The standard molar reaction enthalpies,
, are one of the fundamentals of process optimization. The enthalpies of the dehydrogenation reactions are desired to be as low as possible to decrease the energy demand of hydrogen release [
8]. The reaction takes place under pressure, and the reactants are liquids (except for hydrogen) under the reaction conditions. According to Hess’s Law, the liquid-phase standard molar enthalpies,
(liq), of the reactants are required to calculate the dehydrogenation enthalpy for any hydrogen storage LOHC-system. As a matter of fact, the thermochemical properties of aromatic compounds as the hydrogen-lean counterpart to the LOHC systems have been intensively studied in the past [
9]. In contrast, the partially and fully hydrogenated products–aliphatic cyclic molecules–have received far less attention. With increasing interest in these molecules as hydrogen-rich compounds of LOHC systems, the lack of thermodynamic data becomes problematic. Moreover, the available data are very inconsistent and questionable, which can lead to errors in process engineering. The following example shows the problematic current situation. Only recently an interesting LOHC system based on the eutectic mixture (diphenylmethane + biphenyl) leading to the liquid hydrogen carrier (dicyclohexylmethane + bicyclohexyl) was proposed [
10]. We have extensively studied the thermochemistry of biphenyl and bicyclohexyl, and the data have proven to be reliable [
11,
12]. Therefore, there were no troubles in determining the enthalpy of dehydrogenation of bicyclohexyl [
12]. However, one question remains to be answered reliably: What is the reaction enthalpy for dehydrogenation of dicyclohexyl-methane (see
Figure 1)?
The results calculated in
Table 1 from the data available in the literature for the enthalpy of dehydrogenation,
(liq)/H
2, referred to 1 mole of hydrogen evolved, are not helpful in deciding whether adding diphenyl-methane to the biphenyl is a good idea or not. Indeed, for comparison, the energetic effect
(liq)/H
2 = 65.4 kJ·mol
−1 was determined for bicyclohexyl [
11,
12]. So if the value
(liq)/H
2 = 153.4 kJ·mol
−1 (see
Table 1) is correct, the idea is questionable. However, if the value
(liq)/H
2 = 65.3 kJ·mol
−1 (see
Table 1) is correct, the total energy of dehydrogenation remains more or less the same for the mixture. The only question left to answer is which value is correct. Therefore, the main task of this study is to develop a straightforward algorithm that allows a reasonably accurate assessment of the liquid-phase standard molar enthalpies of formation of the hydrogen-rich compounds of given LOHC systems.
A textbook equation that relates the thermochemical properties relevant to this work is:
where
(g) is the gas-phase standard molar enthalpy of formation and
is the standard molar enthalpy of vaporisation. The
(liq)-values are usually derived from combustion experiments, from solution calorimetry or from chemical equilibrium studies [
20]. The
-values are usually directly measured calorimetrically or derived from the vapour pressure-temperature dependences [
21]. There are at least two aggravating limitations related to the experimental thermochemical measurements. First, the purity of the samples is required to be better than 99.9 %. Secondly, the measurements are material-intensive. As a rule, about 5–10 g of the high-purity sample is needed to measure both contributors to Equation (1). Unfortunately, both of these limitations make systematic thermochemical studies with typical hydrogen-rich LOHC compounds not straightforward, since the most interesting samples are not commercially available. Moreover, for custom synthesis, sample purities are very difficult to achieve, even at levels of 95–98 %, since the naturally occurring cis- and trans-isomers of cyclic compounds are virtually impossible to separate. These apparent limitations force us to conclude that only structure-property correlations and theoretical methods could provide a remedy to obtain the reliable thermochemical data for the hydrogen-rich counterparts. In this work, the focus has been restricted to
(liq) as a goal thermochemical property that is relevant to the energetics of hydrogen storage. In the first step, Equation (1) has been re-written as follows:
The symbols for enthalpies have been extended to include temperatures, since it is common in thermochemistry to refer all enthalpies given in Equations (1) and (2) to an arbitrary but common reference temperature. In this work,
T = 298.15 K was chosen as the reference temperature. Since all three enthalpies are related, in the second step a decision has to be made which enthalpies are better suited for structure-property correlations and theoretical methods. From a practical point of view, the gas phase enthalpies of formation are the best choice. The reason for this is that
(g, 298.15 K)-values reflect the energy content of a single molecule flying freely in vacuum (or gas, but without interacting with other molecules). Therefore, the energetics of this single molecule is easy to understand and correlates with the peculiarities of its structure. In contrast, the
(liq, 298.15 K)-values reflect not only the intrinsic energetics of the molecule but also the intensity of the intermolecular interactions in the liquid phase. Therefore, the interpretation of these combined intra- and inter-molecular interactions is still possible but is influenced by many factors, making it difficult to understand. The enthalpy of vaporisation accounts for the total amount of intermolecular interactions in the liquid phase. According to our experience [
22], the
(298.15 K)-values are subject to a group additivity rule and different correlations with the structural elements. Therefore, to draw the conclusion for the second step, it makes sense to develop appropriate tools to correlate and predict the
(g, 298.15 K) and
(298.15 K) and finally to calculate the
(liq, 298.15 K)-values as their difference according to Equation (2). This guideline helps in following the logic of combining experimental methods for determining thermochemical properties with theoretical methods based on quantum-mechanics for validation as applied in this work.
2.1. Basics of the Group-Additivity Concept (“Centerpiece” Approach)
Group additivity (GA) methods are successfully used to predict both the gas-phase enthalpies of formation and vaporisation enthalpies [
22]. The idea behind conventional GA methods is to split the experimental enthalpies of molecules into relatively small groups in order to obtain well-defined numerical contributions for them. The prediction then proceeds as construction of a framework of a desired model molecule from the appropriate number and type of these contributions. Comprehensive systems of group contributions (or increments) are developed, covering the main classes of organic compounds [
22,
23,
24]. Admittedly, the applicability of GA in the case of cyclic molecules is limited. To overcome this limitation, various ring corrections are implemented in the GA parameterization; however, each correction term is specific only to a particular unsubstituted ring (e.g., for cyclobutane, cyclopentane, cyclohexane, etc.). However, any type of substitution significantly affects ring strain due to intense interactions of the substituent(s) with the ring. This general GA limitation cannot be overcome easily [
25]; however, previous work with the hydrogen-rich LOHC compounds has shown that a variation of the GA method, referred to as the “centerpiece” approach [
26,
27], gives acceptable results for the gas phase enthalpies of formation and for vaporisation enthalpies of these technically important molecules. The idea of the “centerpiece” approach is to select a potentially large “centerpiece” molecule that has a reliable enthalpy and that can generally mimic the structure of the molecule of interest. Then the necessary groups or blocks are attached to the “centerpiece”, resulting in the construction of the desired molecule. A prerequisite for this method is that reliable experimental data are available for the selected “centerpiece” molecule.
For example, for dicyclohexyl-methane, either ethyl-cyclohexane or methyl-cyclohexane can be selected as the suitable “centerpieces”. The enthalpies of formation and enthalpies of vaporisations for both molecules are well-established [
9]. The idea is illustrated in
Figure 2,
Figure 3 and
Figure 4.
First, by cutting off the CH
3 group from ethyl-cyclohexane and from methyl-cyclohexane, the two “fragments” needed to construct the dicyclohexyl-methane were derived (see
Figure 2).
Second, both “fragments” are merged to give the desired molecule dicyclohexyl-methane (see
Figure 3). Alternatively, the molecule dicyclohexyl-methane can be constructed from two “-(cyclohexane)” fragments and the methylene group (see
Figure 4). The numerical values for the species involved in these calculations are given in
Table S1 (electronic supporting materials).
The results obtained by both methods are essentially the same, e.g., the vaporisation enthalpy
(298.15 K) = 64.7 ± 1.0 kJ·mol
−1 according to
Figure 3 and
(298.15 K) = 64.5 ± 1.0 kJ·mol
−1 according to
Figure 4. Therefore, using the gas-phase enthalpy of formation of dicyclohexyl-methane,
(g, 298.15 K) = −242.4 ± 2.0 kJ·mol
−1 assessed as shown in
Figure 3 with the corresponding
(298.15 K) = 64.7 ± 1.0 kJ·mol
−1, the “empirical” liquid-phase enthalpy of formation was estimated to be
(liq, 298.15 K) = −242.4 − 64.7 = −307.1 ± 2.2 kJ·mol
-1, which was further re-calculated into
(liq)/H
2 = 67.3 kJ·mol
−1 (see
Table 1) for comparison with other values available from the literature data. These “empirical” results are now helping to resolve contradictions that are evident from the available data. Furthermore, the estimate value of
(liq)/H
2 = 67.3 kJ·mol
−1 (see
Table 1) could be considered more reliable than the value of
(liq)/H
2 = 65.3 kJ·mol
−1 calculated from Ref. [
18] (see
Table 1), because of the questionable purity of the sample studied by Wise et al. [
18].
To further validate the “centerpiece” approach by a more challenging task, the enthalpies of formation and enthalpies of vaporisation of the perhydro-dibenzyltoluene, which is the hydrogen-rich component of the LOHC system based on dibenzyltoluene (Marlotherm SH
®) [
25], have been calculated. Obviously 1,2,3-trimethylcyclohexane is the perfect “centerpiece” to start the calculations. The “fragment” required to construct perhydro-dibenzyltoluene was derived by splitting off two CH
3 groups from 1,2,3-trimethylcyclohexane (see
Figure 2). The desired molecule is built up by fusing this “centerpiece” with two “-CH
2-(cyclohexane)” fragments (see
Figure 5).
It was found that the vaporisation enthalpy
(298.15 K) = 90.9 ± 1.5 kJ·mol
−1 estimated in this way agrees with the experimental value
(298.15 K) = 88.2 ± 1.5 kJ·mol
−1 [
25] within the experimental uncertainty. The “empirical” gas-phase enthalpy of formation of perhydro-dibenzyltoluene,
(g, 298.15 K) = −380.9 ± 2.0 kJ·mol
−1, assessed as shown in
Figure 5, also agrees with the experimental value
(g, 298.15 K) = −387.4 ± 7.7 kJ·mol
−1, within the experimental uncertainty.
It should be mentioned that the conventional GA procedure really failed to correctly predict the energetics of perhydro-dibenzyltoluene properly: the values
(298.15 K) = 94.7 kJ·mol
−1 and
(g, 298.15 K) = −409.2 kJ·mol
−1, as estimated in previous work [
25], differ significantly from the experiment. The secret of the success of the “centerpiece” approach lies in the fact that the main energetic contribution(s) is already stored in the “centerpiece”(s) and the attached groups are precisely parameterized [
21,
22,
23]. The “centerpiece” as a “pen and paper” approach is used frequently to quickly assess the quality of the available literature data, but also to estimate the “expected” value of the ongoing thermochemical experiment, and in this way to avoid possible systematic errors. However, in the 21st century, everyday laboratory work is unthinkable without modern quantum chemical methods.
2.2. Quantum Chemistry: From Doubts to Enthusiasm
Since 2012 [
28] we have applied high-level quantum chemical methods to calculate the gas-phase enthalpies of formation of the hydrogen-lean and hydrogen-rich counterparts of LOHC systems. From this experience, there are few problems with the composite methods (G3MP2 and G4) when applied to the “relatively” large LOHC molecules containing at least 20−30 “heavy” atoms. The first is the time commitment, which is gradually becoming bearable (the G3MP2 calculations with perhydro-dibenzyltoluene in 2015 [
25] took two months with the facilities of the University of Rostock computing center). The second issue is an ambiguity related to the choice of compounds for the different reaction types used to convert the
H298 enthalpies to the standard molar enthalpies of formation. The details of this issue can be found elsewhere [
29]. However, the years of “mutual validation” of the experimental and quantum chemical (theoretical) enthalpies of formation performed in our laboratory allowed the conclusion that only if both results agree within the combined uncertainties can the final result can be considered reliable. If this is not the case, the experimental efforts (additional purification of the sample, variation of experimental conditions) as well as computational work (search for possibly more stable conformers, use of another method) are continued. Furthermore, the independent structure-property correlations (e.g., the “centerpiece” approach) are involved to obtain an unbiased value for the property under investigation. To give an example of such a successful continuity: for perhydro-dibenzyltoluene, the experimental enthalpy of formation
(g, 298.15 K)
exp = −387.4 ± 7.7 kJ·mol
−1 [
25] was combined from the combustion experiments and the vapour pressure measurement, the empirical value
(g, 298.15 K)
emp = −380.9 ± 2.0 kJ·mol
−1 (see above) was calculated according to the “centerpiece” approach, and finally the theoretical G3MP2 result
(g, 298.15 K)
theor = −379.7 ± 4.1 kJ·mol
−1 was calculated earlier [
25] and corrected in this work (as shown in
Section 3.1). It is evident that all three values agree within their combined uncertainties, providing confidence in the thermochemical results for this compound. In this work, the G3MP2 method was applied to calculate the gas phase enthalpies of formation of differently shaped alkyl-substituted cyclohexanes (see
Section 3.1) for “mutual validation” of the experimental and theoretical results useful for calculations of the hydrogen-rich LOHC components.
2.3. Empirical Correlations: Vaporisation Enthalpies vs. Kovats Retention Indices
Kovats retention indices are widely used in analytical chemistry to identify molecules in the reaction mixture. According to Kovats [
30], the index of an analyte
x is its relative time position between the nearest n-alkanes eluting immediately before and after a target analyte. The Kovats retention index,
Jx, is calculated with help of retention times of
n-alkanes used as standards [
30,
31]:
where
x refers to the adjusted retention time
t,
N is the number of carbon atoms of the
n-alkane eluting before, and (
N + 1) is the number of carbon atoms of the
n-alkane eluting after the peak of interest.
The fundamental variable tracked in GC is the retention time, which is a strong function of the experimental conditions. However, the main idea of the Kovats index is to trap the analyte in the network of alkanes and to reduce this dependence on experimental conditions. The retention indices are usually given for non-polar and polar types of GC columns. It is well-known that the vaporisation enthalpies of structurally similar molecules show a linear dependence when correlated with retention indices:
This correlation can be used to estimate the vaporisation enthalpies of compounds for which the retention indices are available. Collections and compilations of Kovats indices are available and ready to use [
31,
32]. The correlation is good for both non-polar and polar columns [
33]; however, the range of “similarly shaped” compounds is less specific and broader when the non-polar column is used. The experimental reproducibility of the retention index is usually very good (within a few units) when measured isothermally or with the temperature raising program. It is noticeable that the retention indices, measured on non-polar columns with different stationary phases, hardly differ. The fluctuations do not exceed 5–10 units when comparing the data at the same column temperature. This feature is particularly valuable for analyte identification. Temperature has a moderate effect on
Jx, and even the
Jx measured on the temperature program are not significantly different from isothermal results. For numerous individual compounds, it has been investigated how the fluctuations in the retention indices affect the vaporisation enthalpies derived from the linear relationships. It was found that fluctuations in
Jx-values of 15–20 units caused the scatter in
(298.15 K)-values to be well below 1.5 kJ·mol
−1, which is comparable to the experimental uncertainties of vaporisation enthalpies measured by conventional methods. Retention indices of alkyl-substituted cyclohexanes related to hydrogen storage were collected and used to correlate with vaporisation enthalpies in
Section 3.2.
2.4. Empirical Correlations: Vaporisation Enthalpies vs. Normal Boiling Temperatures
The normal boiling temperatures,
Tb, of chemicals are one of the mandatory physicochemical properties most frequently reported in the literature, as these values are measured by distillation and are traditionally used to identify compounds. The normal boiling temperatures,
Tb, are directly related to the vaporisation enthalpies
(
Tb), but the robust correlations with the
(298.15 K)-values are also known from the literature [
34]. Such correlations are particularly successful for a series of structurally similar molecules and the linear dependence:
is usually established and used to estimate the vaporisation enthalpies of compounds for which the normal boiling temperatures are available in the literature [
16,
35].
Correlations of vaporisation enthalpies with the retention indices and with normal boiling temperatures are expected to provide the basis for evaluation of the (298.15 K)-values related to hydrogen storage.