Introduction

Processes in condensed state are widely studied by thermoanalytical techniques. Their mechanisms are very often unknown or too complicated to be characterized by a simple kinetic model since they tend to occur in multiple elementary steps with different rates. Besides the elementary reactions, also the heat and mass transfer processes, and phase transitions may take place. For the description of the kinetics of such complex processes, the methods based on the general rate equation (GRE) are often used.

Within the framework of GRE, the rate of the complex process is expressed as follows [1,2,3]:

$$\frac{{{\text{d}}\alpha }}{{{\text{d}}t}} = k\left( T \right)f\left( \alpha \right)$$
(1)

where α is the degree of conversion (or shortly conversion), t is time, k(T) is the temperature function depending solely on temperature T and f(α) is the conversion function, depending solely on the conversion of the process. Hence, GRE employs a supposition that the kinetics of a complex process can be expressed in the factorized form as a product of two functions independent of each other as given by Eq. (1).

Among the methods based on GRE, the isoconversional methods are the most widely applied in thermoanalytical kinetics since their application is simple, and they are considered to provide reliable results. Isoconversional or model-free methods assume that the conversion function, f(α), holds over a certain range of conversions. For a constant level of conversion, the rate of the process becomes just a function of temperature [1,2,3].

In this paper, two ways of interpretation of isoconversional kinetic parameters are presented resulting from two different interpretations of the physical meaning of GRE. The first way is based on the premise that GRE is a real kinetic equation which leads to the concept of variable activation energy. ICTAC Kinetics Committee recommendations for performing kinetic computations on thermal analysis data are based on this approach [1]. However, there is accumulating evidence indicating that the concept of variable activation energy and the Recommendations [1] may not be fully correct [4,5,6,7]. We will analyze here the physical meaning of GRE and the related kinetic parameters, the application of non-Arrhenian temperature functions and the kinetic compensation effect (KCE).

Terminology of the isoconversional methods

The isoconversional methods can be divided into the integral, differential and incremental ones [2] (Fig. 1). The terminology for the integral and differential methods is settled down. On the other hand, as for the incremental methods, the terminology is a little confused since these methods are called also as “advanced” [8] or even “integral” [9] so that it is advisable to recap what is meant under individual terms.

Fig. 1
figure 1

Integral, incremental and differential isoconversional methods. If integral methods are applied in the usual way, i.e., assuming variable activation energy, a paradoxical situation arises with several distinct values of E being assigned to the same part of α(T) curve. For example, within the conversion range (0, α1) the integral method yields value E1. In the next step, a potentially different value E2 is assigned within the whole conversion range (0, α2). However, this cannot be true as part of this range was already evaluated and assigned with E1. In other words, subsequent data may affect the preceding ones. This contradiction is resolved in case of differential and incremental methods where each part of α(T) curve is evaluated exactly once

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As for the integral isoconversional methods, it is assumed that the conversion function in Eq. (3) holds within the conversion range <0; α> . The Arrhenius equation is almost exclusively employed as the temperature function in GRE [1,2,3]:

$$k = A\exp \left[ { - \frac{E}{{{{RT}}}}} \right]$$
(2)

where A and E stand for the preexponential factor and the activation energy, R is the gas constant. Provided that A and E are constant and combining Eqs. (1) and (2), after the separation of variables one can get [1, 6, 7]:

$$\int\limits_{0}^{\alpha } {\frac{{{\text{d}}\alpha }}{f\left( \alpha \right)}} = A\int\limits_{0}^{{t_{\upalpha } }} {\exp \left[ { - \frac{E}{{{{RT}}}}} \right]} {\text{d}}t$$
(3)

where tα is the isoconversional time. Isoconversional methods are mostly applied for the experiments with linear heating where the temperature of the sample is expressed as follows:

$$T = T_{0} + \beta t$$
(4)

Combining Eqs. (3) and (4), one gets [1,2,3]

$$\int\limits_{0}^{\alpha } {\frac{{{\text{d}}\alpha }}{f\left( \alpha \right)}} = \frac{A}{\beta }\int\limits_{0}^{{T_{\upalpha } }} {\exp \left[ { - \frac{E}{{{{RT}}}}} \right]} {\text{d}}T$$
(5)

where Tα is the isoconversional temperature and β is the heating rate.

In the incremental isoconversional methods, the experimental results are treated in the interval of conversions <α;α + Δα> where Δα is the conversion increment [2]:

$$\int\limits_{\alpha }^{\alpha + \Delta \alpha } {\frac{{{\text{d}}\alpha }}{f\left( \alpha \right)}} = \frac{A}{\beta }\int\limits_{{T_{\upalpha } }}^{{T_{{\upalpha + \Delta \upalpha }} }} {\exp \left[ { - \frac{E}{{{{RT}}}}} \right]} {\text{d}}T$$
(6)

For the differential isoconversional methods, in contrast to integral and incremental ones, the treatment of experimental results is carried out along a single isoconversional line (Fig. 1). Combination of Eqs. (1) and (2) leads to the relationship

$$\frac{{{\text{d}}\alpha }}{{{\text{d}}t}} = Af\left( \alpha \right)\exp \left[ { - \frac{E}{{{{RT}}}}} \right]$$
(7)

For the isoconversional line, Eq. (7) is mostly linearized via writing it in the logarithmic form as [10]

$$\ln \left( {\frac{{{\text{d}}\alpha }}{{{\text{d}}t}}} \right)_{\upalpha } = \ln \left[ {Af\left( \alpha \right)} \right] - \frac{E}{{{{RT}}_{\upalpha } }}$$
(8)

When applying Eq. (7) or (8), two experimental curves are needed. The first one is the dependence of conversion on temperature for the determination of isoconversional temperature for the chosen level of conversion. The other one is the curve of the reaction rate on temperature for determining the reaction rate at the isoconversional temperature.

The kinetic parameters in isoconversional methods are obtained from the set of experimental kinetic runs at various heating rates. From isoconversional methods, two parameters are obtained, i.e., the activation energy and the product of the preexponential factor and conversion function (for differential method, Eqs. 7 and 8), or the activation energy and the ratio of preexponential factor and integral of the inverted form of the conversion function (for integral and incremental methods, Eqs. 5 and 6).

Concept of variable activation energy

This concept is inextricably connected with the name of Sergey Vyazovkin [11, 12]. We briefly summarize the concept so that only the most important papers are cited here.

The isoconversional methods originate in the isoconversional principle that allows one to eliminate the reaction model from kinetic computations. The principle can be derived from Eq. (8) by taking the temperature derivative at α = const:

$$\left( {\frac{{\partial \ln \left( {{\raise0.7ex\hbox{${{\text{d}}\alpha }$} \!\mathord{\left/ {\vphantom {{{\text{d}}\alpha } {{\text{d}}t}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${{\text{d}}t}$}}} \right)}}{\partial T}} \right)_{\upalpha } = \frac{E}{{{{RT}}^{2} }}$$
(9)

When deriving Eq. (9), two essential assumptions are tacitly adopted: (i) The rate of a generally complex process taking place in the thermoanalytical investigation can be described by the factorized form of Eq. (1); (ii) The kinetic parameters A and E in Eq. (2) are constant, i.e., they depend neither on temperature nor on conversion.

For the complex process, the kinetic parameters are considered “apparent,” “averaged,” “effective” or “global” to emphasize that the process may be complex [1,2,3]. The temperature function in Eq. (1) is deemed the rate constant and the conversion function is taken for the model function reflecting the mechanism of the process. Equation (9) does not contain the conversion function so that the isoconversional methods are also called the model-free methods [1,2,3]. The evaluation methods are concentrated predominantly on obtaining the values of activation energy; the role of the preexponential factor is mostly neglected. Presenting the values of activation energy is the only justification for publication in some papers. Conclusions on the mechanism of the process are drawn from the dependence of activation energy on conversion, see, e.g., the papers [13,14,15]. In [16], the preexponential factor is explored based on the use of the compensation effect and the activation entropy changes are also calculated.

The need for development of isoconversional methods was recognized in [11] where it was written: “The accumulation of empirical knowledge and attempts to obtain its theoretical generalization should ultimately result in building an adequate conceptual basis for the kinetic analysis of solid-state reactions. In the meantime the acceptance of the concept of variable activation energy seems a reasonable compromise between the actual complexity of solid-state reactions and oversimplified methods of describing their kinetics.”

GRE as a formal mathematical tool

Here, we continue in building up the basis for the treatment of thermokinetic data. This alternative to the concept of variable activation energy is based on the mathematical analysis of properties of GRE.

Single-step approximation and general rate equation

A complex process involves several elementary reactions. During the elementary reaction, products are formed from reactants via the transition state (or the activated complex). With the elementary reactions, such quantities are connected as the enthalpy, entropy and Gibbs energy of activation [17]. The rate constant of the elementary reactions can be expressed by the Arrhenius equation with constant activation parameters.

For a complex process, each reaction step should be described by its own kinetic equation. Equation (1) resembles a single-step kinetic equation, even though it is a representation of the kinetics of a complex condensed-phase process. The single-step approximation resides in substituting a generally complex set of kinetic equations by the sole single-step kinetic equation [18,19,20]. Flynn [21] recognized that Eq. (1) is not a true kinetic equation and speculated that, for a reliable description of the kinetics of complex processes, a cross term between temperature and conversion should be included in Eq. (1).

Verification of the function separability

The verification will be carried out for the simplest cases of complex processes, i.e., for the parallel, opposing and consecutive processes.

Simplest case of a complex process: two parallel reactions

Parallel reactions attract the attention for long time [19, 22,23,24,25]. Here we apply the same approach as used in [24, 25]. Consider the reaction scheme

$$\begin{gathered} \text{A}\mathop{\longrightarrow}\limits^{{k_{1} }}\text{B} \hfill \\ \text{A}\mathop{\longrightarrow}\limits^{{k_{2} }}\text{C} \hfill \\ \end{gathered}$$
(10)

where k1 and k2 are the rate constants of the two elementary reactions. Assume that both reactions are first-order; then, the rate equation of the complex process given by Scheme (10) is:

$$\frac{{{\text{d}}\alpha }}{{{\text{d}}t}} = \left( {k_{1} + k_{2} } \right)\left( {1 - \alpha } \right)$$
(11)

Comparing Eq. (11) with Eq. (1), one can get

$$k\left( T \right) = k_{1} + k_{2}$$
(12)
$$f\left( \alpha \right) = 1 - \alpha$$
(13)

The apparent rate constant is mostly expressed by Eq. (2). The apparent activation energy of the complex process can be obtained by applying the isoconversional principle given by Eq. (9). By logarithming Eq. (12) and combining the result with Eq. (9), we get:

$$E = \frac{{k_{1} E_{1} + k_{2} E_{2} }}{{k_{1} + k_{2} }} = \frac{{A_{1} E_{1} \exp ( - {{E_{1} } \mathord{\left/ {\vphantom {{E_{1} } {{{RT}}}}} \right. \kern-0pt} {{{RT}}}}) + A_{2} E_{2} \exp ( - {{E_{2} } \mathord{\left/ {\vphantom {{E_{2} } {{{RT}}}}} \right. \kern-0pt} {{{RT}}}})}}{{A_{1} \exp ( - {{E_{1} } \mathord{\left/ {\vphantom {{E_{1} } {{{RT}}}}} \right. \kern-0pt} {{{RT}}}}) + A_{2} \exp ( - {{E_{2} } \mathord{\left/ {\vphantom {{E_{2} } {{{RT}}}}} \right. \kern-0pt} {{{RT}}}})}}$$
(14)

In Eq. (14), the rate constants of elementary reactions, k1 and k2, are expressed by the Arrhenius equation, where A1, E1 and A2, E2 are the corresponding preexponential factors and activation energies.

The apparent preexponential factor can be obtained by combining Eqs. (12) and (14):

$$A = \left( {k_{1} + k_{2} } \right)\exp \left[ {\frac{E}{{{{RT}}}}} \right]$$
(15)

As can be seen from Eqs. (14) and (15), both apparent kinetic parameters, A and E, depend on temperature. The dependence of A and E on conversion is commonly known [11, 12]. The kinetic parameters A and E thus depend on both T and α.

The process given by Scheme (10) is the simplest example of a complex process. For this process, both apparent kinetic parameters, A and E, are not constant but, conversely, depend on both α and T. Hence, this can be generalized. If the kinetics of a complex process is described by GRE, Eq. (1), the apparent kinetic parameters, A and E, depend on both α and T.

In Eq. (11) it is assumed that both elementary reactions in the reaction scheme are first-order, i.e., the conversion functions f1(α) = f2(α) = 1 − α. In case that f1(α) ≠ f2(α), the reaction rate is

$$\frac{{{\text{d}}\alpha }}{{{\text{d}}t}} = k_{1} f_{1} \left( \alpha \right) + k_{2} f_{2} \left( \alpha \right)$$
(16)

Applying Eq. (9) to the logarithmic form of Eq. (16), it can be obtained:

$$E = \frac{{k_{1} E_{1} f_{1} \left( \alpha \right) + k_{2} E_{2} f_{2} \left( \alpha \right)}}{{k_{1} f_{1} \left( \alpha \right) + k_{2} f_{2} \left( \alpha \right)}}$$
(17)

This relationship is derived, for example, in [1, 3]. However, Eq. (17) is evidently incorrect since Eq. (16) cannot be expressed in the factorized form of Eq. (1). This is the prerequisite in deriving the isoconversional principle given by Eq. (9) so that the isoconversional principle is inapplicable in this case.

Opposing and consecutive reactions

In the textbooks of physical chemistry, two other simplest complex processes, i.e., the opposing and consecutive reactions are analyzed. For the opposing reactions, the reaction scheme is

$$\text{A}\underset{{k_{ - 1} }}{\overset{{k_{1} }}{\rightleftharpoons}}\text{B}$$
(18)

The process consists of two elementary reactions. Assuming that the elementary reactions are first-order, the rate equation of the process has been derived [19]:

$$\frac{{{\text{d}}\alpha }}{{{\text{d}}t}} = k_{1} \left[ {1 - \left( {1 + K} \right)\alpha } \right]$$
(19)

where K is the equilibrium constant of the reverse reaction in Scheme (18) at actual temperature. For constant temperature, Eq. (19) obtains the factorized form of Eq. (1). Nonetheless, the isoconversional methods are predominantly applied for linear heating (Eq. 4). For non-isothermal conditions, the equilibrium constant is given by the Gibbs isotherm

$$K = \exp \left[ { - \frac{{\Delta_{{\text{r}}} G}}{{{{RT}}}}} \right]$$
(20)

where ΔrG is the standard reaction Gibbs energy. Hence, under the non-isothermal conditions, Eq. (19) cannot be expressed in the factorized form of Eq. (1).

Regarding the consecutive reactions, their reaction rate cannot be expressed in the factorized form of Eq. (1) [26].

Meaning of GRE and of the apparent kinetic parameters

As seen from the section “Verification of the function separability”, for the simplest complex processes, the reaction rate can be expressed in the factorized form of Eq. (1) only for two (or more) parallel reactions with the same conversion function f(α). In the case of parallel reactions with f1(α) ≠ f2(α), for the non-isothermal first-order opposing reactions and for the consecutive reactions, the rate of the process cannot be expressed in the factorized form. For more complex processes, it is highly improbable that the reaction rate, based on the true reaction mechanism, could be expressed in the form of Eq. (1). Hence, the reaction rate expressed in the factorized form of Eq. (1) is more an exception than a rule; reaction rate, based on the true reaction mechanism, cannot be generally expressed in the form of Eq. (1). Immediately, a question arises what is the physical meaning of Eq. (1). To analyze this problem, it is necessary to realize that the rate of the processes in condensed state is generally a function of temperature and conversion [18,19,20]:

$$\frac{{{\text{d}}\alpha }}{{{\text{d}}t}} = \Phi \left( {T,\alpha } \right)$$
(21)

The dependence given by Eq. (21) is manifested in the experimental data; however, its mathematical functional form is unknown. The single-step approximation [18, 19] employs the assumption that the function Φ(T,α) in Eq. (21) can be expressed as a product of k(T) and f(α). Equation (1) thus approximates the unknown functional form of the kinetic hypersurface given by Eq. (21). There may exist variety of the couples k(T) and f(α) approximating the kinetic hypersurface with the same or similar exactness. Thus, it is also a matter of course that both functions in Eq. (1) lack any physical meaning.

Hence, from the mathematical point of view, the temperature and conversion functions in GRE represent a deconvolution of the kinetic hypersurface into the contributions of temperature and conversion. From the point of view of probability, the meaning of Eq. (1) is that temperature and conversion affect the rate of the process independently without any mutual interaction. Such an approach can be encountered in the theory of transformations in metals and alloys, in the prediction of microbial spoilage in foods, and in modeling of concrete curing [27]. If the complex process depends on pressure, irradiation, etc., further terms describing their effects can be added to Eq. (1) [1, 27].

In the concept of variable activation energy, the temperature function in Eq. (1) is mostly interpreted as the rate constant and the conversion function is considered to reflect the mechanism of the process. It was discussed in [18,19,20] that this interpretation of both functions is incorrect. The temperature function k(T) is not the rate constant and the conversion function f(α) does not reflect the mechanism of the complex process. GRE could be a true reaction rate equation only in two special cases: (i) The process is really elementary; (ii) the process involves an elementary step limiting the rate of the complex process. These possibilities should always be proven or at least, soundly reasoned. The processes (i) and (ii) will be further called the simple processes.

Hence, GRE is not a true reaction rate equation in general; it is a mathematical tool enabling to describe the thermoanalytical kinetic data. The functions k(T) and f(α) represent, in general, just the temperature and conversion components of the kinetic hypersurface where the hypersurface is dependence of the degree of conversion (or the rate of the process) on time and temperature [18,19,20, 27]. The temperature and conversion functions do not have any clear physical meaning in general and neither the related parameters occurring in both functions. Strength of this approach is that the kinetic hypersurface can be described without knowing the detailed mechanism. This enables reconstruction of the observed values of reaction rate, conversion, isoconversional temperature and time and, in general, modeling of the process.

Main differences between the simple and complex processes

The above text shows that it is of vital importance to distinguish between the simple and complex processes. Very often, authors of kinetic papers are not aware of the fundamental differences. A simple process comprises a single elementary reaction governing the rate of the process that occurs via the activated complex. The rate equation is based on the true reaction mechanism, the rate constant obeys the Arrhenius equation and meaning of both kinetic parameters, A and E, is clear. Knowing the kinetic parameters, modeling of the process is possible for various time–temperature regimes.

The complex process involves at least two elementary processes. The theory of transition state is inapplicable in this case since nothing like an apparent or average transition state does exist. The rate equation in the form of Eq. (1) is not a true rate equation, it is just a single-step approximation. Meaning of the kinetic parameters A and E is unclear in general. Knowing the kinetic parameters, limited modeling of the process is possible. The main differences between the simple and complex processes are summarized in Table 1.

Table 1 Main differences between the simple and complex processes
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Kinetic compensation effect

As written in Section "Verification of the function separability", the activation energy and pre-exponential factor depend on both T and α for complex processes. Taking this into account, the temperature derivative of Eq. (8) at constant α should obtain the form

$$\left( {\frac{{\partial \ln \left( {{\raise0.7ex\hbox{${{\text{d}}\alpha }$} \!\mathord{\left/ {\vphantom {{{\text{d}}\alpha } {{\text{d}}t}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${{\text{d}}t}$}}} \right)}}{\partial T}} \right)_{\upalpha } = \left( {\frac{\partial \ln A}{{\partial T}}} \right)_{\upalpha } + \frac{E}{{{{RT}}^{2} }} - \frac{1}{{{{RT}}}}\left( {\frac{\partial E}{{\partial T}}} \right)_{\upalpha }$$
(22)

When evaluating the experimental results applying Eq. (9), it is implicitly assumed that

$$\left( {\frac{\partial \ln \,A}{{\partial T}}} \right)_{\upalpha } - \frac{1}{{{{RT}}}}\left( {\frac{\partial E}{{\partial T}}} \right)_{\upalpha } = 0$$
(23)

After integration over a small temperature interval, one can get:

$$\ln \,A = \frac{E}{{{{RT}}}} + b = aE + b$$
(24)

where a and b are constants. From Eqs. (9), (23) and (24) can be seen that evaluation of experimental results by using Eq. (9) can be another source of the kinetic compensation effect besides the random [28] and systematic [29] errors or other reasons [30]. Kinetic compensation effect thus can be brought about, at least partly, by neglecting the complexity of the process.

Applicability of isoconversional methods

As written above in section “Terminology of the isoconversional methods,” for the integral isoconversional methods it is assumed that the conversion function in Eq. (1) holds within the conversion range <0; α> . For a complex process, A and E depend in general on both T and α. If E depends on α, the integral isoconversional methods are mathematically incorrect [7]. For A depending on T, Eq. (5) should take the form

$$\int\limits_{0}^{\alpha } {\frac{{{\text{d}}\alpha }}{f\left( \alpha \right)}} = \frac{1}{\beta }\int\limits_{0}^{{T_{\upalpha } }} {A\exp \left[ { - \frac{E}{{{{RT}}}}} \right]} {\text{d}}T$$
(25)

Nobody uses Eq. (25) for the treatment of experimental results since there is not much known about the temperature dependence of A. Hence, Eq. (5) is mathematically incorrect in the case of non-isothermal processes. The computer obeys the code commands based on the idea of Eq. (5). However, the resulting parameters A and E are mathematically incorrect in general, and their values may differ for various temperature regimes, for example for the isothermal heating and linear heating. The integral isoconversional methods are correct for simple processes where A and E are constant; they should not be applied for the complex processes. The published results obtained by the integral isoconversional methods should be regarded very critically since they may be mathematically incorrect and; hence, they may not be trustworthy. Flynn–Wall–Ozawa [31, 32], Kissinger–Akahira–Sunose [33], Starink [34], Vyazovkin advanced [35], Vyazovkin [36] methods belong to the very popular integral isoconversional methods that should be avoided. Regarding the practice, the standards such as [37], dealing with the determination of the activation energy by the integral isoconversional methods, should be suspended. Further, the Kissinger method is derived under the assumption that A and E are constant [38] so that this method is also applicable for simple processes only.

The treatment of kinetic results should be carried out in the way that A and E could be considered constant. This is fulfilled for the incremental and differential isoconversional methods. In the incremental isoconversional methods, the experimental results are treated in the interval of conversions <α;α + Δα> where Δα is the conversion increment. For a sufficiently small increment, both A and E can be considered approximately constant so that the rate equation obtains the form of Eq. (6). There have been proposed quite many incremental methods [9, 39,40,41,42,43]. The widely applied method [8] is in fact also an incremental one.

Regarding the differential isoconversional methods, the treatment of experimental results is carried out by employing Eq. (8). The method is called the Friedman method [10].

The differential isoconversional methods are known to suffer from noise that possibly brings about great errors in A and E [27].

Use of non-Arrhenian temperature functions

In Eq. (1), the Arrhenius equation (Eq. 2) is almost exclusively applied as the temperature function. In homogeneous kinetics, the validity of the Arrhenius equation is so comprehensively accepted that its application normally requires no justification. As for the condensed-phase kinetics, the values of E and A are usually regarded as possessing significances similar to those envisioned in the theory developed for homogeneous reactions. However, there are quite numerous indications gathered that the Arrhenius equation would not be the right one for the use in thermoanalytical kinetics. In [44, 45], it was reasoned that the application of the Arrhenius model is questionable. In [46, 47], it is substantiated that the energy distribution in solid-state differs from that in gaseous state so that the application of the Arrhenius equation in the solid-state kinetics is improper. Dollimore et al. [48, 49] employed a non-Arrhenian temperature for the treatment of thermoanalytical data. In [50], the Arrhenius equation is ironically named “the equation of choice.”

In our papers it has been justified that, since k(T) is not the rate constant, there is no reason to be confined to the Arrhenius relationship and use of non-Arrhenian temperature functions were suggested [2, 20]. Other temperature functions that can be applied are the Harcourt–Esson and Berthelot–Hood equations:

$$k\left( T \right) = {{AT}}^{\text{{m}}}$$
(26)
$$k\left( T \right) = {{Ae}}^{{{\text{DT}}}}$$
(27)

A great advantage of temperature functions given by Eqs. (26), (27) is that the temperature integral can be expressed in a closed form [20]. The three temperature functions (Arrhenius, Harcourt–Esson and Berthelot–Hood) are equivalent in the temperature range of measurement [51, 52].

In predictions, when extrapolating the data out of the temperature range, the Arrhenius extrapolation frequently provides unrealistically long isoconversional times. The extrapolation based on the non-Arrhenian temperature functions provides material lifetimes much closer to experience [53, 54]. Examples of non-Arrhenian behavior in polymer degradation are presented in [55,56,57,58,59]. In practice, the Berthelot–Hood equation is oftentimes employed such as in the F-value calculation in canning industry [60, 61], for the evaluation of the lifetime of hard polyurethane foams [62] or in the evaluation of the maturity of concrete [63].

Since the kinetic parameters in Eqs. (2), (26) and (27) can be mutually recalculated, the parameters in Eqs. (26) and (27) depend also on α and T for complex processes. The both functions also exhibit the kinetic compensation effect, i.e., there is a linear dependence between ln A and m for Eq. (26) or ln A and D for Eq. (27).

GRE and the Arrhenius function

As seen from sections “Single-step approximation and general rate equation” and “Meaning of GRE and of the apparent kinetic parameters,” the concept of variable activation energy involves an inherent contradiction. The temperature function in Eq. (1), k(T), should depend solely on the temperature T and the conversion function, f(α), should depend solely on the conversion of the process, α. However, A and E in Eq. (2) depend on both α and T, the same holds for the kinetic parameters in Eqs. (26) and (27). Due to the dependence of E on conversion, GRE is thus inseparable in general and cannot be expressed in the factorized form. What does this mean? The only explanation of this entangled situation is that an improper couple of the functions k(T) and f(α) is chosen. Obviously, neither Eq. (2) nor Eqs. (26) and (27) represent a proper choice of temperature function in the case of a complex mechanism. The right temperature function in Eq. (1) is not the Arrhenius equation in general. Hence, the concept of the variable activation energy becomes objectionable.

A properly chosen temperature function should depend on temperature only. Equations (2), (26) and (27) cannot describe the right temperature function correctly in the whole range of conversions < 0;1 > . However, they can approximate the right temperature function for infinitesimal or sufficiently small increments dα or Δα. The unknown right temperature function is interpolated in small increments by temperature functions given by Eqs. (2), (26) or (27). This outcome is in favor of the differential and incremental isoconversional methods and points out again that the integral isoconversional methods are mathematically incorrect in general and their use should be omitted.

Comparison of both concepts

Treatment of experimental data by isoconversional methods represents a feasible way to describe the kinetics of complex processes since it is simple and easy. However, interpretation of the results may not be so simple and straightforward. There exist two ways of interpreting the results obtained by the isoconversional methods that are in sharp contradiction and are mutually incompatible. The calculations are more or less the same in both approaches. The main difference is in understanding of the physical meaning of GRE and the kinetic parameters included in the functions k(T) and f(α).

The concept of variable activation energy assumes that GRE is a true or real kinetic equation. The temperature and conversion functions are considered the rate constant and the reaction model. With a few exceptions [48, 49], the Arrhenius equation is invariably chosen as the temperature function in Eq. (1). For complex processes, the activation energy depends on conversion in general; from this dependence, conclusions on the mechanism of the process are frequently deduced.

The concept of understanding GRE just as a formal mathematical tool leads to substantially different outcomes. From the analysis of the simplest complex mechanisms it follows that the complex mechanisms cannot be expressed in the factorized form of Eq. (1) and that the temperature and conversion functions do not possess a clear physical meaning in general. The temperature function, k(T), is not the rate constant in general and the conversion function, f(α), does not reflect the mechanism of the process. Both, k(T) and f(α), are functions enabling to describe the kinetic hypersurface. Neither the meaning of parameters occurring in both functions is clear for the complex processes. Variable activation energy is just a sign of complexity of the process, it has no other meaning.

Since the physical meaning of parameters is misty, no mechanistic conclusions should be drawn from the values of an individual kinetic parameter; particularly, just from the values of activation energy. GRE is understood as an approximation of the unknown kinetic hypersurface within the range of conversions and temperatures of the experimental data. Choice of the approximation in the factorized form of Eq. (1) enables the development of isoconversional methods. The temperature functions in the form of Eqs. (2), (22) and (23) can approximate the real, yet unknown, temperature function. This can be approximated for infinitesimal or small increments. The two mentioned approximations again underline the fact that the parameters in k(T) and f(α) do not have any physical meaning in general. The conclusions might be done from the quantities with clear physical meaning, i.e., from the values of isoconversional times, isoconversional temperatures, conversion, reaction rate, etc., i.e., the quantities that can be accessible experimentally. These quantities can be recovered and modeled from the kinetic parameters obtained. A great advantage of the isoconversional methods is that only two kinetic parameters are needed for the modeling. Any mechanistic conclusion should be verified by an independent method [27].

Conclusions

It is proved here that, for complex processes, the right temperature function is not necessarily the Arrhenius function. In the light of this statement the concept of the variable activation energy becomes doubtful. The variable activation energy is just a construct without a clear content. In [47] Galwey wrote: “Before describing a quantity as an activation energy, make sure that it involves activation and that it is an energy.” To avoid blunders and overinterpretation, it would be advisable to introduce the definition of temperature coefficient, B, as a ratio of the activation energy and the gas constant [27]. In addition, other temperature functions provide equivalently good description of the experimental data.

The concept of variable activation energy represented a great progress at the time of its origin; however, it is full of controversies. No conclusions should be drawn on the basis of mathematically and logically incorrect thoughts. When considering GRE as a formal mathematical tool for the description of experimental thermoanalytical results, the theory gets the mathematical and logical consistency. Finally, it is essential to realize that the general rate equation is just a mathematical tool, it is not a real rate equation in general. Its reliability depends on the reliability of the single-step approximation given by Eq. (1). If there exist other models and methods better physically justified, it is advisable to apply them instead of the methods based on Eq. (1) [27].