10}^3 - 155·T, \mathrm{J}·{\mathrm{mol}}^{-1}$$

(24)

$${∆}_{*}G=50·{10}^3-86·T, \mathrm{J}·{\mathrm{mol}}^{-1}$$

(25)

Equation (24) is associated with the condition that the activities of the stable solid reactants are equal to 1, whereas the activity of the metastable CaO* form is given by the following [39,47]:

$$a^{*}=\exp {\displaystyle \frac{{∆}_{*}G}{RT}},$$

(26)

It should be noted that $a^{*}=$ 1 for $T=$ 581.40 K, but it approaches 0 as the temperature increases.

By adopting Equations (24) and (25) and taking k₁ as before (Table 1) together with Arrhenius parameters [48], Equation (23) can be expressed as follows:

$$r=(1-m){\displaystyle \frac{\mathrm{e}\mathrm{x}\mathrm{p}\left(15.4-\frac{22973.3}{T}\right)}{1+m·\exp \left(28.987-\frac{31032}{T}\right)}}$$

(27)

The compilation of functions according to Equation (27) is shown in Figure 3a in the form $\ln r \mathrm{v}\mathrm{s}.1/T,$ whereas Figure 3b corresponds to the form $\mathrm{l}\mathrm{n}{\displaystyle \frac{r}{1-m}} \mathrm{v}\mathrm{s}. 1/T$.

From a formal point of view, identical procedures were applied in this case for the H-CL model, with the necessity of accounting for the critical temperature T_eq. Figure 3a suggests that at higher temperatures, in the vicinity of the equilibrium temperature T_eq, the reaction/process rate decreases, as E < 0, as observed in [39]. In contrast, Figure 3b allows for the determination of new Arrhenius parameters. Graphically, the KCE equation (lnA vs. E) defines the isokinetic temperature, here T_iso = 884.40 K, which lies outside the analyzed temperature range because m > 1. Similar to the H-CL model, the differentiation corresponding to the linearity of the KCE indicates that the activation energy varies with the conversion degree.

3.3. Comparison of the H-CL and V Models

In the comparison of the models, Equation (19), discussed in [45,46], was used. From this perspective, the expression used here $\left(1-m\right)={\displaystyle \frac{{\alpha }_{eq}-\alpha }{{\alpha }_{eq}}}$ corresponds to a first-order kinetic function with a thermodynamic constraint, and thus it is necessary to enforce the condition ${\alpha }_{eq}-\alpha >0.$ This requirement limits the range of temperatures, similar to the constraints applied in the H-CL model. On the other hand, the $m={\displaystyle \frac{P_i}{P_{eq}}}=$ constant can also be considered, as presented in [39].

Returning to the argumentation of Equation (19), the linkage between partial pressure and conversion degree follows from comparing the ratio expression in a mathematical form identical to that of the equilibrium constant [49,50]: ${\displaystyle \frac{K_{\alpha }}{K}}={\left({\displaystyle \frac{\alpha }{{\alpha }_{eq}}}\right)}^{\nu }$ (for calcite $\nu =$ 1). Under isothermal conditions, the denominator is a function of temperature only. For example, the equilibrium constant of thermal dissociation of SO₃ is expressed by a three-parametric temperature-dependent equation [51], similarly to what is proposed for conversion degree in [44] (to be used later). In endothermic decomposition processes, the same negative signs are retained for the temperature-dependent terms.

Calculations according to Equations (21) and (27) were performed for constant conversion degree ($\alpha$ = 0.05–0.9) and temperatures in the range 900–1300 K, followed by selection of the data set. According to Equation (20), the equality $\alpha =m$ holds only for $T\ge T_{eq}$. Consequently, below this temperature the conversion degree is very small, whereas above it $\alpha$ > 1. The application of isolines for $\alpha =$ constant under isothermal conditions defines the upper limit of the analysis as the equilibrium temperature $(T_{eq}=$ 1166.13 K$)$ while the lower limit is determined individually from Equation (20). Taking into account the temperature profiles shown in Figure 1 and Figure 3, both cases exhibit approximately straight lines with similar slopes. According to Equation (15), this behavior determines the activation energy of an apparent or appearing type, which differs from the assumed value (here $E=$ 191 kJ·mol⁻¹).

In the case of the H-CL model, above the equilibrium temperature the activation energy decreases significantly, tending toward $E=$ 0 and for $\alpha =m=$ 0.95, $\ln r=$ constant = −6.2 (s⁻¹) and this value was determined as the maximum for the nucleation rate constant $\mathrm{l}\mathrm{n}{k}_n$ [45].

Adopting, in accordance with [43], the simplification ${\displaystyle \frac{k_{-n}}{k_n}}\to 0$ and reducing the denominator by 1, we obtain

$$r\cong {\displaystyle \frac{1-m}{1+m}}{k}_n{│}_{T_{eq}}$$

(28)

and thus $\ln r=\ln k_n$ + constant, which confirms the previous calculations [45,46] and implies $E\to 0$ (Figure 1).

In contrast, in the case of the V model, in the temperature range above the equilibrium temperature the slope changes, indicating a negative activation Energy $E<$ 0, as demonstrated in [39].

For the present considerations at high temperatures (by reducing the denominator by 1), Equation (27) can be expressed in the form:

$$r\cong {\displaystyle \frac{1-m}{m}}{A}_1\mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{-E_1+{∆}_{1}G+{∆}_{*}G}{RT}}\right)$$

(29)

and finally, for the enthalpies determined in Equations (24) and (25), we obtain ${E=E}_{app}=$ 191 − (208 + 50) = −67 kJ·mol⁻¹, in [44] for denucleation $E=$ −52.4 kJ·mol⁻¹ (Table 1 and Figure 3a). In this case, at the equilibrium temperature, according to Equation (29), $r=0,$ when $m=$ 1.

An identical result is obtained after applying similar simplifications using the form expressed with specific values, and from Equation (27) the same result follows ${(E}_{app}=$ −67 kJ·mol⁻¹).

As follows from the formal comparison of both models, in the high-temperature range the H-CL model favors the nucleation process, whereas the V model favors desorption, i.e., the evacuation of CO₂ into the space above the solid phase, which in turn initiates the nucleation process.

A formal comparison of the logarithms of the reaction/process rates is presented in Figure 4.

In the low-temperature range, both models yield similar reaction/process rates; the correlation equation shown in Figure 4 is characterized by a similar slope and a nonzero intercept. At higher temperatures, the V model significantly exceeds the rate values obtained for the H-CL model. In the V model, according to Equation (28), $\ln r \propto {\displaystyle \frac{E_{app}}{RT}},$ in contrast to the H-CL model, where, from the linkage of Equation (21) with Equation (27), $\ln r \propto$ constant, and in the temperature range approaching the equilibrium temperature the activation energy tends to $E=0$.

The comparison shown in Figure 5 presents the activation Energy $E_{app} \mathrm{v}\mathrm{s}. \alpha$ under isothermal conditions. The specific values are given in Figure 1b (H-CL model) and Figure 3b (V model) in the form of the KCE. In this work, $E_{app}$ denotes the activation energy, which was determined for the constant conversion degree isoline.

In a limited range of conversion degree, Figure 5 resembles process of the “forwards–backwards” type, where both curves tend toward a common horizontal asymptote. Moreover, this asymptote is, to a very good approximation, the arithmetic mean of the two determined apparent activation energies, $E_{app}\cong$ 216 kJ·mol⁻¹. Thus, the sum for both models is constant and close to the assumed activation energy, here $E=$ 191 kJ·mol⁻¹. Consequently, Figure 5 shows that the H-CL and V models provide a graphical representation of the same reaction/process, with a varying contribution of the individual pathways assigned to the imposed conversion degree.

According to this interpretation, the apparent activation energy depends not only on the conversion degree but also on the physical processes accompanying the chemical reaction; that is, the transformation processes determine it.

Assessing the variability of the activation energy under isothermal conditions, when $m={\displaystyle \frac{\alpha }{{\alpha }_{eq}}}$, and for the assumed activation energy $E=$ 191 kJ·mol⁻¹ in the range 0.05 $\le$ $\alpha \le$ 0.9, the H-CL model shows a monotonic increase in the activation energy up to approximately 94 kJ·mol⁻¹. In contrast, for the V model the activation energy decreases monotonically below the assumed value and reaches a minimum of about 116 kJ·mol⁻¹. With good approximation, the reduced values of the activation energy are also observed for the KCE equations calculated here (Figure 1b and Figure 3b) and for the comparison presented in [45], where the proposals given in [3,43] were used. Similar fluctuations were observed using the Vyazovkin method [17].

According to the balance in [46], the Gibbs energy equation with the extended term ${∆}_{n}G$ $\left({∆}_r{H}^{\varnothing }=E_1-E_2, {∆}_r{S}^{\varnothing }=R\ln {\displaystyle \frac{A_1}{A_2}}\right)$ is valid:

$${∆}_r{G}^{\varnothing }=E_1-E_2-RT\ln {\displaystyle \frac{A_1}{A_2}}+{∆}_{n}G$$

(30)

and for the data from [44], $E_2=$ 70.6 kJ·mol⁻¹ and $\ln A_1=$ 15.4 and $\ln A_2=$ 4.95 ($A_{1,2}$ in s⁻¹) and

$${∆}_{n}G=52410-61·T, \mathrm{J}·{\mathrm{mol}}^{-1}$$

(31)

We obtain

$${∆}_r{G}^{\varnothing }=172730-148·T, \mathrm{J}·{\mathrm{mol}}^{-1}$$

(32)

For another complete set of data, $E_1=$ 191 kJmol⁻¹, $E_2=$ 17.52 kJmol⁻¹ and $\ln A_1=$ 15.4 and $\ln A_2=$ −3.238 ($A_{1,2}$ in s⁻¹) and ${∆}_{n}G=0$, and we obtain

$${∆}_r{G}^{\varnothing }=177480-155·T, \mathrm{J}·{\mathrm{mol}}^{-1}$$

(33)

Equations (32) and (33) correspond to the thermodynamic equilibrium relationship for the thermal dissociation of calcite, in agreement with [48]:

$${∆}_r{G}^{\varnothing }=174920 - 150·T, \mathrm{J}·{\mathrm{mol}}^{-1}$$

(34)

According to the considerations in [39], in the V model, by adopting the relationship in Equation (34), we assume

$${∆}_r{G}^{\varnothing }={∆}_{1}G+{∆}_{d}G-{∆}_{*}G$$

(35)

from which we determine (${∆}_{1}G$ is given by Equation (23) and ${∆}_{d}H=$ 20 kJ·mol⁻¹, ${∆}_{d}S=$ 92 J(mol·K)⁻¹):

$${∆}_{*}G= 53100 - 97·T, \mathrm{J}·{\mathrm{mol}}^{-1}$$

(36)

Equations (31) and (36), as well as the literature data [39], i.e., Equation (25), mainly differ in terms of entropy. However, they are close in evaluating the same phenomenon from a thermodynamic perspective ${∆}_{n}G\cong {∆}_{*}G$, albeit for two different descriptions of the same observation: nucleation and the simultaneous transformation of metastable CaO* into the stable form.

3.4. Discussion

Based on the equations presented in the introduction, models are distinguished for both isothermal and dynamic conditions. The present study, in further considerations, focuses on two models under isothermal conditions, designated as H-CL and V. The fundamental equations concern Equation (21) for the H-CL model and Equation (27) for the V model, in which the quantity of interest is the reaction/process rate at a given isoconversion degree, according to the assumption in Equation (19), $\alpha =$ constant. The two models considered, based on the fundamental kinetic equations, Equations (21) and (27), are characterized by the inclusion of thermodynamic constraints in the form of the equilibrium conversion degree α_eq. In isoconversional considerations, lower conversion degrees span a wide temperature range, and the thermodynamic constraint shifts to higher values with increasing temperature. Consequently, these values correspond to a broad spectrum of reaction/process pathways, understood as the coexistence of chemical decomposition and nucleation. In the second case, the temperature range becomes narrower and, simultaneously $\alpha \to {\alpha }_{eq}$; thus, $r={\displaystyle \frac{d\alpha }{dt}}\to$ 0 and the transformation process vanishes.

Proceeding further, the comparison for the two models first leads to an equation in mathematical form, where $r={\displaystyle \frac{d\alpha }{dt}}$ represents the reaction/process rate, and the function symbol $f\left({\displaystyle \frac{\alpha }{{\alpha }_{eq}}}\right)$ dependent on temperature is included (the conversion degree is also dependent on it):

$${\displaystyle \frac{r}{1-m}}=k_1{\displaystyle \frac{1}{1+f\left(\frac{\alpha }{{\alpha }_{eq}}\right)}}$$

(37)

Subsequently, the comparison addresses the consistency of the mechanism. Two stages are distinguished: the kinetic stage initiating the reaction/process (index 1) with a reversible reaction (index 2), followed by the simultaneous stage of gas product release (index d) and nucleation (index n), where the equilibrium elements providing resistance are both adsorption (index a) and denucleation (index “–n”). CO₂ sorption slows down nucleation, and the region near the equilibrium temperature plays a particularly important role.

Nevertheless, the negative apparent activation energy decreases the reaction/process rate, and nucleation ceases (Figure 2) because the rate constant of nucleation $k_n\to 0$, which is a consequence of the free energy vanishing, $∆G{|}_{T_{eq}}=0$. This indicates that the metastable form of the product (CaO) has transformed into its crystallized form.

An important issue is the relationship between activation energy and conversion degree. Figure 5 shows a monotonic decrease/increase in E_app with increasing α within a limited range, with the H-CL model positioned below the V model. In both cases, approaching the end of the reaction/process, the activation energies fall below the assumed value (E in kJ·mol⁻¹), reaching approximately 94–116 kJ·mol⁻¹. This demonstrates a shift in the relative contributions of the chemical reaction and the transformation process for the same conversion degree. The complete termination of the reaction/process leads to pointwise zero activation energy (H-CL model) or negative activation energy (V model). The presented analysis results correspond to a condition of r² ≥ 0.99. Due to both the limitations $\alpha \to {\alpha }_{eq}$ and $\left(T_{eq}-T\right)\to 0$, the newly determined Arrhenius parameters are highly dependent on the number of computational nodes and the reliability of the coefficient of determination. Reducing the tolerance produces variable characteristics for the studied problem.

By expressing Equation (37) in logarithmic form, it can be written as

$$\ln {\displaystyle \frac{r}{1-m}}=\ln k_1-\mathrm{l}\mathrm{n}\left[1+f\left({\displaystyle \frac{\alpha }{{\alpha }_{eq}}}\right)\right]$$

(38)

in which the basis is the rate constant for the completed reaction. It is represented by $k_1$, and for $f\left({\displaystyle \frac{\alpha }{{\alpha }_{eq}}}\right)<1$ this term does not significantly affect the activation energy, because $\ln \left[1+ f\left({\displaystyle \frac{\alpha }{{\alpha }_{eq}}}\right)\right]\cong f\left({\displaystyle \frac{\alpha }{{\alpha }_{eq}}}\right)$. This simplification can be used to interpret low conversion degrees.

For the H-CL model, the term $\ln k_1$ is reduced stepwise by specific values of the function $f\left({\displaystyle \frac{\alpha }{{\alpha }_{eq}}}\right)$, which is clearly visible in Figure 1a; for example, for $\alpha =$ 0.05 and $T=$ 1000 K, $f\left({\displaystyle \frac{\alpha }{{\alpha }_{eq}}}\right)=$ 0.852. In the case of the V model, $E_{app}$ decreases smoothly from the assumed level $E=$ 191 kJ·mol⁻¹, and the function $f\left({\displaystyle \frac{\alpha }{{\alpha }_{eq}}}\right)$ vanishes at $\alpha =$ 0 (Figure 3a). At higher temperatures, however, it significantly exceeds unity, and then this function strongly contributes to a reduction in the activation energy. For the H-CL model a continuous trend $E\to 0$ is observed, whereas for the V model $E<0$. Under isothermal conditions, the model description yields a clear relationship, $E=E\left(\alpha \right);$ however, under dynamic conditions and within an isoconversional framework, the thermodynamic constraints change radically. Decomposition in an atmosphere of an inert gas flow continuously removes the limitation ${\alpha }_{eq}\to 1$; thus, for each level of $\alpha =$ constant over a wide temperature range, a specific combination of kinetic elements and physical processes is formed. These are observed here as nucleation accompanied by a simultaneous transformation of the metastable oxide into the stable form. Constancy of the activation energy implies that either only the chemical reaction is being analyzed or that the proportion of the contributions remains constant with the conversion degree. A useful relation is Equation (30), which, through a thermodynamic balance, normalizes the kinetic elements with the transformation process.

Consideration of the $E=E\left(\alpha \right)$ relationships for the isoconversional variant under dynamic conditions indicates the possibility of interpretation using isothermal models. By applying the Friedman and Vyazovkin/Ortega equations with the use of the orthogonal collocation method, a distinctly oscillatory character was even demonstrated (PMMA decomposition under dynamic conditions) [52]. In the more recent literature, both increasing dependencies [21,53,54,55] and decreasing ones [2,20,53], decreasing followed by increasing trends [56,57], as well as alternating relationships [2,16,53,55,57,58] have been reported. Particularly interesting are the theoretical analyses under isothermal conditions, which describe the relationship $E=E\left(\alpha \right)$ in linear or parabolic form, from which it follows that the most favorable effect occurs when $E=$ constant [53]. A characteristic feature is that, in the case of plant biomass, the dependencies are generally increasing [21,54,55], and in [21] the validity of the empirical non-linear relationship ($p_2=-$1) was confirmed:

$$E=p_1+p_2 {\displaystyle \frac{\alpha }{\ln \alpha }}+ p_3\ln \alpha ,\mathrm{where} p_{1\dots 3} \mathrm{are} \mathrm{constant}$$

(39)

with the simultaneous determination of lnA. The increase in activation energy with increasing conversion degree is a consequence of the thermal stability of lignin, which decomposes with increasing temperature after hemicellulose and cellulose [59].

Relationships that are practically regarded as a constant activation energy with increasing conversion degree apply both to specific experimental conditions enriched with selected kinetic functions for CaCO₃ [2,3,29], although there are also curves that can be classified as decreasing and then approaching constant values [17]. The decomposition of complex substances, such as polystyrene [60] or higher alcohols [22], suggests relationships of the type $E\cong$ constant.

Continuing the considerations presented in [17] for the same CaCO₃ decomposition data obtained in nitrogen, the possibilities for determining the activation energy were extended. Thus, within the framework of the Vyazovkin method, analyses were performed using the abandoned “two-point” principle and successive pairs of heating rates with repetitions and averaging using the geometric mean, including repeated pairings, as well as several additional variants not previously described in the literature for this purpose. A compilation of the corresponding equations is provided in

Table 2. Compilation of several possibilities for determining the activation energy from successive pairs of heating rates under isoconversional conditions (α = constant and f(α) = constant); the indices “j” correspond to a higher temperature than those denoted by index “i”—see Appendix A.

E=	Comment	Reference	No. of Eq.	Average Value of E, kJ·mol−1
$R·{\displaystyle \frac{T_i{·T}_j}{T_j-T_i}}·\left(2\ln {\displaystyle \frac{T_i}{T_j}}+\ln {\displaystyle \frac{q_j}{q_i}}\right)$	The simplest version of the Vyazovkin concept using the Coats–Redfern equation	[17]	(14)	191.55 $\pm$ 1.64
$R·{\displaystyle \frac{T_i{·T}_j}{T_j-T_i}}·\left(\ln {\displaystyle \frac{T_i}{T_j}}+\ln {\displaystyle \frac{q_j}{q_i}}\right)$	In Equation (9), the temperature integral is approximated by the form  $I\left(E,T_{\alpha }\right)=T_{\alpha }\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{E}{R{T}_{\alpha }}}\right)$	Solution acc. [50]	(40)	200.12 $\pm$ 1.55
${\displaystyle \frac{R}{1.0516}}·{\displaystyle \frac{T_i{·T}_j}{T_j-T_i}}·\left(\ln {\displaystyle \frac{q_j}{q_i}}\right)$	The commonly used Doyle approximation, applied in Equation (8)		(41)	198.46 $\pm$ 1.44
$R·{\displaystyle \frac{T_i{·T}_j}{T_j-T_i}}·\left(\ln {\displaystyle \frac{s_j}{s_i}}+2\ln {\displaystyle \frac{T_i}{T_j}}+\ln {\displaystyle \frac{q_j}{q_i}}\right)$	The derivative of the three-parametric equation, here $s=-{\displaystyle \frac{dln\alpha }{d\left(\frac{1}{T}\right)}}$, or equivalently ${\displaystyle \frac{d\alpha }{dT}}={\displaystyle \frac{\alpha }{T^2}}s$, in comparison with the reference equation and the linear equation,  $s=a_1-a_{2}T$	[44]	(42)	192.39 $\pm$ 3.91
$R·{\displaystyle \frac{T_i{·T}_j}{T_j-T_i}}·\left[\ln {\displaystyle \frac{n_j}{n_i}}+\ln {\displaystyle \frac{\left(T_i-T^{\varnothing }\right)}{\left(T_j-T^{\varnothing }\right)}}+\ln {\displaystyle \frac{q_j}{q_i}}\right]$	The basic relation is based on an isotherm, $\alpha ={\left({\displaystyle \frac{t}{t_f}}\right)}^n,$ where $t_f$ denotes the finite time for $\alpha =$ 1, and n = var is the exponent, corresponding to the slope in the double-logarithmic representation (Figure 6)	The new approach uses $T^{\varnothing }=$ 298.15 K, taking into account the variable slope for two heating rates	(43)	191.58 $\pm$ 1.50
$R·{\displaystyle \frac{T_i{·T}_j}{T_j-T_i}}·\left[\ln {\displaystyle \frac{\left(T_i-T^{\varnothing }\right)}{\left(T_j-T^{\varnothing }\right)}}+\ln {\displaystyle \frac{q_j}{q_i}}\right]$		The new approach:  $T^{\varnothing }=$ 298.15 K,  $n=$ constant	(44)	196.62 $\pm$ 1.53
$R·{\displaystyle \frac{T_i{·T}_j}{T_j-T_i}}·\left[\ln {\displaystyle \frac{\left(T_i-T_{i,0}\right)}{\left(T_j-T_{j,0}\right)}}+\ln {\displaystyle \frac{q_j}{q_i}}\right]$		The new approach:  $T_{i,0}, T_{j,0}$ denote the temperatures for each heating rate for $\alpha \approx$0.05,  $n=$ constant	(45)	197.64 $\pm$ 3.00

. For Equations (14), (40) and (41), the basic reference equation is Equation (13) and the procedure is consistent with [17], whereas Equation (6) was used for Equations (42)–(45). The determination of the variable exponent n is based on the double-logarithmic representation shown in Figure 6, where n corresponds to the slope of the lnα vs. lnt relationship across the studied heating rates. Further details are provided in Appendix A.

Figure 7 shows the analyzed relationship of activation energy vs. conversion degree for cyclic changes in two heating rates and the equations given in Table 2. In the procedure, the individual quantities were used following the sequence established in [17], and the geometric mean was adopted for each varying pair of heating rates $q_j>q_i.$ Additionally, arithmetic mean values for all constant conversion degrees are reported (Appendix A), ranging from 191.55 to 200.12 kJ·mol⁻¹. All curves for initially higher values reach a minimum and then approach a plateau, except for the model in Equation (42), for which this effect occurs at higher conversion degrees. Surprisingly, however, the models according to Equation (14) and the previously unproposed one expressed by Equation (43) lead to a similar $E=E\left(\alpha \right)$ relationship. The models expressed by Equations (40), (41), (44) and (45) are located above the approach similar to Vyazovkin’s method, showing higher values than assumed. Apart from the representation according to Equation (42), the curves exhibit a similar pattern of changes, differing practically by a constant value.

4. Conclusions

• In the analysis of two kinetic models under isothermal conditions, defined as kinetic–nucleation (the H-CL, as kinetic model) and kinetic–desorption (the V, as thermodynamic model), it was shown under simplifying assumptions that both models yield similar reaction/process rates up to the equilibrium temperature (T_eq) (Figure 4). At higher temperatures, nucleation becomes dominant: the H-CL model leads to activation energy E → 0, whereas the V model reaches a negative activation energy E < 0. In both cases, the sequence of activation energies has an apparent character.
• The H-CL and V models, in terms of apparent activation energy, represent the same reaction/process, with a varying contribution of the individual pathways of calcite decomposition and the transformation of the resulting oxide. These energies were assigned to the imposed conversion degree as isoconversional values. In the H-CL model, this energy is significantly lower than the assumed value for the overall reaction ($E=$ 191 kJ·mol⁻¹) and complements the energy in the V model. In this case, the sum of the apparent activation energies is constant and close to the assumed value. In other words, the constant sum of the apparent activation energies of the H-CL and V models corresponds to the activation energy of the completed reaction/process (Figure 5).
• In contrast to the apparent activation energy obtained from isothermal analyses of the H-CL and V models, the thermodynamic constraint in the form of an equilibrium conversion degree causes the two models to become complementary. Once this constraint is removed under dynamic conditions and, additionally, within the isoconversional computational procedure, low conversion degrees span a very wide temperature range. Consequently, a common observation is the simultaneous occurrence of chemical reactions and solid-state product transformation.
• In the context of the isoconversional reaction variant, for dynamic conditions several approaches in this area were analyzed (Table 2). A close agreement was observed between Vyazovkin’s method using the Coats–Redfern equation and a new approach based on a power approximation (Equation (43)).
• From the CaL perspective, an important aspect is balancing the free energy of the reaction/process using the Arrhenius parameters determined, together with evaluation of the free energy of nucleation according to Equation (30). In thermodynamic terms, the H-CL model, which emphasizes the nucleation process, and the V model, describing the transformation of the metastable oxide, represent the same phenomenon, differing only slightly in entropy.