Introduction

The kinetic analysis of solid-state processes allows obtaining fundamental information that can be used for modelling a myriad of processes with direct application to industry [1,2,3,4,5,6,7,8,9,10,11]. Kinetic analysis is quite challenging not only in terms of obtaining reliable experimental data [12,13,14], but also from the point of view of extracting the kinetic parameters from these experimental curves [9, 15]. The main purpose of a kinetic analysis is to determine the kinetic parameters that describe the process and allow predicting its thermal behaviour with accuracy in a wide range of conditions, usually different from those used for the analysis. Recently, it has been recognized that predictions are compromised if the selection of the kinetic model is not properly performed [16]. Moreover, ideal kinetic models in the literature have been proposed assuming ideal conditions that are rare in real systems that present intrinsic features. For example, these ideal models assume homogeneous materials in terms of particle size, while ideal samples usually present a broad particle size distribution (PSD). Previous studies have shown that the PSD might influence the results of kinetic analysis [17]. For instance, it has been demonstrated that interface reaction models can be confused with diffusion or nucleation models if the PSD is not taken into account [18].

The experimental data used in the kinetic analysis can be recorded under a wide range of conditions and using different thermal analysis techniques, including differential scanning calorimetry (DSC), thermogravimetric analysis (TGA) and high-temperature X-ray diffraction. The experimental conditions are often limited by the process studied and the method employed. Concerning this, the isothermal kinetic analysis presents certain advantages as compared to other conditions employed to study solid-state processes. For instance, kinetic studies conducted under isothermal conditions are more effective when discriminating the kinetic model as compared to other approaches, since the shape of the extent of conversion versus time curves directly reflects the kinetic model obeyed by a process [19]. The isothermal kinetic analysis continues to be amply employed nowadays to study a wide range of processes. Thus, it is the most feasible option for reactions studied under realistic conditions of operation [19, 20], long-term curing [21, 22], oxidation and reduction processes [23, 24], hydration and dehydration processes [25,26,27], reactions followed through X-ray diffraction [28] or those studies intended to discriminate the kinetic model under isothermal conditions [29,30,31,32].

Here, we demonstrate mathematically that the value of activation energy determined from kinetic analysis of isothermal curves is independent of the selected model used to fit the data. To the best of our knowledge, this demonstration is unprecedented and has significant implications, as reported values of activation energy are valid independently of the assumed kinetic model. Furthermore, taking advantage of some of the mathematical arguments that led to that conclusion, we present a simple methodology to obtain the values of the activation energy through analysis of two curves recorded under isothermal conditions. The method is validated by analysing different experimental curves, corresponding to different processes, namely the thermal degradation of ethylene–propylene–diene, the thermal decomposition of limestone and the cure of a clear laminating epoxy resin, and comparing the results with those reported in the literature.

Theoretical foundation

The equation that describes the rate of solid-state processes is [7]:

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

where \(t\) is the time, \(\alpha\) is the extent of conversion, \(f\left( \alpha \right)\) is the kinetic model, and \(k\left( T \right)\) is the Arrhenius function given by:

$$ k\left( T \right) = A\exp { }\left( { - \frac{{E_{{\text{a}}} }}{RT}} \right) $$
(2)

Being \(A\) a pre-exponential factor, Ea the activation energy, R the gas constant and T the temperature. In isothermal conditions, \(k\left(T\right)\) becomes a constant and Eq. 1 can be integrated as follows:

$$\underset{0}{\overset{\alpha }{\int }}\frac{\mathrm{d}\alpha }{f\left(\alpha \right) }=k\underset{0}{\overset{t}{\int }}\mathrm{d}t \to g\left(\alpha \right)=kt $$
(3)

where \(g\left(\alpha \right)=\underset{0}{\overset{\alpha }{\int }}\mathrm{d}\alpha /f\left(\alpha \right)\) is the integral form of the kinetic model. Let us consider a process studied under isothermal conditions at two different temperatures, \({T}_{1}\) and \({T}_{2}\). According to Eq. 2, different values of the rate constant \(k\) will be obtained, so that:

$${\left.\frac{\mathrm{d}\alpha }{\mathrm{d}t}\right|}_{{T}_{1}}={k}_{1}f\left(\alpha \right) $$
(4a)
$${\left.\frac{\mathrm{d}\alpha }{\mathrm{d}t}\right|}_{{T}_{2}}={k}_{2}f\left(\alpha \right)$$
(4b)

Therefore:

$${\left.\frac{\mathrm{d}\alpha }{\mathrm{d}t}\right|}_{{T}_{2}}={k}_{2}f\left(\alpha \right)={\frac{{k}_{2}}{{k}_{1}}k}_{1}f\left(\alpha \right)= \frac{{k}_{2}}{{k}_{1}}{\left.\frac{\mathrm{d}\alpha }{\mathrm{d}t}\right|}_{{T}_{1}} $$
(5)

From this equation, we can derive the relationship between the time invested in reaching a given value of \(\alpha \) in the two isothermal experiments.

$${\left.\mathrm{d}t\right|}_{{T}_{2}}=\frac{{k}_{1}}{{k}_{2}}{\left.\mathrm{d}t\right|}_{{T}_{1} } \to \int {\left.\mathrm{d}t\right|}_{{T}_{2}}=\frac{{k}_{1}}{{k}_{2}}\int {\left.\mathrm{d}t\right|}_{{T}_{1}} \to {\left. t\right|}_{{T}_{2}}=\frac{{k}_{1}}{{k}_{2}}{\left.t\right|}_{{T}_{1}} $$
(6)

Figure 1 illustrates the proportionality relationship between the times required to reach a certain value of \(\alpha \) at two different temperatures, being the constant of proportionality equal to the ratio between the rate constants. Data shown in Fig. 1a were calculated using a Runge–Kutta method to solve Eq. 1 for \({T}_{1}=\) 750 K and \({T}_{2}=\) 800 K, assuming \(f\left(\alpha \right)=3{(1-\alpha )}^{2/3}\) (contracting volume model R3) [7], \({E}_{a}=\) 180 kJ mol−1 and \(A=\) 109 s−1. As might be observed in Fig. 1b, both curves overlap when the values of time corresponding to the curve calculated for \({T}_{2}\) are multiplied by the ratio between the two rate constants \({k}_{1}/{k}_{2}\).

Fig. 1
figure 1

a Curves simulated assuming a contracting volume model and isothermal conditions with T1 = 750 K and T2 = 800 K, Ea = 180 kJ mol−1 and A = 109 s−1. b Overlapping of both curves when the values of time calculated for \(T_{2}\) are multiplied by \(k_{2} /k_{1}\)

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The standard method based on the fitting to ideal models to determine the activation energy from isothermal experiments consists of recording several curves at different temperatures and representing some selected kinetic models \(g\left(\alpha \right)\) as a function of \(t\) to estimate which of them best linearizes the data [7]. The process is supposed to obey the model that provides the value of the correlation coefficient closest to one, and the slope of the best-fitting line to the data is assumed to be the rate constant \(k\). Then, the activation energy and the pre-exponential factor can be determined by plotting the logarithm of \(k\) versus \(1/T\).

Let’s suppose that we have a set of \(n\) isotherms recorded at different temperatures: \({T}_{1}\),\({T}_{2}\), \({T}_{3}\)….\({T}_{\mathrm{n}}\). If we represent \(g\left(\alpha \right)\) as a function of \(t\) for an arbitrary given model and apply the linear least squares approximation, the slope for the isotherm corresponding to \({T}_{\mathrm{i}}\) is given by:

$${s}_{i}=\frac{N\sum_{j=1}^{N}g\left({\alpha }_{\mathrm{j}}\right){\left.{t}_{\mathrm{j}}\right|}_{{T}_{\mathrm{i}}}-\sum_{j=1}^{N}g\left({\alpha }_{\mathrm{j}}\right)\sum_{j=1}^{N}{\left.{t}_{\mathrm{j}}\right|}_{{T}_{\mathrm{i}}}}{N\sum_{j=1}^{N}{\left({\left.{t}_{\mathrm{j}}\right|}_{{T}_{\mathrm{i}}}\right)}^{2}-{\left(\sum_{j=1}^{N}{\left.{t}_{\mathrm{j}}\right|}_{{T}_{\mathrm{i}}}\right)}^{2}}$$
(7)

Being \(N\) the number of points \(\left(t,\alpha \right)\) recorded during the experiment. According to Eq. 6, each value \({\left.{t}_{\mathrm{j}}\right|}_{{T}_{\mathrm{i}}}\) is related to \({\left.{t}_{\mathrm{j}}\right|}_{{T}_{1}}\) through the following equation:

$${\left.{t}_{\mathrm{j}}\right|}_{{T}_{\rm i}}=\frac{{k}_{1}}{{k}_{\rm i}}{\left.{t}_{\mathrm{j}}\right|}_{1} $$
(8)

Substituting this into Eq. 7, we get:

$${s}_{\mathrm{i}}=\frac{N\sum_{\rm j=1}^{N}g\left({\alpha }_{\rm j}\right)({{k}_{1}/{k}_{\rm i})\left.{t}_{\mathrm{j}}\right|}_{1}-\sum_{\rm j=1}^{N}g\left({\alpha }_{\mathrm{j}}\right)\sum_{\rm j=1}^{N}({{k}_{1}/{k}_{\mathrm{i}})\left.{t}_{\mathrm{j}}\right|}_{1}}{N\sum_{\rm j=1}^{N}{\left(({{k}_{1}/{k}_{\rm i})\left.{t}_{\mathrm{j}}\right|}_{1}\right)}^{2}-{\left(\sum_{\rm j=1}^{N}({{k}_{1}/{k}_{\mathrm{i}})\left.{t}_{\mathrm{j}}\right|}_{1}\right)}^{2}}=\left(\frac{{k}_{\mathrm{i}}}{{k}_{1}}\right)\frac{N\sum_{\rm j=1}^{N}g\left({\alpha }_{\mathrm{j}}\right){\left.{t}_{\rm j}\right|}_{1}-\sum_{\rm j=1}^{N}g\left({\alpha }_{\mathrm{j}}\right)\sum_{\rm j=1}^{N}{\left.{t}_{\mathrm{j}}\right|}_{1}}{N\sum_{\rm j=1}^{N}{\left({\left.{t}_{\mathrm{j}}\right|}_{1}\right)}^{2}-{\left(\sum_{\rm j=1}^{N}{\left.{t}_{\mathrm{j}}\right|}_{1}\right)}^{2}}=\left(\frac{{k}_{\mathrm{i}}}{{k}_{1}}\right){s}_{1}\;\;\mathrm{ for }\;\;i>1$$
(9)

By taking logarithms, considering that \(k_{{\text{i}}} = A\exp \left( { - E_{{\text{a}}} /RT_{{\text{i}}} } \right),\) and rearranging terms it follows:

$$ \ln s_{{\text{i}}} = \ln \left( {\frac{{s_{1} }}{{k_{1} }}} \right) - \frac{{E_{{\text{a}}} }}{{RT_{{\text{i}}} }} + \ln A\quad {\text{for}} \;\;i > 1 $$
(10)

Thus, regardless of the kinetic model used, \(s_{i}\) linearly depends only on \(1/T_{\text{i}}\) with slope \(- E_{\text{a}} /R\). Equivalently, the activation energy determined by this approach will be the same for any kinetic model employed. Note that when the kinetic model actually obeyed is selected, \(s_{1} = k_{1}\) and Eq. 10 becomes:

$$ \ln k_{\rm i} = - \frac{{E_{a} }}{{RT_{\rm i} }} + \ln A $$
(11)

Thus, the activation energy can be determined from the slope and the pre-exponential factor from the intercept of the line logarithm of \(k\) versus \(1/T\). Figure 2a shows the extent of conversion as a function of time for four temperatures: 750 K, 775 K, 800 K and 825 K. Data were simulated assuming isothermal conditions, the kinetic model R3, \(E_{a}\)=180 kJ mol−1 and \(A\) = 109 s−1. Table 1 collects the results obtained from fitting these data to Eq. 3 assuming different kinetic models. As expected, the value of the correlation coefficient obtained is maximum for the kinetic model R3. This analysis illustrates that the value of the activation energy does not depend on the kinetic model assumed; the slope of the line \(\ln s\) versus \(1/T\) is the same for all the kinetic models studied, consistently with Eq. 10.

Fig. 2
figure 2

a Curves simulated assuming a contracting volume model (R3) and isothermal conditions: 750 K, 775 K, 800 K and 825 K. \(E_{{\text{a}}} =\) 180 kJ mol−1 and A = 109 s−1. b \(\ln s\) as a function of \(1/T\) for different ideal kinetic models

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Table 1 Values of activation energy, rate constants and correlation coefficients obtained from fitting the data plotted in Fig. 2a to Eq. 3 assuming different kinetic models
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The relationship expressed by Eq. 6 enables us to develop a simple method for determining the activation energy with two curves recorded under isothermal conditions, which significantly simplify the typical procedure. Indeed, by plotting the time required to attain a certain value of the extent of conversion at \(T_{2}\) against the time needed when the isotherm is conducted at \(T_{1}\), we obtain a line which slope is given by \(m = k_{1} /k_{2}\). Thus, taking into account Eq. 2 for the Arrhenius function, the activation energy can be calculated as:

$$ E_{{\text{a}}} = R\frac{{T_{1} \cdot T_{2} }}{{\left( {T_{1} - T_{2} } \right)}}\ln m $$
(12)

Let us apply this method to data plotted in Fig. 1. Figure 3 shows the values of \(\left. t \right|_{800K}\) as a function of \(\left. t \right|_{750K}\) for different values of the extent of conversion. The slope of the best-fitting line is m = 0.16445, and applying Eq. 12 we obtain Ea = 180 kJ mol−1 as expected, which validate the method for determining the value of the activation energy with two of the isothermal curves shown in Fig. 2.

Fig. 3
figure 3

Values of \(\left. t \right|_{800K}\) as a function of \(\left. t \right|_{750K}\) for different values of the extent of conversion

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Experimental section

The method described in Sect. 2 was applied to the study of three different reactions: the pyrolysis of ethylene–propylene–diene (EPDM), the thermal decomposition of limestone and the curing of an epoxy resin. The sample of (EPDM) used was supplied by ExxonMobil (Vistalon Rubbers): 69 mass% ethylene, 5 mass% ethylidene norbornene and 26 mass% propylene. The limestone (ESKAL 60) was provided by KSL Staubtechnik GmbH (Germany). Figure 4 shows a scanning electron micrograph and the particle size distribution (PSD) of the sample of limestone used. As might be observed, the sample exhibits a narrow PSD with a peak around 60 μm. Both the thermal degradation of EPDM and the decomposition of limestone were studied in isothermal conditions using a thermogravimetric analyser Q5000IR from TA instruments. The experiments started with a heating ramp at 300 °C min−1 from room temperature to the temperature of the isotherm. Then, the temperature was maintained constant for two hours in the experiments with EPDM and seven hours in the case of limestone. The experiments were carried out in nitrogen with a flow rate of 200 mL min−1. On the other hand, the data corresponding to the curing of the epoxy resin were taken from the literature [22].

Fig. 4
figure 4

SEM micrograph and particle size distribution of the sample of limestone used in this work

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Results and discussion

Figure 5a shows the time evolution of the extent of degradation of EPDM corresponding to two isothermal experiments carried out at 683 K and 693 K. As can be observed in Fig. 5b, there is a linear relationship between the values of time needed to reach a certain value of \(\alpha\) at the two different temperatures employed. The value of apparent activation energy calculated from the slope, after applying Eq. 12, is \(E_{{\text{a}}} = \left( {253.4 \pm 0.7} \right)\) kJ mol−1, which is in agreement with that reported in the literature \(E_{a} = \left( {252 \pm 8} \right)\) kJ/mol [33]. Figure 5c illustrates that both isotherms overlap when Eq. 6 is applied using the value of \(E_{a}\) calculated from the slope of the line plotted in Fig. 4b.

Fig. 5
figure 5

a Curves recorded under isothermal conditions at 683 K and 693 K for the pyrolysis of EPDM. b Linear relationship between the values of time needed to reach a certain value of \(\alpha\) at the two different temperatures employed. c Overlapping of both curves after applying Eq. 6. The value of activation energy obtained is \(E_{\text{a}} = \left( {253.4 \pm 0.7} \right)\) kJ mol−1

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Data corresponding to the thermal decomposition of limestone are plotted in Fig. 6. These curves were recorded at 873 K and 893 K. The value of activation energy obtained in this case is \(E_{{\text{a}}} = \left( {182.1 \pm 0.6} \right)\) kJ mol−1. This reaction has received a great deal of interest in recent years because it is in the core of the so-called “calcium looping” technology for CO2 capture and the storage of concentrated solar power [34, 35]. Thus, the reaction has been amply studied and the values of activation energy reported in the literature are in good agreement with the one determined in this work [19, 36,37,38].

Fig. 6
figure 6

a Time evolution of the extent of decomposition of limestone at 873 and 893 K. b Linear relationship between the values of time needed to reach a certain value of \(\alpha\) at the two different temperatures employed. c Overlapping of both curves after applying Eq. 6. The value of activation energy obtained is \(E_{a} = \left( {182.1 \pm 0.6} \right)\) kJ mol−1

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The same method was applied to data corresponding to experiments in which clear laminating epoxy resin curing was monitored by DSC at two distinct temperatures: 338 K and 343 K. As aforementioned, the experimental results, shown in Fig. 7a, were published by Tziamtzi et al. [22]. The value of activation energy determined using the method proposed, \(\left( {55 \pm 7} \right)\) kJ mol−1, is in agreement with the values reported in the work from which the data were obtained. Besides, it is also in line with other results previously reported in the literature for this type of material [39, 40].

Fig. 7
figure 7

a Degree of curing as a function of time, recorded under isothermal conditions in a DSC, at 338 K and 343 K. b Linear relationship between the values of time needed to reach a certain value of \(\alpha\) at the two different temperatures employed. c Overlapping of both curves after applying Eq. 6

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The three cases analysed above served us to validate the method with data recorded under different experimental conditions, demonstrating its general applicability.

Conclusions

In this work, it has been demonstrated that the apparent activation energy calculated from isothermal analysis based on the fitting to ideal kinetic models is independent of the model selected to do the fitting. Furthermore, a method to calculate the activation energy from two isothermal experiments has been presented. The method was applied to the analysis of data corresponding to experiments conducted in a TGA, namely the pyrolysis of EPDM and the thermal decomposition of limestone. The comparison with the apparent activation energies previously reported for these reactions demonstrated the reliability of the method. Moreover, to extend its range of applicability, the method has been also validated through its application to data from the literature recorded by differential scanning calorimetry.