Isokinetic and Compensation Temperature in the Analysis of Thermal Dissociation of the Solid Phase under Dynamic Conditions

Abstract

Sets of Arrhenius parameters, determined according to known different equations for dynamic conditions, in the vast majority form the Kinetic Compensation Effect (KCE). Converting these data to the simplified components of the Eyring equation comes down to Enthalpy–Entropy Compensation (EEC), which is consistent with the second law of thermodynamics. It has been proved that the impact of the generally known Coats−Redfern solution on the equation in differential form results in an isokinetic form of the equations and a very important coordinate $\left. [T_0;{\alpha }_0\right]$ (initial temperature and conversion degree), depending on the heating rate. This makes it possible to determine the parameters of Arrhenius’ law for both in silico and experimental data. An analytical method for determining this coordinate has been proposed. These considerations have given rise to an analysis of the relationship between two temperatures: initial and isokinetic. The sense of isokinetic temperature has been verified by the parameters CQF and K. Going further, it was found that the effects of EEC can be transformed into KCE and vice versa, which means that the two temperatures should be identical, i.e., $T_{iso}=T_c.$ However, the experimental data indicate that the analyzed temperatures form a sequence ${T_0\leftrightarrow T}_{iso}\leftrightarrow T_c\le T_{eq}$, where $T_{eq}$ is the equilibrium temperature.

1. Introduction

The compensation effects described in the literature occur during both kinetic and thermodynamic studies. In the case of kinetic studies, they are related to the Arrhenius law and are described by the Kinetic Compensation Effect (KCE). KCE is sometimes called the Iso-Kinetic Relation (IKR) [1] or less commonly IE (Isokinetic Effect) [2]. In thermodynamic analysis, compensation effects called EEC (Enthalpy Entropy Compensation) result from van’t Hoff’s law. Freed [3] lists 54 references relating to various possibilities of EEC occurrence. The common feature in both cases (for both types of compensation effects) is the occurrence of characteristic temperatures as constant values: isokinetic for KCE ($T_{iso}$) and compensation for EEC ($T_c)$, respectively.

1.1. Outline of the Problem

Taking the most general course of consideration according to [4], the mentioned effects can be presented in the form of Equations (1)–(3) for $\beta ,\gamma =\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$:

$$\ln A_i=\gamma +\frac{E_i}{R\beta }$$

(1)

$$∆H_i^{+}=\gamma +\beta ∆S_i^{+}$$

(2)

or

$$∆H_i^{\varnothing }=\gamma +\beta ∆S_i^{\varnothing }$$

(3)

In Equations (1)–(3), the $\beta$ symbol stands for $T_{iso}$ (Equation (1)) and $T_c$ (Equations (2) and (3)) suggesting that these are the same temperatures in a value sense. Provided between Equations (1) and (2) there are approximate, although reliable relationships, the reference of the same equations to thermodynamic standard functions is not obvious. The formal notation of Equation (3) has been demonstrated for many chemical, biochemical or physicochemical reactions/processes [1,4,5,6,7,8,9].

In the Gold Book [10] Equation (2) is presented as isokinetic relationship:

$$∆H_i^{+}-\beta ∆S_i^{+}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$$

(4)

and Equation (3) is related to chemical reaction as the isoequilibrium relationship:

$${∆}_{r}H-\beta {∆}_{r}S=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$$

(5)

Equations (4) and (5) and also (1) express the view of Freed [3], who (briefly) postulates that the compensation or isokinetic temperature is determined by the identity of the chemical equilibrium constants or kinetic constants, respectively. In the first case, van’t Hoff’s isobar together with the Gibbs free energy is used in the proof and reduced to EEC, and the second is commonly known and accepted as KCE, and is related to the Arrhenius equation.

By way of theoretical considerations, Starikov proved the validity of Equation (3) based on Bayesian statistical mechanics [11]. Equations (2) and (4) contain thermodynamic activation functions derived from Eyring’s theory from 1935 [12]. The theory has been adapted for modeling complex reaction/process pathways, as well as for the practical use of selected relationships captured by Equations (2)–(5) [1,3,5,8,9,13,14,15,16,17,18,19,20]. Some of the mentioned works combine Eyring’s theory with the Arrhenius equation [13,14,18].

Analyzing the experimental data obtained under isothermal and dynamic conditions, a large discrepancy in the values of determined isokinetic temperatures can be observed. The application of isothermal, dynamic or isoconversional kinetic functions for modeling does not lead to one common isokinetic temperature value [20]. Thus, it can be expected that the reliability of Equation (2) or Equation (4) is dependent on the parameters determined from kinetic equations or reduced to thermokinetics, consistent with the classical Arrhenius equation, i.e., $\ln A \mathrm{v}\mathrm{s}. E$. Consequently, the possibility of confronting the isoconversion and compensation temperature relationships arises.

In this type of analysis, temperatures defining the initial temperature [21] or [22] and isokinetic are still useful. The equilibrium temperature follows from Equation (5) when we introduce $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}=0$, i.e.,

$$T_{eq}=\frac{{∆}_r{H}^{\varnothing }}{{∆}_r{S}^{\varnothing }} \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n} {∆}_r{G}^{\varnothing }=0$$

(6)

In this work, an attempt is made to use the isokinetic $(T_{iso}$) and compensation $(T_c$) temperatures as quantities representative for considered kinetic functions $f\left(\alpha \right)$ to demonstrate isokinetics, which is largely understood as the closeness of the Arrhenius law parameter values determined by the known equations.

1.2. Aim of the Paper

Kinetic considerations provide an opportunity to assess isokinetic relationships based on general kinetic equations [23,24] and selected initial conditions that depend on the heating rate. This creates a temperature sequence in observing the progression of the reaction/process from an initial state to an irreversible state with respect to temperature under conditions of its linear increase $q>0$:

$$T_0=\mathrm{v}\mathrm{a}\mathrm{r} \mathrm{v}\mathrm{s}. q;T_m=\mathrm{v}\mathrm{a}\mathrm{r} \mathrm{v}\mathrm{s}. q;T_{iso}, T_c \mathrm{a}\mathrm{n}\mathrm{d} T_{eq}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t} \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n} P = \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$$

(7)

Temperatures with symbols $T_0,T_m,T_{iso}$ are determined by kinetic experiments. Temperature $T_c$ can be of both kinetic and thermodynamic meaning, while $T_{eq}$ is only thermodynamic. The temperature $T_m$ denotes the temperature of the maximum reaction/process rate characteristic for dynamic conditions.

The following equation is the starting point for the analysis of kinetics of processes/reactions tested under non-isothermal conditions, also called dynamic conditions:

$$\frac{\mathrm{d}\alpha }{\mathrm{d}T}=\frac{A}{q}f\left(\alpha \right)e^{-\frac{E}{RT}}$$

(8)

Approximations in the form presented in the works [21,23,25,26,27,28,29,30] are most often used as solutions of Equation (8), while other known solutions will be omitted here.

In practice, the Coats−Redfern approximation is most commonly used in the form of a temperature integral (or, less commonly, an incomplete gamma function):

$$g\left(\alpha \right)=\frac{AR{T}^2}{qE}{e}^{-\frac{E}{RT}},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}q=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$$

(9)

Presenting the occurrence of the isokinetic temperature at low conversion degrees $\left(\alpha =0.01-0.05\right)$ [21] will mean there is an initial state:

$${\left(\frac{\mathrm{d}\alpha }{\mathrm{d}T}\right)}_{T_0}=\frac{1}{T_0}$$

(10)

The aim of this study is to determine from Equations (8)–(10) in what form we observe the isokinetic relationship of the selected models described symbolically as $f\left(\alpha \right)$ or $g\left(\alpha \right)$, in order to further analyze the variability of the temperature series presented in Equation (7) and the mutual relations between these temperatures.

2. Methods

2.1. General Kinetic Equation for Dynamic Conditions

If we introduce Equation (9) into Equation (8), we obtain the equation also found in the literature [27], and earlier in [28]:

$$\frac{\mathrm{d}\alpha }{\mathrm{d}T}=f\left(\alpha \right)·g(\alpha )·\frac{E}{R{T}^2}$$

(11)

Presenting Equation (11) in form:

$$\int \frac{\mathrm{d}\alpha }{f\left(\alpha \right)·g\left(\alpha \right)}=-\frac{E}{RT}+C$$

(12)

the integral on the left-hand side of the equation after substitution: $\mathrm{d}\alpha =f\left(\alpha \right)·\mathrm{d}g\left(\alpha \right)$ leads to the form:

$$\mathrm{l}\mathrm{n}\left. [g\left(\alpha \right)\right]=-\frac{E}{RT}+C$$

(13)

Equation (13) satisfies the criteria of the mathematical formalism for different kinetic functions $g\left(\alpha \right)$–however, deviations from the assumed activation energy can be expected.

Table A1. List of kinetic functions as in [31] and approximation of the first term in Equation (14) ($0<\alpha <1)$.

Code	$\mathit{f}\left(\mathit{\alpha }\right)$	$\mathit{g}\left(\mathit{\alpha }\right)$	$\mathit{f}\left(\mathit{\alpha }\right)·\mathit{g}\left(\mathit{\alpha }\right)$	$\mathbf{ln}\mathit{g}\left(\mathit{\alpha }\right)$$\mathbf{for} {\mathit{\alpha }}_0^{}*$
P4	$4{\alpha }^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$	${\alpha }^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$	$4\alpha$	$\frac{1}{4}\ln {\alpha }_0$
P3	$3{\alpha }^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$	${\alpha }^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$	$3\alpha$	$\frac{1}{3}\ln {\alpha }_0$
P2	$2{\alpha }^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$	${\alpha }^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$	$2\alpha$	$\frac{1}{2}\ln {\alpha }_0$
P2/3	$\frac{2}{3}{\alpha }^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$	${\alpha }^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$	$\frac{2}{3}\alpha$	$\frac{3}{2}\ln {\alpha }_0$
D1	$\frac{1}{2}{\alpha }^{-1}$	${\alpha }^2$	$\frac{1}{2}\alpha$	$2\ln {\alpha }_0$
F1	$1-\alpha$	$-\ln \left(1-\alpha \right)$	$-\left(1-\alpha \right)\ln \left(1-\alpha \right)$	$\ln {\alpha }_0$
A4	${4\left(1-\alpha \right)\left. [-\ln \left(1-\alpha \right)\right]}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$	${\left. [-\ln \left(1-\alpha \right)\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$	$4\left(1-\alpha \right)\left. [-\ln \left(1-\alpha \right)\right]$	$\frac{1}{4}\ln {\alpha }_0$
A3	${3\left(1-\alpha \right)\left. [-\ln \left(1-\alpha \right)\right]}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$	${\left. [-\ln \left(1-\alpha \right)\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$	$3\left(1-\alpha \right)\left. [-\ln \left(1-\alpha \right)\right]$	$\frac{1}{3}\ln {\alpha }_0$
A2	${2\left(1-\alpha \right)\left. [-\ln \left(1-\alpha \right)\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$	${\left. [-\ln \left(1-\alpha \right)\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$	$2\left(1-\alpha \right)\left. [-\ln \left(1-\alpha \right)\right]$	$\frac{1}{2}\ln {\alpha }_0$
D3	${\frac{3}{2}\left(1-\alpha \right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{\left. [{1-\left(1-\alpha \right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right]}^{-1}$	${\left. [1-{\left(1-\alpha \right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right]}^2$	$\frac{3}{2}{\left(1-\alpha \right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left. [1-{\left(1-\alpha \right)}^{1/3}\right]$	$2\ln ({\alpha }_0/3)$
R3	$3{\left(1-\alpha \right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$	$1-{\left(1-\alpha \right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$	$3{\left(1-\alpha \right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left. [1-{\left(1-\alpha \right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right]$	$\ln ({\alpha }_0/3)$
R2	$2{\left(1-\alpha \right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$	$1-{\left(1-\alpha \right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$	$2{\left(1-\alpha \right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\left. [1-{\left(1-\alpha \right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right]$	$\ln ({\alpha }_0/2)$
D2	${\left. [-\ln \left(1-\alpha \right)\right]}^{-1}$	$\alpha +\left(1-\alpha \right)\ln \left(1-\alpha \right)$	$\frac{\alpha +\left(1-\alpha \right)\ln \left(1-\alpha \right)}{\left. [-\ln \left(1-\alpha \right)\right]}$	$2\ln \left(\frac{{\alpha }_0}{\sqrt{2}}\right)$

* Maclaurin expansion was used, for model D3–after squaring. Stopped at the first term of the equation.

uses the proposition from the book [31] having in mind explicitly given variables and coefficients depending on the definition of the function $f\left(\alpha \right)$. It can also be considered that these are the most common kinetic models. It should be noted that Table A1 (see: Appendix A) does not present all known functions–0th kinetics (R1) and higher orders, e.g., Johnson-Mehl-Avrami [27], D4 Ginstein-Brouhnstein [32], Dahme-Junker [33], or more complicated ones like Šesták-Berggren (SB) [32,34], are missing.

Thus, the intercept in Equations (12) and (13) can be represented as ($\ln \left. [g\left(\alpha \right)\right] \mathrm{v}\mathrm{s}. 1/T)$:

$$C=\ln \left. [g\left({\alpha }_0\right)\right]+\frac{E}{R{T}_0}$$

(14)

As in the work [23], the functions g(α) were replaced with sufficient accuracy by the first term of the expansions into a power series ($\ln \left. [g\left({\alpha }_0\right)\right]$ for small ${\alpha }_0$ in Table A1).

The constant written in the form of Equation (14) contains the sought after two unknowns as the coordinate [$T_0;{\alpha }_0$] at the initial point and requires a special solution, when the kinetic function is a priori unknown.

Expressions for the simplified constant C $\left(C=n\ln \left(\frac{{\alpha }_0}{m}\right)+\frac{E}{R{T}_0}, n, m=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\right)$ are included in Table A1.

For arbitrary data, the combination of Equation (8) to Equation (10) was proposed for the coordinate $\left. [T_0,{\alpha }_0\right]$ establishing the relation:

$$f\left({\alpha }_0\right)·g\left({\alpha }_0\right)=\frac{R{T}_0}{E}$$

(15)

The integration constant in Equation (14) can be transformed into a complex form with the following components:

$$C=\ln \left. [g\left({\alpha }_0\right)\right]+[f\left({\alpha }_0\right)·g\left({\alpha }_0\right){]}^{-1}$$

(16)

The pre-exponential factor is determined from Equation (9) after reference to temperature T = T₀.

$$\ln A=\ln \left. [g\left({\alpha }_0\right)\right]+\frac{E}{R{T}_0}+\ln \left(\frac{qE}{R{T}_0^2}\right)$$

(17)

According to Equation (14) the first two terms to the right of Equation (17) are the integration constant, hence we obtain:

$$\ln A=C+\ln \left(\frac{qE}{R{T}_0^2}\right)$$

(18)

Equation (18) is valid for all kinetic models, including those for which $f\left({\alpha }_0\right)\ne 1$, as can be shown by inserting the logarithmized Equation (9) into Equation (14) for $T=T_0$. In Equation (17) after rearranging the products under the logarithms (or substituting into Equation (18) the integration constant from Equation (14)) we obtain Equation (19):

$$\ln A=\ln \left. [g\left({\alpha }_0\right)·\frac{E}{R{T}_0}\right]+\frac{E}{R{T}_0}+\ln \left(\frac{q}{T_0}\right)$$

(19)

and after using Equation (15) and performing operations we obtain:

$$\ln A=-\ln \left. [f\left({\alpha }_0\right)\right]+\frac{E}{R{T}_0}+\ln \left(\frac{q}{T_0}\right)$$

(20)

For the most commonly used models it can be assumed that for F1 $f\left({\alpha }_0\right)\cong 1$ (for R1: $f\left({\alpha }_0\right)=1$, so Equation (15) takes the form:

$$g\left({\alpha }_0\right)=\frac{R{T}_0}{E}$$

(21)

which for $f\left({\alpha }_0\right)=1$, $g\left({\alpha }_0\right)={\alpha }_0$ and so Equation (21) leads to the next form:

$${\alpha }_0=\frac{R{T}_0}{E}$$

(22)

After introducing Equation (17) into Equation (16), we obtain an expression containing only the unknown temperature $T_0$:

$$C=\frac{E}{R{T}_0}-\ln \left(\frac{E}{R{T}_0}\right)$$

(23)

In turn, introducing Equations (21) and (22) we obtain an expression containing only the unknown conversion degree ${\alpha }_0$:

$$C=\frac{1}{{\alpha }_0}+\ln {\alpha }_0$$

(24)

Knowing the intercept in the relationship $\ln \left. [g\left(\alpha \right)\right] \mathrm{v}\mathrm{s}. 1/T$, the initial conversion degree can be determined from Equation (24). In Equation (20), the inversion of the coordinate system for $T=T_0$ and after double-sided logarithmizing leads to:

$$\ln \left(\frac{q}{T_0}\right)=\ln A+\ln \left. [f\left({\alpha }_0\right)\right]-\frac{E}{R{T}_0},\hspace{1em}\hspace{1em}\frac{q}{T_0}=\mathrm{v}\mathrm{a}\mathrm{r}$$

(25)

Such inversion is similar to the transformation of Equation (9) into the well-known KAS equation. For most kinetic models, the term $\ln \left. [f\left({\alpha }_0\right)\right]$ can be neglected, and if the analytical form of the kinetic function and ${\alpha }_0$ is known, it is not a problem to determine this value. It directly affects the accuracy of the determination of $\ln A$.

2.2. Data Set for Verification of Kinetic Models

Two data sets were used in this study, the first resulting from the simulation of the assumed Arrhenius parameters and for Equation (9) and $g\left(\alpha \right)=〈\alpha 〉$, i.e., $E=200 \mathrm{k}\mathrm{J}·{\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}$, $A={10}^{14}{\mathrm{s}}^{-1}$ and five heating rates $q=0.01; 0.03; 0.1; 0.3; 0.5K·{\mathrm{s}}^{-1}$, respectively. The calculations were carried out for the temperature range $T=298-1000 \mathrm{K}$ with a step $∆T=1 \mathrm{K}$, choosing the $\alpha$ values from the range $\left. [0; 1\right]$, similarly to what was adopted in [21]. To distinguish the notation for individual cases of the function $g\left(\alpha \right)$ values, a single-column matrix is marked as:

$$g\left(\alpha \right)=〈\alpha 〉 \mathrm{o}\mathrm{r} g\left(\alpha \right)=g〈\alpha 〉, 〈\alpha 〉\in \left. [0;1\right]$$

(26)

when $〈\alpha 〉$ is an invariant of the kinetic models.

The second set of experimental data was taken from the work [19], which was extensively analyzed in full by the ICTAC congress in 2000 [35]. They concern the dissociation of calcite in a nitrogen atmosphere, the most preferably proposed kinetic model being R1, F1 or fractional.

3. Results

3.1. Results of the Calculations

For kinetic models (Table A1) 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 (0th order, n = 1), the left-hand side of Equation (13) is as follows:

$$n\ln 〈\alpha 〉=-\frac{E}{RT}+C$$

(27)

which reduces the listed models to 0th order kinetics.

Similarly for models F1 (n = 1), A4 (n = 1/4), A3 (n = 1/3), A2 (n = ½):

$$n\ln \left. [-\ln \left(1-〈\alpha 〉\right)\right]=-\frac{E}{RT}+C$$

(28)

For models R2 or R3 ($n=1$) we obtain, respectively:

$$\ln \left(1-\sqrt{1-〈\alpha 〉}\right)=-\frac{E}{RT}+C$$

(29)

or:

$$\ln \left(1-\sqrt[3]{1-〈\alpha 〉}\right)=-\frac{E}{RT}+C$$

(30)

For the popular D3 model:

$$g\left(\alpha \right)={\left. [1-\sqrt[3]{1-〈\alpha 〉}\right]}^2$$

(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 $\left(E, A, q\right)$, Figure 1.

Table 1. Summary of calculated parameters of Equation (27) for five assumed heating rates and $n=1\left(r^2=1\right)$.

Heating Rate $\mathit{q}$, K s−1	$\mathit{E}$, kj mol−1 acc. Equation (27)	Intercept C acc. Equation (27)	${\mathit{T}}_0$ K from Equation (23)	${\mathit{\alpha }}_0$ from Equations (22) or (24)	$\mathbf{ln}\mathit{A}$ $(\mathit{A}$ in s−1) acc. Equation (18)
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 ± standard deviation	207.22 ± 0.14	38.807 ± 1.569	586.3 ± 22.4	0.02353 ± 0.00088	33.718 ± 0.029

shows the calculated activation energies, integration constants (intercept of linear equations), temperatures and the corresponding conversion degrees as $\left. [T_0; {\alpha }_0\right].$

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:

$$\ln \left(\frac{q}{T_0}\right)=-\frac{23400}{T_0}+32.144\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(r^2=1.0000\right)$$

(32)

The activation energy $E=199.53 \mathrm{k}\mathrm{J}·{\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}$ and the pre-expotential 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 ($A={10}^{14} {\mathrm{s}}^{-1}, f\left({\alpha }_0\right)=1$).

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: $\ln \left(1-\sqrt[3]{1-〈\alpha 〉}\right)$ (R3).

Table 2. Analysis of the kinetic function D3 as R3 by means of Equation (30), $r^2$= 0.9999.

$\mathbf{Heating} \mathbf{Rate}$$\mathit{q}$, K s−1	Slope in Equation (30)	Intercept in Equation (30)	$\mathit{E},$ kJ mol−1	Intercept acc. Equation (14)	$\mathbf{ln}\mathit{A}$ $(\mathit{A}$ in s−1) acc. Equation (20)
0.01	25043	40.115	208.21	40.054 *	34.017
0.03	25069	39.072	208.42	38.999	34.010
0.1	25069	39.072	208.42	37.804	33.962
0.3	25106	36.843	208.73	36.776	33.980
0.5	25119	36.356	208.84	36.285	33.976
average ± standard deviation	-	38.29 ± 1.31	208.52 ± 0.23	37.984 ± 1.392	33.989 ± 0.021

* For $\left. [T_0; {\alpha }_0\right]$ of Table 1 and slope in Equation (30).

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.

Thus, if this model accepts the coordinates $\left. [T_0; {\alpha }_0\right]$ 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.

The consequences of the considerations for the assumed values of kinetic parameters $E=200 \mathrm{k}\mathrm{J}·{\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}$ and $A={10}^{13.966} s^{-1}$ also provide an opportunity to analyze the isoconversion models.

The adopted methodology and the performed cycle of calculations indicate the possibility of extending the interpretation of Equation (9) it can be written in a form as KAS equation:

$$\ln \left(\frac{q}{T_i^2}\right)=-\frac{E}{R{T}_i}+\ln A-\ln \left(\frac{E·g\left({\alpha }_i\right)}{R}\right)$$

(33)

For each point with coordinates $\left. [T_i; {\alpha }_i\right]$, an intercept is created: $\ln A-\ln \left(\frac{E·g\left({\alpha }_i\right)}{R}\right)=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$, and for the presented considerations the relationship $g\left({\alpha }_i\right)={\alpha }_i$ has been assumed, what follows from considerations on Equation (13) and further ones. Figure 4 and

Table 3. Summary of calculation results for five different heating rates and for constant values of the conversion degree ${\alpha }_0$ log⁡  A = 14.00 ln A = 32.236 r² = 1.0000.

$\mathbf{Conversion} \mathbf{Degree}, {\mathit{\alpha }}_0$	$\frac{\mathit{E}}{\mathit{R}}$ in Equation (33)	Intercept in Equation (33)	$\mathbf{ln}\mathit{A}$ $(\mathit{A}$ in s−1)	$\mathit{E},$ kJ mol−1
0.1	24056	24.451	32.236	200.00
0.2	24052	24.451	32.231	199.97
0.3	24053	23.347	32.231	199.97
0.4	24054	23.062	32.234	199.99
0.5	24055	22.840	32.235	199.99
0.6	24062	22.668	32.246	200.05
0.7	24055	22.504	32.235	199.99
0.8	24055	22.371	32.236	200.00
average ± standard deviation	-	-	32.236 ± 0.004	200.00 ± 0.02

summarize the results of calculations for five different heating rates and for constant values of the conversion degree ${\alpha }_i$ from 0.1 to 0.8.

The data presented in Table 3 show that with the relative error below 0.02% we determine the activation energy $E$ and with a similar approximation, the pre-exponential factor $A$. Despite the very small differences in values, we can derive KCE as an isoconversional effect $(\alpha =\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t})$:

$$\ln A=0.1864E-5.0363, E \mathrm{i}\mathrm{n} \mathrm{k}\mathrm{J} {\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}, T_{iso}=645.27 \mathrm{K}\left(r^2=0.9923, sl=0.0\left(6\right)\right)$$

(34)

The determined isokinetic temperature is higher than that provided in Table 1 even for heating rate $q=0.5 \mathrm{K}·{\mathrm{s}}^{-1}$, but more important is the fact that there appears a KCE that is statistically significant only for the isoconversion method.

The starting point of the calculation is the integration constant Equation (13), which initiates the possibility of confronting Equation (25), since formally in both cases $E=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$. Considerations for simulated data show that the integration constant $\left(C\right)$ in Equations (14), (16), (19), (20) divides the kinetic data into a range of very low reaction/process rates from those corresponding to the higher temperature range and heating rates $\frac{\mathrm{d}\alpha }{\mathrm{d}\mathrm{l}\mathrm{n}T}>1$.

The KCE phenomenon, conventionally written as Equation (1) for $\beta =T_{iso}$ is characterized by the fact that $\frac{\mathrm{d}\ln A}{\mathrm{d}E}=(R{T}_{iso}{)}^{-1}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$, so there is no simple way to relate to Equation (25) due to the variability of the ratio $\frac{q}{T_0}=\mathrm{v}\mathrm{a}\mathrm{r}$ for $E=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$. Note that even in the simplest kinetic case for the simulated R1 model this effect does not occur (Table 1). This fact is also confirmed by the data for model R3 (Table 2). A small increase in $E$ leads to a small decrease in $\ln A$, which contradicts the classical Arrhenius law [2], but is possible for physical processes. An example is the relationship between the two parameters of viscosity in the Arrhenius-like equation, such as the energy and the pre-exponential factor; see Figure 3b in [36].

At this stage of consideration, the importance of the initial temperature $\left(T_0\right)$ is confirmed, but there is not enough strong evidence to consider the isokinetic temperature $\left(T_{iso}\right)$ as an undisputed quantity. From the data considered, it appears that $T_{iso}>T_0$.

3.2. Thermokinetic Analysis of Experimental Data

The starting point for kinetic considerations is Equation (13) and the assumption that $g\left(\alpha \right)=\alpha$ (e.g., Anderson in [35]). This is supported by Equation (33) as well as the calculations presented in

Table 4. Analysis of experimental data of thermal dissociation of calcite in nitrogen based on Equation (36)–Doyle approximation.

$\mathbf{Heating} \mathbf{Rate}, \mathit{q}$, K·(60s)−1	$\frac{\mathit{E}}{\mathit{R}}$ in Equation (36)	Intercept in Equation (36)	${\mathit{r}}^2$	$\mathit{E},$ kJ·mol−1	$\mathbf{ln}\mathit{A}$ ( $\mathit{A}$ in s−1)	${\mathit{T}}_0$, K acc. Equation (23)
1	24,576	24.712	0.9994	204.32	15.81	876.3
3	24,651	23.798	0.9986	204.95	15.99	909.7
5	24,222	22.849	0.9988	201.38	15.57	927.6
7.5	24,612	22.837	0.9991	204.62	15.94	943.0
10	24,565	22.484	0.9992	204.23	15.88	954.6
15	24,685	22.176	0.9997	205.23	15.97	971.4
25	24,836	21.850	0.9997	206.49	16.15	990.6
average± standard deviation	-	-		204.46 ± 1.44	15.90 ± 0.17	939.0 ± 35.6

, although it will be necessary to verify this assumption. The calculations were based on the experimental data presented in [19] (Table 1a in [19]) for thermal dissociation of calcite in nitrogen under dynamic conditions.

Bearing in mind the need to determine $E$ and $\ln A$, it was found that there are at least two possibilities to analyze the experimental data. The first option is to use the relationship:

$$\ln \left(\frac{\alpha }{T^2}\right)=\ln \left(\frac{A·R}{q·E}\right)-\frac{E}{R·T},\hspace{1em}q=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$$

(35)

while the second one uses the Doyle approximation for the intercept presented in [20]:

$$\ln \alpha =\ln \left(\frac{0.005·A·E}{q·R}\right)-\frac{E}{R·T},\hspace{1em}q=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$$

(36)

Equations (35) and (36) can be applied to isoconversional analyses in the form:

$$\ln \left(\frac{q}{T_0^2}\right)=\ln \left(\frac{A·R}{E·{\alpha }_0}\right)-\frac{E}{R·T_0},\hspace{1em}{\alpha }_0=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$$

(37)

when Equation (37) is a special case of Equation (33) and Equation (38):

$$\ln q=\ln \left(\frac{0.005·A·E}{R·{\alpha }_0}\right)-\frac{E}{R·T_0},\hspace{1em}{\alpha }_0=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$$

(38)

Experimental data of the thermal dissociation of calcite in nitrogen for the seven heating rates in the coordinate system of Equation (35) are presented in Figure 5.

Equations (35)–(38) generate KCE and therefore the individual estimators were statistically verified–all relationships are significant with a probability of at least 99.99%. The relevant graphs are shown in Appendix A in Figure A1, Figure A2, Figure A3 and Figure A4, while whether the determined isokinetic temperatures can be considered authoritative will be presented later in the paper.

Equation (36) can be taken as analogous to Equation (13) for $g\left(\alpha \right)=\alpha$, 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:

$$\ln \left(\frac{q}{T_0}\right)=-\frac{23212}{T_0}+15.6671\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(r^2=0.9989\right)$$

(39)

from which $E=192.99 \mathrm{k}\mathrm{J}·{\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}$ and $A={10}^{6.80} s^{-1}\left(\ln A=15.67\right)$ were calculated assuming $f\left({\alpha }_{iso}\right)=1$ (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 Figure A1, Figure A2, Figure A3 and Figure 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 $\left. [\ln A; E\right]$ 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 $\left. [T_0; {\alpha }_0\right]$ for any kinetic data is complicated.

An interesting approach to the problem of determining this values $\left. [T_0; {\alpha }_0\right]$ 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]:

$$\ln \alpha =A_0-\frac{A_1}{T}-A_2 \ln T$$

(40)

after differentiation of Equation (40) relative to temperature, for $T=T_0$ and introducing Equation (10) we obtain:

$${\frac{1}{T_0}=\alpha }_0\left(\frac{A_1}{T_0^2}-\frac{A_2}{T_0}\right)$$

(41)

and after multiplying both sides by $T_0^2$ and after ordering, we get:

$$T_0=\frac{{\alpha }_0·A_1}{1+{\alpha }_0·A_2}$$

(42)

The approximate methods can be used to determine the coordinates $\left. [T_0; {\alpha }_0\right]$ from the system of Equations (40) and (42).

After introducing Equations (42)–(40) for $T=T_0$ we obtain:

$$\frac{1}{{\alpha }_0}+\ln {\alpha }_0+A_2\ln \left(\frac{{\alpha }_0{A}_1}{1+{\alpha }_0{A}_2}\right)=A_0-A_2,\hspace{1em}\hspace{1em} A_2\ne 0$$

(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. Results of the analysis of experimental data on the thermal decomposition of CaCO₃ in nitrogen for seven heating rates using the three-parameter Equation (40).

Heating Rate q, K·(60s)−1	${\mathit{A}}_0$	${\mathit{A}}_1·{10}^{-5},$ K	${\mathit{A}}_2$	${\mathbf{r}}^2$	${\mathsf{\alpha }}_0$ acc. Equation (43)	${\mathit{T}}_0,$ K acc. Equation (42)	${\mathit{T}}_{\mathit{\alpha }=0.05},$ K [5]
1	893.70	1.302	110.52	0.9993	0.0266	879.05	894.95
3	552.88	0.911	66.97	0.9999	0.0303	911.24	925.14
5	530.03	0.889	64.05	1.0000	0.0315	928.01	942.14
7.5	405.68	0.741	48.25	0.9999	0.0330	943.31	956.64
10	463.43	0.824	55.46	0.9999	0.0325	955.59	969.13
15	278.84	0.587	32.23	1.0000	0.0354	970.59	983.63
25	366.81	0.717	43.17	0.9999	0.0343	991.37	1004.62
average± standard deviation					0.0319 ± 0.0027	939.88 ± 34.88

summarizes the results of the experimental data analysis of the thermal decomposition of CaCO₃ 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 Table 4 and Table 5 is surprisingly consistent, despite their determination on two independent paths.

3.3. Consequences of the Analysis of Experimental Data

It should be noted that the relations $\ln \alpha \mathrm{v}\mathrm{s}. \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$T$}\right.$ are known in the literature as the temperature criterion [39] and are also discussed in works [40,41]. Equations (40) with (42), (43) are new elements in this respect and allow the determination of the isokinetic temperature.

As is known, the temperature criterion is valid for low conversion degrees. In the meantime, in this work, both in the part concerning model analyses (Figure 1) and experimental data of the thermal dissociation of calcite (Table 4), a linear relationship was obtained for all data, i.e., for the analyzed range of variation of conversion degree: as well for total range, as with its limitation, respectively.

While for model considerations this is acceptable, then for other more complicated solid decompositions after an initial stage with a high activation energy its value decreases with the conversion degree. The observations provided in [31] show that, in practice, different kinds of dependences of $E$ on $\alpha$ can occur $\left(E=E\left(\alpha \right)\right).$ On the other hand, in Equations (35)–(38) with assumed Arrhenius parameters, the 0th order kinetics, $g\left(\alpha \right)=\alpha$ was used.

Nevertheless, what distinguishes this work from previous considerations is the possible range of variation of $\alpha \in \left. [0.05; 0.9\right]$ to be analyzed, which omits the extremely important initial range for low conversion degree, as well as the end of the process. The first remark that comes to mind for comparison is the value of initial temperature–for the model it is equal to average $T_0=586.3 \mathrm{K}$, while for the experimental data (Figure A1, Figure A2, Figure A3 and Figure A4) it depends on the adopted kinetic function and, see Table 4, on average it is equal to $T_0=939.0 \mathrm{K}$ (depending on $q$). When the experimental data for $q=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$ are analyzed using Equations (35) and (36), then $T_{iso}=950-1061 \mathrm{K}$. On the other hand, when we use isoconversion methods $\left(\alpha =\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\right)$ using Equations (37) and (38), we observe a wider range of variation $T_{iso}=1074-1298 \mathrm{K}$, which authorizes the conclusion that isoconversion methods are characterized by a much weaker increase in the $\ln A$ factor in relation to activation energy.

It should be noted that the parameters of the Arrhenius equation determined in this work from experimental data of the thermal decomposition of CaCO₃ in nitrogen have similar values and are in agreement with the data collected in [35] (e.g., Anderson): $E=191 \mathrm{k}\mathrm{J}·{\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}$, $\ln A=15.4/ A\mathrm{i}\mathrm{n} s^{-1}$.

For the four cases resulting from Equations (35)–(38), the determined Arrhenius law parameters can be treated as average values, as evidenced by the very small standard deviations. Nevertheless, these small variations generate an observable KCE.

Comparing the results of the analysis of the model and experimental data, it is clear that, in both cases, the values of the activation energies are similar, however, the frequency constant calculated for the experimental data reaches a much lower value: $A={10}^{6.80} s^{-1}\left(\ln A=15.67\right)$, which makes the determined isokinetic temperatures take much higher values.

The kinetic Equations (35–38) were derived according to the presented model considerations, taking into account that the most important functional factor is the expression $\left(-\frac{E}{RT}\right)$. However, since in order to create the single column matrix (26) the relation (9) was used, in the derived equations also the occurring variable $T^2$ had to be taken into account. Therefore, it can be assumed that going from Equations (35) and (36) this variable weaves into the free expression and only slightly changes the values of the estimators, remaining in accordance with KCE, i.e., the values of estimators $\ln A$ and $E$ increase or decrease simultaneously.

4. Discussion

4.1. Isokinetics and Initial Temperature

Accepting the general thermokinetic Equation (11), it was proved that the great majority of kinetic equations can be generalized to a single relationship (13), which contains the most important component $\left(-\frac{E}{RT}\right)$. Using the relationship (13) and forming the one-column matrix according to (26) one obtains a very practical for analysis of experimental data, Equations (27)–(31) with the intercept Equations (16), (19), (20), allowing us to determine the Arrhenius equation parameters and the initial (Equations (17) and (19)) or isokinetic temperature (Equation (1)).

We have complete agreement, when for the kinetic models presented in Table A1 the $n$ factors provided there will be used as proposed in [23,24]. Using Equation (25) and the single-column matrix $〈\alpha 〉$ we obtain the basis for inferring the substitutability of kinetic functions of thermal dissociation processes of the solid phase. By forming the single-column matrix for the assumed Arrhenius law parameters, we can reproduce these parameters using a modification of Equation (25), e.g., in the form of Equation (34)–Table 3. The situation is even more favorable for experimental data, which are illustrated in Figure 6.

Despite the observed variation of the initial temperature values $T_0$ with heating rate during analysis of both model and experimental data (Table 2, Table 4 and Table 5), the averaged values of $E$ and $\ln A$ can be taken as correct. For these reasons, Equations (13) and (19), from which we determine the activation energy and the initial temperature, become significantly important.

In the case of experimental studies there is a possibility that Equations (13) and (19) will be valid in the whole experimentally determined range of conversion degree increase or in its selected range.

With the heating rate, the initial temperature $T_0$ increases, but the changing parameters of the Arrhenius law lead to a constant isokinetic temperature, $T_{iso}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$.

The general forms of the Equation (13) can be reduced to the practical notation of Equation (24), which clearly indicates an extremely important element in thermogravimetric analysis for dynamic conditions, which is $T_0$. Its determination is much more convenient and simpler, when we use the three-parameter Equation (40) along with Equations (42) and (43).

4.2. Isokinetic Temperature

As demonstrated by Lente in his book [42], the problem of determining kinetic parameters can lead to incorrect or unjustified quantities. Generally, justified considerations on the correctness of the determined parameters of the Arrhenius equation may undoubtedly include another element connected with it–the isokinetic temperature $T_{iso}$ or compensation temperature $T_c$, to which many works have been devoted, among others [4,21,41,42,43,44,45,46,47].

The temperature $T_{iso}$ is determined by both Equation (1) and the classical KCE equation in the form, where intercept is expressed by the Arrhenius law, which formally makes it a tautology:

$$\ln A=\frac{E}{R{T}_{iso}}+\mathrm{l}\mathrm{n}\left(k_{iso}\right)$$

(44)

where:

$$k_{iso}=A·e^{-\frac{E}{R{T}_{iso}}}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$$

(45)

When, in Equation (45), for each coordinate pair $[E;\ln A]$, the determined $T_{iso}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$ leads to $k_{iso}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t},$ then this is the “true” isokinetic temperature. The KCE in terms of (44) with (45) exhibits features of self-consistent equations. As will be shown further on, the fluctuation of the constant is relatively high despite $r^2>0.99$.

Thus, it can be considered that this leads to a KCE called horizontal (isoconversional). According to the analyses presented in [43,44,45,46], the isoconversional KCE, characteristic of isoconversional methods, can be described as the statistical compensation effect [45] characterized by high isokinetic temperatures.

The fundamental problem, however, is whether the general notation of KCE (Equation (1)), and specifically in the form of Equation (34) and in Figure A1, Figure A2, Figure A3 and Figure A4, makes physicochemical sense for a significant statistical level, as defined by high value of determination coefficient.

For this purpose, two criteria: CQF (Compensation Quality Factor) and K according to [48,49] were used. The analytical distinction in the KCE (and EEC) relations in both physicochemical and statistical sense for high determination coefficients $\left(r^2\right)$ for a specific number of $N$ measurements was proposed. The K parameter can be called the position parameter.

For KCE assessment it is proposed:

$$CQF=1-\sqrt{\frac{1-r^2}{\frac{1}{r^2}{\left(\frac{T_{iso}}{T^{*}}\right)}^2-2\left(\frac{T_{iso}}{T^{*}}\right)+1}},\hspace{1em}0\le CQF\le 1$$

(46)

$$K=2\frac{\frac{1}{T_{hm}}-\frac{1}{T_{min}}}{\frac{1}{T_{low}}-\frac{1}{T_{high}}},\hspace{1em}\hspace{1em}{T}_{min}=\frac{T_{iso}}{r^2}$$

(47)

For a probability of 99% [48]:

$${CQF}_{99}=0.29+\frac{1.41}{N^{0.57}}$$

(48)

The meaning of the symbols and more methodological and interpretive details are included in Appendix A.

Starting from the evaluation of the veracity of the determined isokinetic temperature in Equation (34), an important issue is the determination of the temperature designated as $T_{low}$. On the one hand, according to the data presented in Table 1, this is the average initial temperature $\left(T_{low}=586.3 \mathrm{K}\right)$, but simulation analyses were carried out starting from a temperature of 298 K. The average temperature for $\alpha =1$ was taken as the $T_{high}$ temperature, i.e., $T_{high}=643.2 \mathrm{K}$.

For the first case using Equation (46) we get: $CQF=0.37 \mathrm{f}\mathrm{o}\mathrm{r} N=8$ and according to Equation (48) ${CQF}_{99}=0.721$ and Equation (47) $K=1.22$, that is, the isokinetic temperature does not meet the criterion Equation (48). On the other hand, for the second version of $T_{low}=298 \mathrm{K}$ we similarly obtain: $CQF=925 \mathrm{K};K=1.019$ and accept the physicochemical sense of the isokinetic temperature.

In this paper, the position parameter according to Equation (47) can be transformed to the form:

$$K=\left(1-\frac{T_{hm}}{T_{iso}}{r}^2\right)·\frac{T_{high}+T_{low}}{T_{high}-T_{low}}$$

(49)

which makes it possible to carry out a position analysis according to the characters in the expression shown in brackets, i.e., $\left(1-\frac{T_{hm}}{T_{iso}}{r}^2\right)\le or\ge 0$, and $K\le 0 or K\ge 0.$ The favourable case can be defined as the inequality: $T_{hm}>T_{iso}$ when $r^2=1$, because $T_{iso}\to T_{low}$, which means approaching to the initial temperature $T_o$. When $T_{hm}\ll T_{iso}$, then this condition [43] characterizes isoconversional methods $(\alpha =\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t})$ relative to classical methods $(q=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t})$.

For simplicity, still assuming $r^2=1$, this position parameter informs us about the relation between the isokinetic temperature and the range of tested temperatures in which it was determined, that is (Figure A5):

(a)

${K=0; T}_{iso}=T_{hm}$;

(b)

${K<0\left(K=-1\right); T}_{iso}<T_{low}(T_{iso}=T_{low})$;

(c)

${K>0\left(K=+1\right); T}_{iso}>T_{high}(T_{iso}=T_{high})$.

Case (c) can lead to excessively high isokinetic temperatures.

Returning to the calculations presented in Figure A1, Figure A2, Figure A3 and Figure A4, it can be noted that, despite statistically satisfactory significance levels, none of the items meet the CQF criteria. Thus, the isokinetic temperatures are experimentally determined quantities that do not satisfy the CQF in the physicochemical sense.

These considerations indicate that an important issue is the temperature range in which we observe KCE, and it is advantageous that it is as wide as possible. Due to the “quality” of KCE, when $T_{hm}>T_{iso}$, then $T_{iso}$ is closer to $T_0$ than to the final temperature of the reaction/process, which is relatively unfavorably much larger.

The criteria proposed in the works [48,49] may make it necessary to revisit many already published works.

4.3. Compensation Temperature

Eyring has derived an equation describing the reaction rate constant as a function of temperature and the free enthalpy of activation $∆G^{+}\left(p=p^{\varnothing }=0.1\mathrm{M}\mathrm{P}\mathrm{a}\right)$ depending on the reaction mechanism limited by the formation of the active complex $\mathrm{A}+\mathrm{B}\to ({\mathrm{A}\mathrm{B})}^{+}$ in gaseous phase [12]. The theses proposed in work [12] were adapted into the form of relations, in which the transmission coefficient and threshold energy distribution functions of the reactants were omitted [16,50,51,52,53,54] and only the “universal frequency term” $\frac{k_{B}T}{h}$ was left [51]. This expression is directly proportional to the product of $\left(R·T\right)$ half equal to potential and kinetic energy [16]. The equation in its most common form is as follows:

$$k_{TST}={BTe}^{-\frac{∆G^{+}}{RT}}, B=\frac{k_B}{h}=2.08364·{10}^{10}{\left(\mathrm{K}·\mathrm{s}\right)}^{-1}$$

(50)

where:

$$∆G^{+}=∆H^{+}-T∆S^{+}$$

(51)

For the solid phase dissociation process following the reaction scheme: ${\mathrm{A}}^{\left(\mathrm{s}\right)}\to {\mathrm{B}}^{\left(\mathrm{s}\right)}+{\mathrm{C}}^{\left(\mathrm{g}\right)}$ or ${\mathrm{A}}^{\left(\mathrm{s}\right)}\to {\mathrm{C}}^{\left(\mathrm{g}\right)}$ and the value of the stoichiometric coefficient for the substrate ${\mathrm{A}}^{\left(\mathrm{s}\right)}$ $\left({\nu }_A=1,\sum {\nu }^{+}=1-{\nu }_A=0\right)$ the equation is known in the form of Equations (50) and (51) [53]. The use of the presented approach seems to be common for thermal dissociation of the solid phase, such as for the complete decomposition of calcium oxalate (CaC₂O₄·H₂O) to CaO [16].

Using the concepts of a certain convergence between the Arrhenius $\left(k\right)$ and Eyring $\left(k_{TST}\right)$ equations, from the equation $\frac{\mathrm{d}\ln k}{\mathrm{d}T}=\frac{\mathrm{d}\ln k_{TST}}{\mathrm{d}T}$ we obtain:

$$∆H^{+}=E-RT$$

(52)

while from the direct comparison of the constant rates $(k=k_{TST})$ we obtain:

$$∆G^{+}=E-RT \ln \frac{A}{BT}$$

(53)

Therefore, from Equation (51) and Equations (52) and (53) we determine the entropy of activation:

$$∆S^{+}=R\left. [\ln \left(\frac{A}{BT}\right)-1\right]$$

(54)

However, it should be noted that, despite the widespread acceptance of the presented comparison, there are premises which do not accept the concept of using the same set of input data [52].

On this basis, relationships between both activation thermodynamics and kinetics (parameters of the Arrhenius equation) are derived by taking the temperature of the maximum reaction rate as the activation temperature for dynamic conditions [53,54]:

$$∆H^{+}=E-R{T}_m$$

(55)

$$∆S^{+}=R\left. [\ln \left(\frac{A}{B{T}_m}\right)-1\right]$$

(56)

Suppose that Equations (55) and (56) satisfy the EEC, with a slope called the compensation temperature $T_c$ (see Equation (2)), which directly follows from the second law of thermodynamics:

$$∆H^{+}=T_c∆S^{+}+H$$

(57)

or:

$$H=∆H^{+}-T_c∆S^{+}=∆G^{+}\left(T_c\right)=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$$

(58)

By substituting Equations (55) and (56) into the form (57) and dividing the sides by $\left(R·T_C\right)$, after transformation we get:

$$\frac{E}{R{T}_c}=\ln A-\ln \left(B{T}_c\right)+\frac{H}{R{T}_c}-\ln \left(\frac{T_m}{T_c}\right)+\frac{T_m}{T_c}-1$$

(59)

The sum of the last three components of Equation (59) is close to 0, which is evident either for $T_m=T_c$ or by taking the approximation: $\ln \left(\frac{T_m}{T_c}\right)\cong \frac{T_m}{T_c}-1.$ The derivation is also valid by taking ${T=T}_m$, so it shows features of generality, and Equation (59) takes the form KCE:

$$\ln A=\frac{E}{R{T}_c}+\ln \left(B{T}_c\right)-\frac{H}{R{T}_c}$$

(60)

The intercept in Equation (60) is:

$$\ln k\left(T_c\right)=\ln \left(B{T}_c\right)-\frac{H}{R{T}_c}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$$

(61)

and is the logarithmic form of Equation (50) containing the EEC expressed by Equation (58).

Ultimately, Equation (60) takes the typical form for KCE:

$$\ln A=\frac{E}{R{T}_c}+\ln k\left(T_c\right)$$

(62)

which means the identity of the isokinetic and compensating temperatures, $T_{iso}=T_c.$ The reverse possibility of transition from KCE (Equations (44) or (1)) to EEC (Equations (57) or (2)) can also be demonstrated.

From Equations (52) and (54) we extract the relevant Arrhenius parameters from the activation function at temperature $T=T_{iso}$:

$$E=∆H^{+}+R{T}_{iso}$$

(63)

$$\ln A=\frac{∆S^{+}}{R}+\ln \left(B{T}_{iso}\right)+1$$

(64)

and after inserting Equations (63) and (64) into the KCE structure we obtain:

$$∆H^{+}=T_{iso}∆S^{+}+R{T}_{iso}\ln \left(\frac{B{T}_{iso}}{k_{iso}}\right)$$

(65)

Equation (65) satisfies the EEC expression of Equations (57) and (58). In addition, by substituting Equation (65) in the intercept: $T_{iso}=T_c$ and using the Arrhenius equation for $k_{iso}=k_c$, we again obtain Equation (53).

The treatment of the presented considerations is a continuation of papers [1,4,5,20] and is based on the relationship between the parameters of the Arrhenius law and Eyring’s theory, from which it follows that the isokinetic temperature is equal to the compensation temperature. However, in the light of propositions to verify the physicochemical sense of the first mentioned [48,49] and one presented in Figure A1, Figure A2, Figure A3 and Figure A4 (see Appendix A), the determined quantities are reliable only from the point of view of statistical evaluation.

4.4. Isokinetic Temperature Again

In order to clarify the methodology of the experimental determination of isokinetic temperature, the parameters of the Arrhenius equation were used.

For the assumptions: $f\left({\alpha }_0\right)=1$, $g\left({\alpha }_0\right)={\alpha }_0$, Equation (22) determines the activation energy:

$$E=R\frac{T_0}{{\alpha }_0}$$

(66)

and using both Equation (19) and Equation (66) we obtain:

$$\ln A=\frac{1}{{\alpha }_0}+\ln \left(\frac{q}{T_0}\right)$$

(67)

The data illustrated in Figure A1 $(N=7)$ and according to Equations (66) and (67) $(N=7)$ were combined with an additional coordinate considered in this work as a reference in [35] (e.g., Anderson) $(N=15)$ and are presented in Figure 7.

The determined activation energies presented in Figure 7 form two resultant groups, in fact, they are combined in the form of one simple KCE. Results of the scoring method (much higher values–red) can be classified as quasi-isoconversional $\left({\alpha }_0\cong \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\right)$. On the other hand, values that are lower (blue) or close to the reference data (E = 191 kJ/mol, lnA = 15.4/ A in s⁻¹) [35], e.g., Anderson) are determined for individual heating rate-dependent curves by the Coats−Redfern equation for a range of changes $\alpha \in \left. [0.1-0.9\right]$.

For the data presented in Figure 7 for Equation (46) and $T_{iso}=914.29 \mathrm{K}$, $r^2=0.9980$ and temperatures: $T_{low}=976 \mathrm{K}$, $T_{high}=1069.27 \mathrm{K}$ (for $\alpha \in \left. [0.1-0.9\right],\mathrm{s}\mathrm{e}\mathrm{e}$ [5]) and setting up $T^{*},T_{high},T_{hm}=1020.51 \mathrm{K}$ for $N=15$ and probability at least 99% (sl = 0.01) (Equation (48)):

$$CQF=0.7017,\hspace{1em}\hspace{1em}{CQF}_{99}=0.29+\frac{1.41}{{15}^{0.57}}0.591$$

and from Equation (49):

$$K=2.50$$

This result shows the reliability of the determined isokinetic temperature, which is ahead of the range of tested temperatures $\left(T_{iso}<T_{low}\right)$. It therefore has the character of an extrapolated quantity.

4.5. Summary of Equations for Experimental Data

The resolution of the significance of the determined temperatures is described by three equations:

• typical KCE equation (Figure 7):

$$\ln A=\frac{E}{R·914.29}+\ln \left(5.455·{10}^{-5}\right), r^2=0.9980$$

(68)

• slope in Equation (57) provided in [43] concerning calcite dissociation from different references:

$$∆H^{+}=1164.6·∆S^{+}+342.22·{10}^3$$

(69)

• from the linear relationship between the parameters of the three-parameter equation, Table 5 (slope in Equation (70)):

$$A_1=896.29·A_2+31440\hspace{1em}\hspace{1em}{r}^2=0.9977 \mathrm{for} 7 \mathrm{heating} \mathrm{rate}$$

(70)

However, according to Equation (45), for the constant temperature provided in Equation (68) $T_{iso}=914.29 \mathrm{K}$, a relatively large range is obtained: ${\stackrel{-}{k}}_{iso}=5.541·{10}^{-5}\pm 9.899·{10}^{-6} s^{-1}$.

In turn, we calculate only one compensation constant: $k_c=0.0108 s^{-1}$ from Equation (61) for the constants given in Equation (69). The value of this constant is much higher than the calculated average value of the isokinetic constant ${\stackrel{-}{k}}_{iso}$. Thus, the intercept in KCE is literally a quantity that averages a wide range of variation in the kinetic constant at the isokinetic point.

Synoptic Figure 8 explains the discrepancy between $T_0$ and the experimentally obtained KCE. The analytical considerations take into account a very wide range of conversion degree variations up to the co-ordinate $\left. [T_0; {\alpha }_0\right]$ while the research scope focuses on higher conversion degrees. By making selections in the base of single column matrix $〈\alpha 〉$ towards $\alpha \in \left. [0.1-0.9\right]$, the slope, or activation energy, increases.

According to

Table 6. Summary of temperatures determined for experimental data ($T_{\mathrm{e}\mathrm{q}}$ = 1172.4 K [19]).

Equations/Table or Figures	${\mathit{T}}_0,$ K	${\mathit{T}}_{\mathit{i}\mathit{s}\mathit{o}},$ K	${\mathit{T}}_{\mathit{c}},$ K	Remarks
(65)	-	914.29	-	-
(66)	-	-	1164.6	-
(67)	896.29	-	-	-
Table 4	876.3–990.6 (939.0)	-	-	-
Table 5	879.05–991.37 (939.88)	-	-	-
Figure A1, Figure A2, Figure A3 and Figure A4 for q = const	-	949.5–1060.6	-	$T_{iso}$ is only of correlation−statistical meaning
Figure A1, Figure A2, Figure A3 and Figure A4 for α = const	-	1074.0–1298.4	-	$T_{iso}$ is higher for isoconversional method (α = const.)

, the relationships between the obtained temperature quantities are not very different from each other. The identity between $T_{iso}$ and $T_c$ is not confirmed (according to Equation (62)), and in turn $T_c=1164.6 \mathrm{K}$ approaches $T_{eq}=1172.4 \mathrm{K}$ [19]. The initial temperatures are consistent for the snapshots given in Table 4 and Table 5 and, as it were, in their interval “absorb” the isokinetic temperature of Equation (68), which fulfills the indicator criterion of CQF, K [48,49].

The experimental data in terms of Equations (44) and (68) form a sequence of inequalities $T_c>T_{iso}$, while the exact isokinetic temperature is calculated from the formula: $\mathrm{s}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}=(R{T}_{iso}{)}^{-1}$. In the correlation calculus using the ratio $1/T$ for the exact calculation of $T$ was emphasized in the work [43], by which the accuracy of the numerical values between KCE and EEC (Equations (68) and (69)) should be distinguished, which was also pointed out in [55].

However, when too large a fluctuation of the assumed constancy of $T_c$ is observed, it becomes necessary to carry out selection according to the proposal of Perez-Benito et al. [5,20]. The simplest approach is represented by inequality (71):

$$T_c>T$$

(71)

where $T$ corresponds to temperature $T_m$ in Equations (55) and (56) and is determined in [20] as a working temperature.

According to the findings of (71), the inequality is satisfied by Equations (68) and (69) in relation to the temperature of the maximum reaction rate, equal to $T=T_m=1125.52 \mathrm{K}$ for a heating rate of 25 K/min (Table 5). In contrast, Equations (10) and (70) are closer to the definition of initial temperature. These findings remain consistent with the proposal of [43].

Following the concepts already used in Equations (46)−(49), the physicochemical sense of isokinetic and compensation temperature, which according to Equation (62) should be the same, it is most advantageous to verify in two steps based on the work [48,49] for statistically significant correlations.

It was proved experimentally that for chemical reactions of four chemically similar compounds (dissociation of hexacene dimers) the Eyring plot $\left(\ln \left(k/T\right) \mathrm{v}\mathrm{s}. 1/T\right)$ indicates the fact of occurrence of $T_c$ several times in a narrow temperature range [56].

EEC expands our understanding of the little-studied quantitative aspects of organic reactions. Experimental evidence of its physical reality in a number of cross reaction series is considered through non-catalytic and pyridine-catalyzed reactions of aryloxiranes with organic acids of various classes. In the context of the compensation effect, there are transitions from one state of reaction systems in which the enthalpy term of the free activation energy acquires a zero value ($∆H^{+}=0$, $∆G^{+}=-T∆S^{+}$), to another, in which the contribution to the free energy of activation of the entropy term disappears ($∆S^{+}=0$, $∆G^{+}=∆H^{+}$) [57]. Mentioned in parentheses is that the thermodynamic states of the activation process are included in Equation (58).

On the other hand, from a physicochemical point of view, a very interesting example is presented in [46] concerning temperature programmed desorption (TPD), when the surface coverage decreases with increasing temperature. Analyzing this phenomenon as a reversible process, while the isosteric heat of adsorption–once the entropy is determined–leads to EEC [6], the constant surface coverage condition is not respected–as is the case in isothermal sorption kinetic measurements–we observe a rather large scatter in terms of KCE [47].

4.6. Some Remarks to All the Elements

For dynamic conditions, the equation in differential form, Equation (8), is commonly recognized as the starting point for consideration. If the approximate form of the solution of Equation (8), referred to by the authors as the Coats−Redfern equation (Equation (9)), interacts with this equation, it leads to Equation (13). This equation has an identical mathematical structure, independent of the heating rate and pre-exponential factor. The consequences of analyzing the adoption of specific analytical forms of the integration constant open up new possibilities for comparison of the Arrhenius parameters obtained through different paths, including Vyazovkin’s, discussed in detail in the book [31]. In contrast to isoconversion concepts, in this work one still stays with the classical variant $\left(q=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\right)$ and for this possibility and for simulated “in silico” data the very important significance of the coordinate $\left. [T_0,{\alpha }_0\right]$ is presented, thanks to which Equation (25) reproduces the activation energy $E$ and also the constant $\ln A·f\left({\alpha }_0\right)$. For many models, $f\left({\alpha }_0\right)$ can be calculated or simply ignored.

In relation to the proposal to determine the “elbow” point coordinate, based on the knowledge of higher order derivatives [38], the methodology has been simplified, which boils down to determining the isokinetic temperature from the constant in Equation (19) and calculating the initial transformation step from Equation (27) or (40).

On the other hand, consideration of simulated data does not provide a basis to demonstrate the existence of the KCE, which appears for experimental data. It is necessary not only to verify it statistically, but also using the CQF and K indicators via Equations (46)−(49) to give the isokinetic temperature a physicochemical sense.

In the association of works [5,20,21,43], the experimentally determined considered temperatures form a sequence, where $T_{eq}$ denotes the thermodynamically determined equilibrium temperature of the reaction (Equation (6)), sometimes also called the inversion temperature:

$${T_0\leftrightarrow T}_{iso}\leftrightarrow T_c\le T_{eq}$$

(72)

The current work proved that the temperatures $T_{iso}$ and $T_c$ are theoretically equal (compare Equations (44) and (62)), while the temperature sequence given in Equation (72) follows from many factors. These are: sample quality (mass, degree of purity, moisture content, crystalline/amorphous form, analysis conditions: isothermal/dynamic, heating rate, atmosphere, flow, reactor type) and workshop issues (variation of $\alpha$, calculation techniques, distinguishing variants: $q=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}, \alpha =\mathrm{v}\mathrm{a}\mathrm{r}$ or $q=\mathrm{v}\mathrm{a}\mathrm{r}, \alpha =\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$). In light of these considerations, the isokinetic temperature $T_{iso}$ is of lesser reliability than the initial temperature $T_0$ or even the compensation temperature $T_c$, which is implicit in the kinetic relationship.

As noted in [52], Equation (69) is derived from various sources, and Equation (68) is composed of correlation and point analysis–both equations involve the same reaction/process of thermal decomposition of calcite.

The omitted analysis of Equation (3) in conjunction with Equation (2) was hinted at in the paper [43].

5. What’s Next?

From the point of view of the equations used, the starting point is the generally used Equation (8) for dynamic conditions and its most commonly used approximate Coats−Redfern solution (Equation (9)). The relation of Equations (8) with (9) leads to the differential Equation (11) with the analytical solution Equation (13). The condition of Equation (10) allows the determination of several equivalent forms of the integration constant in Equations (23) and (24). The final form of Equation (25) determines the activation energies using the heating rate (q) and initial temperature (T₀). In further steps, a new way of determining the coordinates [T₀;α₀] in Equations (40)−(43) is proposed. Further procedures use the relations of KCE, EEC and their constant characteristic temperatures against a narrowly variable initial temperature (T₀).

6. Conclusions

1. The relationship between KCE and EEC is mutually possible and it has been shown that the relationship $T_{iso}=T_c$ theoretically should occur. In light of these considerations, the isokinetic temperature $T_{iso}$ is of less reliability than the initial temperature $T_0$ or even the compensation temperature $T_c$. The analysis for the same data of the thermal dissociation of calcite in nitrogen comes down to a sequence of temperatures ${T_0\leftrightarrow T}_{iso}\leftrightarrow T_c\le T_{eq}$.

2. The effect of the generally known Coats−Redfern solution influence on the differential kinetic equation for dynamic conditions (Equation (8)) and on the resulting isokinetic mathematical forms (Equation (13)) has been used. For in silico and experimental data in the case of thermal dissociation of simple solid chemical compounds, a relationship between the initial temperature $T_0$ and the heating rate has been demonstrated, which enables the determination of the Arrhenius equation parameters. An analytical way of determining the coordinate $\left. [T_0, {\alpha }_0\right]$ based on the three-parameter equation considering the derivative at temperature $T_0$, $\mathrm{d}\alpha /\mathrm{d} \ln T\equiv 1$ has been proposed.

3. In solid-phase thermokinetic analysis, isokinetic relationships have been known for years. Equation (13) is independent of the heating rate and the activation energy is in turn invariant with the conversion degree. The integration constant in this equation can be expressed analytically in various ways. The possibility of its determination by quantities included in the coordinate $\left. [T_0;{\alpha }_0\right],$ when $E=$ const, has been demonstrated. This type of viewpoint presents a different approach from the isoconversion methodology, for which $E=E\left(\alpha \right)$ is very often observed.

4. Using the parameters CQF and K as in [48,49], it has been shown that KCE in the physicochemical sense favors wide ranges of parameter variations of the Arrhenius law. The CQF index determines the reliability of even a very high probability of a linear relationship of the KCE. On the other hand, the position parameter K, assuming $r^2=1$, indicates that, for −1 ≤ K ≤ 1, the isokinetic temperature is within the range of tested temperatures and for the other values it has an extrapolative character.