Misinterpretation of Thermodynamic Parameters Evaluated from Activation Energy and Preexponential Factor Determined in Thermal Analysis Experiments

Abstract

Thermogravimetry (TGA) and differential scanning calorimetry (DSC) are used broadly to study the kinetics of thermally stimulated processes such as thermal decomposition (pyrolysis) or thermal polymerization. These studies typically yield the activation energy (E) and preexponential factor (A). The resulting experimental values of E and A are oftentimes used to determine the so-called “thermodynamic parameters”, i.e., the enthalpy, entropy, and Gibbs free energy. Attention is called to the persistent and mistaken trend to interpret the resulting quantities as the thermodynamic parameters of the conversion of reactants to products. In fact, these quantities are specific to the conversion of reactants to the activated complex and, as such, provide no insights into the thermodynamics of the conversion of reactants to products. The basics of the activated complex (transition state) theory are provided to explain the meaning of the equations used for evaluating the thermodynamic parameters from the experimental values of E and A. Typical examples of misinterpretation are highlighted and discussed briefly. The applicability of the theory to the systems studied by the thermal analysis kinetics is also discussed.

1. Introduction

Thermal analysis methods such as thermogravimetry (TGA) and differential scanning calorimetry (DSC) are used broadly to explore the kinetics of thermally stimulated processes. The process rate is typically treated in terms of the following rate equation:

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

(1)

where α is the extent of the reactant conversion, t is the time, f(α) is the reaction model, T is the temperature, and k(T) is the rate constant. The latter is usually parameterized via the Arrhenius equation:

$$k\left(T\right)=A\exp \left({\displaystyle \frac{-E}{RT}}\right)$$

(2)

where R is the gas constant, A and E are the Arrhenius parameters, i.e., the preexponential factor and activation energy, respectively. There is no need to reiterate the importance of proper ways of determining the kinetic triplets (i.e., the model, activation energy, and preexponential factor) because they are covered adequately in a set of detailed recommendations by the Kinetics Committee of the International Confederation for Thermal Analysis and Calorimetry (ICTAC) [1,2,3,4,5]. Therefore, for the purpose of the present discussion, it is taken for granted that any application of the kinetic triplets is necessarily based on properly determined values.

One such application that requires immediate attention is associated with a strongly growing trend of employing the experimentally determined E and A for evaluating the so-called “thermodynamic parameters”. These include the enthalpy, entropy, and Gibbs free energy. A Scopus search indicates that in the area of biomass pyrolysis alone this type of applications was presented in 62 papers in 2023 and in 53 papers of the first seven months of 2024 [6]. Unfortunately, such evaluations commonly overlook the fact that the equations used for this purpose originate from the activated complex (transition state) theory [7,8], which is discussed in sufficient detail in Section 2. This theory stipulates that the resulting thermodynamic parameters should be interpreted as the parameters for the conversion of reactants to the activated complex. Ignoring this fact causes the resulting thermodynamic parameters to be misinterpreted as the parameters for the conversion of reactants to reaction products. Regrettably, such misinterpretation has become widespread, as can be seen from a multitude of publications on the kinetics derived from thermal analysis data. As an illustration, one can consider three recent review papers [9,10,11] referencing nearly 500 articles, which can serve as representative examples of the aforementioned misinterpretation.

The present article is intended to assist in proper interpretation of the thermodynamic parameters of the activated complex. This intention is accomplished by providing the bare minimum of the activated complex theory and discussing some typical examples of misinterpretation as well as the issue of the applicability of the theory.

2. Theory

The activated complex (transition state) theory suggests [7] that reactants convert to products via a reversible step of forming the so-called activated complex or transition state. These two concepts are used interchangeably and, henceforth, are referred to as “activated complex” for brevity. Its formation requires significant energy expenditure that manifests itself as an energy barrier to the reaction. Consider a reaction between A and BC that yields AB and C:

$$A+BC\rightleftarrows A..B..C⟶AB+C$$

(3)

where A..B..C is the activated complex. In Equation (3), the reversible step is characterized by the equilibrium constant K^≠. The superscript symbol “≠” is employed traditionally to denote parameters related to the activated complex.

The theory demonstrates that the rate constant can be presented as follows:

$$k\left(T\right)=\kappa {\displaystyle \frac{k_{B}T}{h}}{K}^{\ne }$$

(4)

where k_B is the Boltzmann constant, h is the Planck constant, and κ is the transmission coefficient. Taken to be temperature independent, this coefficient is routinely omitted (i.e., set as 1) in the derivations of the thermodynamic parameters related to the activated complex, which is also applied in the present work. Because K^≠ is treated as a regular equilibrium constant, the following holds true:

$$K^{\ne }=\exp \left({\displaystyle \frac{-\Delta G^{\ne }}{RT}}\right)$$

(5)

where ΔG^≠ is a positive quantity called the free energy of the activated complex formation. With regard to the positive sign of ΔG^≠, one needs to recognize two interrelated facts. First, because of the aforementioned energy expenditure, the activated complexes must have markedly larger energy than the reactants. That is, their free energy is necessarily higher than that of the reactants. Therefore, ΔG^≠ is unavoidably positive and the process of their formation is non-spontaneous. Second, for statistical reasons (the Boltzmann distribution), the fact that the energy of the activated complexes is significantly greater than that of the reactants means that the fraction of the former is much smaller than that of the latter. That is, for Reaction (6):

$$A+BC\rightleftarrows A..B..C$$

(6)

equilibrium is shifted towards the reactants, i.e., K^≠ < 1. This in accord with Equation (5) is equivalent to ΔG^≠ > 0, and other way around.

Figure 1 gives a schematic representation of the Gibbs free energies for Reaction (3). As already explained, ΔG^≠ is necessarily positive and represents the free energy barrier to the formation of the activated complex. On the other hand, ΔG_rxn is the free energy change for the conversion of the reactants to products:

$$A+BC⟶AB+C$$

(7)

It can be either negative or positive depending on the values of the Gibbs free energies for AB and C relative to those for A and BC. Figure 1 displays an example of a reaction for which the Gibbs free energies of the products AB and C are lower than those of the reactants A and BC. As a result, ΔG_rxn is negative and the reaction is spontaneous. Of course, this is not always the case, and many important reactions have positive ΔG_rxn, i.e., non-spontaneous. Perhaps, the most common example of such reactions is dissociation of weak acids in water. Most importantly, ΔG^≠ and ΔG_rxn represent different processes so that determining ΔG^≠ (Reaction (6)) does not provide any insights into the free energy change associated with the conversion of reactants to products (Reaction (7)).

An especially important accomplishment of the theory is that it links the theoretical value of the activation enthalpy ΔH^≠ to the experimentally determined activation energy E. This is caried out as follows: First, the experimental activation energy is defined via the temperature derivative of k(T) from Equation (2):

$${\displaystyle \frac{d\ln k(T)}{dT}}={\displaystyle \frac{E}{R{T}^2}}$$

(8)

Then, the usual thermodynamic relation:

$$\Delta G^{\ne }=\Delta H^{\ne }-T\Delta S^{\ne }$$

(9)

(where ΔS^≠ is the entropy of activation) is inserted into Equation (5), which is then substituted into Equation (4) to yield the following:

$$k\left(T\right)={\displaystyle \frac{k_{B}T}{h}}\exp \left({\displaystyle \frac{\Delta S^{\ne }}{R}}\right)\exp \left({\displaystyle \frac{-\Delta H^{\ne }}{RT}}\right)$$

(10)

Taking the temperature derivative of k(T) from Equation (10) yields the following:

$${\displaystyle \frac{d\ln k(T)}{dT}}={\displaystyle \frac{1}{T}}+{\displaystyle \frac{\Delta H^{\ne }}{R{T}^2}}$$

(11)

Comparing the right hand sides of (8) and (11) gives rise to:

$$E=RT+\Delta H^{\ne }$$

(12)

This result is arrived at if the equilibrium constant K^≠ is defined in the units of pressure, i.e., $K^{\ne }=K_p^{\ne }$. Such definition is only suitable for reactions of gases. For the condensed phase reactions, a proper definition is in the units of concentration, i.e., $K^{\ne }=K_c^{\ne }$. The concentration and pressure based equilibrium constants are linked to each other as below [12]:

$$K_c^{\ne }=K_p^{\ne }{\left(RT\right)}^{-\Delta n^{\ne }}$$

(13)

Then, following the same algebra as before leads to a slightly more complex equation:

$$E=RT+\Delta H^{\ne }-\Delta n^{\ne }RT$$

(14)

where Δn^≠ is the change in the number of molecules when the activated complex is formed from the reactants.

Note that Equation (14) becomes identical to Equation (12) when Δn^≠ = 0, i.e., only in the case of a monomolecular reaction. This should be kept in mind considering that regardless of the reaction molecularity, Equation (12) appears to be used preferentially for estimating ΔH^≠ from the experimental E values [10]. Clearly, the usage of Equation (12) entails an implicit assumption that a reaction is monomolecular, which cannot be considered as being generally valid.

Equation (12) permits to easily determine ΔH^≠ by subtracting RT from the experimentally determined value of E. It is noteworthy that the RT value is normally quite small. For example, the thermal decomposition of cellulose, some wood samples, vinyl polymers, and certain carbonates occur in the temperature range 300–800 K and demonstrate activation energies in the neighborhood of ~200 kJ/mol [5]. In this situation, RT is 2–6 kJ/mol, which amounts to only 1–3% of the experimental E. The thermal polymerization typically occurs around 300–500 K with the respective E values from 50 to 100 kJ/mol [4] so that RT does not exceed 5% of E. Considering that the experimental uncertainty in E is typically about 5–10% [1,3], the RT value can be considered negligible.

Once ΔH^≠ is determined it is important to recognize its meaning (see Figure 2). Clearly, this parameter is associated with the formation of the activated complex from the reactants, i.e., with Reaction (6). By its meaning, it is the enthalpic barrier to this reaction, and its value is nearly equal to the activation energy E. The latter represents the minimum amount of kinetic energy that is needed for reaction to occur [13]. This is obviously a positive quantity. In turn, the ΔH^≠ value is also always positive, or, in other words, Reaction (6) is invariably endothermic. This is in contrast to the conversion of the reactants to products, i.e., Reaction (7). The latter is characterized by the reaction enthalpy ΔH_rxn, which can be negative or positive. By way of example, ΔH_rxn is negative for thermal polymerization, which typically is a strongly exothermic process [4]. The thermal decomposition of inorganic solids and polymers is commonly endothermic, i.e., its ΔH_rxn is positive [5]. In all, just as in the case of ΔG^≠, ΔH^≠ cannot offer any insight into the enthalpy change associated with the conversion of reactants to products (Reaction (7)). ΔH_rxn should either be measured experimentally by calorimetry or estimated theoretically from the enthalpies of the formation of the reactants and products.

Further, replacing ΔH^≠ with E in Equation (10) and comparing it to Equation (2) suggests that the preexponential factor in the Arrhenius equation has the following form:

$$A={\displaystyle \frac{k_{B}T}{h}}\exp \left({\displaystyle \frac{\Delta S^{\ne }}{R}}\right)$$

(15)

The utility of Equation (15) is that it provides a convenient way of determining the entropy of activation ΔS^≠ from the experimentally determined value of the preexponential factor A. This is accomplished by rearranging Equation (15) into Equation (16):

$$\Delta S^{\ne }=R\ln \left({\displaystyle \frac{Ah}{k_{B}T}}\right)$$

(16)

Again, the resulting quantity ΔS^≠ is related to Reaction (6), which is the formation of the activated complex from the reactants and thus does not provide any useful information about the entropy change in Reaction (7), which is the conversion of the reactants to products. The latter reaction is characterized by the quantity ΔS_rxn, which should be determined from the thermodynamic properties of the reactants and products. The sign of ΔS^≠ can be either positive or negative. Monomolecular reactions frequently demonstrate positive values [14]. For reactions of higher molecularity, a negative sign is most common because two or more molecules of the reactant merge into a single activated complex. Then, a decrease in ΔS^≠ results from the loss of degrees of freedom that occurs when the reactant molecules combine into the activated complex. For instance, if we consider the translational motion only, each of two molecules of the reactant has 3 degrees of translational motion, i.e., 6 degrees freedom for the total of two molecules. However, the two molecules merged into a single activated complex have only 3 degrees of translational motion, meaning that the activated complex is more ordered than the respective reactant molecules and thus possesses lower entropy.

Finally, once ΔH^≠ and ΔS^≠ are determined from the experimental values of E and A, one can readily evaluate ΔG^≠. This is trivially carried out by using Equation (9). The meaning of this quantity and its fundamental difference from ΔG_rxn are discussed at the beginning of this section (Figure 1).

To conclude this section, a comment should be made concerning the physical significance of evaluating ΔH^≠, ΔG^≠, and ΔS^≠. There is little, if any, merit in estimating ΔH^≠ because for most practical purposes, its value is equivalent to the experimentally estimated value of E. That is, ΔH^≠ does not carry any information beyond the information already available from E. However, estimating ΔG^≠ is of more interest. Even though ΔG^≠ is known a priori to be positive, its value can differ significantly from E determined in thermal analysis experiments and, therefore, from ΔH^≠ estimated from this E. As per Equation (9), this difference arises from the TΔS^≠ term, which can be very significant because for some reactions, the absolute values of ΔS^≠ can reach rather large values. Examples include trimerization of potassium and rubidium dicyanamide (ca. −200 J/(mol K)) [15], the melt polymerization of tricyanate ester (ca. −110 J/(mol K)) [16], azide–alkyne cycloaddition (ca. −120 J/(mol K)) [17], and denaturation of a large variety of proteins (from ~100 to ~2200 J/(mol K)) [18]. Therefore, it should be possible and illuminating to discover that two reactions have similar values of E but markedly different values of ΔG^≠ due to considerably different entropic conditions of forming the activated complex.

Perhaps the most interest lies in estimating ΔS^≠. In addition to the role it can play in controlling ΔG^≠, analysis of its sign and value can provide important mechanistic insights, as shown earlier [15,16,17]. For instance, an increase in ΔS^≠ was indicative of the process of the reactant ordering under the conditions of nanoconfinement [15]; similarly, the process of solvation demonstrated diminished ΔS^≠ due to restricting the reactant mobility [16], and comparison of the actual ΔS^≠ values against the values expected for two-step and concerted mechanisms supported the occurrence of the latter [17].

3. Typical Examples of Misinterpretation

As stated earlier, the following examples are taken from three recent publications [9,10,11]. Because these are review articles, it is recognized that certain claims are not necessarily those of the reviews’ authors. Rather, they could be claims made by the authors of the reviewed papers.

The following discussion presents direct quotes from said review papers. As seen from these papers, many workers report the thermodynamic parameters of the activated complex without using the traditional superscript symbol “≠”, i.e., as ΔH, ΔS, and ΔG. This is an unfortunate practice because it can add to confusing the parameters of the activated complex formation with the parameters used for the conversion of the reactants to products.

Regarding the enthalpy of activation, ΔH^≠, one encounters the following claims: “represents the endothermic or exothermic behavior… it is the amount of energy transferred during a chemical reaction” [10] and “represents the total energy consumed by biomass during its conversion to yield host of product” [11]. Clearly, ΔH^≠ is here misinterpreted as ΔH_rxn. Related to this issue are the following claims: “the similarity of ΔH with Ea… promotes the generation of reactions in the thermal process” [9] and “the smaller the difference of ΔH with the Eα is, the more favourable for the reaction to occur” [10]. Of course, such claims have no physical grounds because, as shown in Section 2, the difference between the experimental activation energy and the enthalpy of activation is simply RT (Equation (12)). This RT value has no individual thermodynamic meaning and the only reason it enters Equation (12) is that the preexponential part of Equation (10) includes T in the numerator.

On the subject of the free energy of activation, ΔG^≠, the claim that “positive values of… ΔG reveals non-spontaneity of the pyrolysis of millet residues” [11] indicates that ΔG^≠ is misinterpreted as ΔG_rxn. This issue propagates even further, as seen from claims such as “the higher the ΔG is, the more bioenergy can potentially be attained” [10], “(ΔG) is calculated to determine the amount of energy present in a certain biomass” [9], or “(ΔG = 146–189 kJ/mol) showed that the biomass has great potential for producing bioenergy” [9]. Indeed, ΔG_rxn can be used as an estimate of the maximum energy available to a chemical reaction to perform work on surroundings [13]. However, to do such work, the reaction must necessarily be spontaneous, i.e., ΔG_rxn must be negative. On the contrary, as discussed in Section 2, ΔG^≠ is unavoidably positive (as also reported in the above-referenced review papers), and thus the reaction of the activated complex formation is incapable of performing any work on surroundings.

Concerning the activation entropy, ΔS^≠, the claim that “Entropy (ΔS) … measures the randomness of pyrolysis system” [11] appears to relate to the process of pyrolysis of the reactant material to products that cannot be described by ΔS^≠, and ΔS_rxn must be determined instead. Also, there are no doubts that ΔS^≠ cannot quantify “the degree of arrangement of the carbon in the waste and biomass” [10].

4. Applicability of Theory

Naturally, before estimating and interpreting the parameters of the activated complex theory, it is necessary to consider whether the theory can be applied, at least in principle, to a system under study. The thermal analysis kinetics explores the widest range of experimental systems, some of which have a very complex composition and structure. In contrast, the activated complex theory is oftentimes seen as a theory applicable primarily to a single elementary step of the gas phase reaction. Expectedly, the gas phase allows for the most accurate theoretical treatment. However, already in the initiatory publications on the theory, it was stated that “We here simply sketch the procedure for the gas phase and indicate where the modifications for reactions in solution will come” [19] and that “The systems we can study by this method… will include gas, liquid and heterogeneous reactions” [20]. Indeed, the theory was advanced to include heterogeneous processes, reactions in solution, diffusion in liquids and solids [8], and the thermal decomposition of solids [21]. More recent progress in the applications of the activated complex theory to reactions in the condensed phase, including solids, is reviewed elsewhere [22].

To sum up, the applicability of the theory to condensed phase reactions is not in question. In turn, the occurrence of a reaction via more than a single elementary step does not immediately invalidate the application of the theory. For instance, it has been applied successfully to the thermal denaturation of proteins, a process known to proceed via simultaneous breaking of multiple hydrogen bonds, i.e., an ensemble of elementary steps [18]. Similarly, heterogeneous reactions tend to occur as a sequence of elementary steps having their own activated complexes, whereas the overall kinetics can be driven by a single rate-determining step and, thus, by the formation of the respective activated complex [8]. This brings about the following important point. The application the activated complex theory can be justified in principle when a multi-step processes manifests the rate-determining step. In the case of the thermal analysis kinetics this situation is readily revealed by means of an isoconversional method when it yields the activation energy, which shows no significant variation with conversion [1,3,4,5]. Such behavior is not very rare even for complex systems. For example, the isoconversional activation energy is well established [23,24] to be constant for pyrolysis of cellulose, the major component of biomass. In this circumstance, the resulting values of E and A can be sensibly converted to the thermodynamic parameters of the activated complex and interpreted accordingly (see Section 2).

Furthermore, the absence of a single rate-determining step (e.g., detected as a significant variation in the isoconversional activation energy) is not necessarily a stumbling block to applying the activated complex theory. In general, a multi-step processes can still be described rationally in terms of a proper mechanistic multi-step model, which would yield meaningful values of E and A for the individual steps [3,4,5]. Then, the resulting values can be converted to the thermodynamic parameters of the activated complexes of the individual steps. Of course, for a multi-step process the situation is incomparably more problematic than for a process, whose kinetics is driven by a single rate-determining step. The root of the problem is in establishing a proper mechanistic model. Note that the latter does not mean just some multi-step model that fits the rate data, no matter how accurately. It means a model that can be justified by mechanistic data. Establishing such models can be practically impossible in the case of the systems having a very complex composition and structure. Consequently, the application of the activated complex theory can be impossible to justify in such cases.

Biomass is certainly a very complex system whose pyrolysis is likely to manifest the multi-step behavior. It appears that, at least in certain cases, the kinetics of biomass pyrolysis can be treated by the mechanistically sensible model of the three reaction steps associated with pyrolysis of the main pseudo-components of biomass, i.e., cellulose, hemicellulose, and lignin [25,26]. Therefore, if one must apply the activated complex theory to biomass pyrolysis, it would probably make most sense to try applying it to the E and A values estimated for the aforementioned three steps.

5. Conclusions

Workers involved in thermal kinetics studies should be alerted to the persistent trend of misinterpretation of the enthalpy, entropy, and Gibbs free energy values estimated from the activation energy and preexponential factor determined in thermal analysis experiments. It is emphasized that such estimates employ basic equations of the activated complex (transition state) theory and give rise to the thermodynamic parameters of the formation of the activated complex from the reactants. Although this fact precludes one from interpreting the resulting parameters as the parameters of the conversion of the reactants to products, such erroneous interpretation is rampant. Typical examples of such misinterpretation are provided by using three recent review papers.