Thermodynamic Exercises for the Kinetically Controlled Hydrogenation of Carvone

Abstract

Carvone belongs to the chemical family of terpenoids and is the main component of various plant oils. Carvone and its hydrogenated products are used in the flavouring and food industries. A quantitative thermodynamic analysis of the general network of carvone hydrogenation reactions was performed based on the thermochemical properties of the starting carvone and all possible intermediates and end products. The enthalpies of vaporisation, enthalpies of formation, entropies and heat capacities of the reactants were determined by complementary measurements and a combination of empirical, theoretical and quantum chemical methods. The energetics and entropy change in the hydrogenation and isomerisation reactions that take place during the conversion of carvone were derived, and the Gibbs energies of the reactions were estimated. It was shown that negative Gibbs energies are recorded for all reactions that may occur during the hydrogenation of carvone, although these differ significantly in magnitude. This means that all these reactions are thermodynamically feasible in a wide range from ambient temperature to elevated temperatures. Therefore, all these reactions definitely take place under kinetic and not thermodynamic control. Nevertheless, the numerical Gibbs energy values can help to establish the chemoselectivity of catalysts used to convert carvone to either carvacarol or to dihydro- and terahydrocarvone, either in carvotanacetone or carveol.

1. Introduction

Renewable raw materials have received increasing attention as alternatives to petrochemical raw materials with a view to modern sustainable industrial development. The fragrance and flavour industry is based on terpenes. Carvone belongs to this chemical family and occurs naturally as the main component of caraway, dill, gingergrass, and spearmint oils. The various applications of carvone in the flavour and food industries as a starting material for pharmaceutical compounds, antimicrobial agents, etc. justify research into increasing the global production of this monoterpene [1]. Carvone is obtained at low cost by the steam distillation of these oils or recovered from citrus-derived limonene [2]. The production of carvone is also possible with specialised microorganisms using biotechnological techniques; e.g., the biosynthesis of (R)-(˗)-carvone from glucose was possible with the help of genetically modified Escherichia coli [3].

The upgrading of biomass can improve the sustainability of processes, but it is still at an early stage of development. For example, the catalytic hydrogenation of carvone produces valuable chemicals (Figure 1 and Figure 2) for the food, pharmaceutical, and agricultural industries.

However, the general good selectivity of these reactions is challenging, and most of the catalysts used are not selective, leading to product mixtures [4,5]. To address this problem, a thermodynamic analysis of the reactions shown in Figure 1 and Figure 2 is required. Carvone is an interesting substrate for the study of thermodynamic stability and catalytic activity in stereo- and chemoselective hydrogenation because it contains an asymmetric centre and three types of double bonds that can be hydrogenated: carbonyl C=O, a conjugated endo-double bond C=C (incorporated within the six-membered ring) and an isolated exo-double bond C=C outside the six-membered ring.

As shown in Figure 1, there are four possible directions in the hydrogenation of carvone (reactions 1,4,6) and isomerisation reaction 12. Each of these directions determines a cascade of subsequent reactions that lead to products of different chemical classes. The focus of this study is to quantify the energetics of the reactions shown in Figure 1 and Figure 2 to understand which of the hydrogenation directions is preferable from a thermodynamic perspective. These preferences are usually helpful in selecting specific catalysts that will drive the process in the desired direction.

The thermodynamic analysis of chemical reactions usually involves a complex of basic thermodynamic properties such as enthalpies, entropies, heat capacities, etc. of the reactants shown in Figure 1 and Figure 2. The thermochemical experiments to measure these properties require large quantities of high-purity (99.9%) samples. Since the carvone derivatives have very similar physico-chemical properties, the separation and purification of the individual compounds is a very challenging and demanding task. In order to possibly reduce the experimental effort, a hybrid thermodynamic method for evaluating the feasibility of chemical reactions is outlined in this paper. This hybrid approach combines the experimental and theoretical methods in an appropriate balance that leads to a realistic thermodynamic analysis of the feasibility and energetics of the network of chemical reactions depicted in Figure 1 and Figure 2. The thermodynamic workflow of this hybrid approach is as follows.

2. Thermodynamic Workflow

A chemical reaction is generally feasible if the sign of the standard Gibbs energy of the reaction, ${{∆}_{r}G}_{\mathrm{m}}^{\mathrm{o}}$, is negative (e.g., for a reaction in the ideal gas state):

$${{∆}_{r}G}_{\mathrm{m}}^{\mathrm{o}}={{∆}_{r}H}_{\mathrm{m}}^{\mathrm{o}}-T\times {∆}_r{S}_{\mathrm{m}}^{\mathrm{o}}=-RT\times \ln K_p$$

(1)

where ${{∆}_{r}H}_{\mathrm{m}}^{\mathrm{o}}$ is the standard molar reaction enthalpy, ${{∆}_{r}S}_{\mathrm{m}}^{\mathrm{o}}$ is the change in the standard molar entropy of a chemical reaction and $K_p$ is the gas-phase thermodynamic equilibrium constant. If the reaction takes place in the liquid phase, $K_p$ in Equation (1) is replaced by $K_a$, which represents the liquid-phase thermodynamic equilibrium constant. The latter Equation (1) is known in textbooks as the Gibbs–Helmholtz equation. The $K_p$ and $K_a$ values are directly related to the yield of the desired reaction products. The equilibrium constants that are ≥1 promise the already acceptable yields at a particular temperature. The negative and large ${{∆}_{r}G}_{\mathrm{m}}^{\mathrm{o}}$ values are an indication of a large thermodynamic driving force for the chemical reactions of interest. As a consequence, the reaction to the product is favorable and proceeds under thermodynamic control. However, this does not exclude the need to search for a suitable catalyst and temperature, as kinetics is also important to achieve high reaction rates in order to reach equilibrium.

There are two contributions to the Gibbs–Helmholtz equation, the enthalpic, ${∆}_r{H}_{\mathrm{m}}^{\mathrm{o}}$, and entropic, ${{∆}_{r}S}_{\mathrm{m}}^{\mathrm{o}}$. In the first contribution, the standard molar reaction enthalpy can be calculated for each chemical reaction according to Hess’s Law:

$${∆}_{\mathrm{r}}{H}_{\mathrm{m}}^{\mathrm{o}}\left(T\right)=\mathsf{\Sigma }[{∆}_{\mathrm{f}}{H}_{\mathrm{m}}^{\mathrm{o}}(T,\mathrm{products})]-\mathsf{\Sigma }[{∆}_{\mathrm{f}}{H}_{\mathrm{m}}^{\mathrm{o}}(T,\mathrm{educts})]$$

(2)

using the standard molar enthalpies of formation, ${∆}_{\mathrm{f}}{H}_{\mathrm{m}}^{\mathrm{o}}$(T), of reactants. Most chemical reactions in industry take place either in the gas or liquid phase, so the corresponding enthalpies of formation must be used in Equation (2). These values are linked using the following equation:

$${∆}_{\mathrm{f}}{H}_{\mathrm{m}}^{\mathrm{o}}(\mathrm{liq},T)={∆}_{\mathrm{f}}{H}_{\mathrm{m}}^{\mathrm{o}}(\mathrm{g},T)-{∆}_{\mathrm{l}}^{\mathrm{g}}{H}_{\mathrm{m}}^{\mathrm{o}}\left(T\right)$$

(3)

where ${∆}_{\mathrm{f}}{H}_{\mathrm{m}}^{\mathrm{o}}$(liq, T) is the liquid-phase enthalpy of formation, ${∆}_{\mathrm{f}}{H}_{\mathrm{m}}^{\mathrm{o}}$(g, T) is the gas-phase enthalpy of formation and ${∆}_{\mathrm{l}}^{\mathrm{g}}{H}_{\mathrm{m}}^{\mathrm{o}}$(T) is the standard molar enthalpy of vaporisation. The standard molar enthalpies of formation and enthalpies of vaporisation can either be of experimental origin or obtained using various empirical methods, e.g., from group-additivity rules or structure–property correlations.

The second contribution to the Gibbs–Helmholtz equation, the change in the standard molar entropy of the reaction, is derived as follows from the standard molar entropies, $S_{\mathrm{m}}^{\mathrm{o}}$(T), of the reactants:

$${∆}_{\mathrm{r}}{S}_{\mathrm{m}}^{\mathrm{o}}\left(T\right)=\mathsf{\Sigma }[S_{\mathrm{m}}^{\mathrm{o}}(T,\mathrm{products})]-\mathsf{\Sigma }[S_{\mathrm{m}}^{\mathrm{o}}(T,\mathrm{educts})]$$

(4)

The liquid and gas phase entropies are linked using the following equation:

$$S_{\mathrm{m}}^{\mathrm{o}}(\mathrm{liq},T)=S_{\mathrm{m}}^{\mathrm{o}}(\mathrm{g},T)-{∆}_{\mathrm{l}}^{\mathrm{g}}{S}_{\mathrm{m}}^{\mathrm{o}}\left(T\right)$$

(5)

where $S_{\mathrm{m}}^{\mathrm{o}}$(liq, T) is the liquid-phase standard molar entropy at temperature T, which can be measured using cryogenic adiabatic calorimetry or estimated using empirical methods; $S_{\mathrm{m}}^{\mathrm{o}}$(g, T) is the gas-phase entropy, which is usually calculated using statistical thermodynamics or a suitable quantum chemical (QC) method; and ${∆}_{\mathrm{l}}^{\mathrm{g}}{S}_{\mathrm{m}}^{\mathrm{o}}$(T) is the standard molar entropy of vaporisation, which can be derived from the slope of the temperature dependence of the experimental vapour pressures as follows:

$${∆}_{\mathrm{l}}^{\mathrm{g}}{S}_{\mathrm{m}}^{\mathrm{o}}\left(T\right)={∆}_{\mathrm{l}}^{\mathrm{g}}{H}_{\mathrm{m}}^{\mathrm{o}}/T+R\times \mathrm{l}\mathrm{n}\left(p_i/p^{\mathrm{o}}\right)$$

(6)

where $p_i$ is the absolute vapour pressure of a pure compound i at a given temperature T and $p^{\mathrm{o}}$ = 0.1 MPa is the standard pressure.

It is common practice to carry out thermodynamic calculations at the reference temperature T = 298 K. However, the typical temperature range for carvone hydrogenation reactions is between 293 and 523 K [4,5]. The variation in the reaction enthalpies and entropies with temperature is given by the thermodynamic relations:

$${∆}_{\mathrm{r}}{H}_{\mathrm{m}}^{\mathrm{o}}\left(T\right)={∆}_{\mathrm{r}}{H}_{\mathrm{m}}^{\mathrm{o}}\left(298\mathrm{K}\right)+{∆}_{\mathrm{r}}{C}_{p,\mathrm{m}}^{\mathrm{o}}\times (T-298\mathrm{K})$$

(7)

$${∆}_{\mathrm{r}}{S}_{\mathrm{m}}^{\mathrm{o}}\left(T\right)={∆}_{\mathrm{r}}{S}_{\mathrm{m}}^{\mathrm{o}}\left(298\mathrm{K}\right)+{∆}_{\mathrm{r}}{C}_{p,\mathrm{m}}^{\mathrm{o}}\times \ln (T/298\mathrm{K})$$

(8)

where ${∆}_{\mathrm{r}}{C}_{p,\mathrm{m}}^{\mathrm{o}}$ is the change in standard molar heat capacity for the reaction, which is calculated from the standard molar heat capacities, $C_{p,\mathrm{m}}^{\mathrm{o}}$(liq or g), of educts and products. The latter values can be easily measured using differential scanning calorimetry (DSC) or estimated using empirical methods.

The final input properties of the thermodynamic workflow according to Equations (2)–(6) are ${∆}_{\mathrm{f}}{H}_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K), ${∆}_{\mathrm{l}}^{\mathrm{g}}{H}_{\mathrm{m}}^{\mathrm{o}}$(298 K), and $S_{\mathrm{m}}^{\mathrm{o}}$(liq), for all the reactants shown in Figure 1 and Figure 2. Since most of these compounds are not commercially available and their synthesis and purification is not trivial, only a reasonable combination of experimental, empirical and quantum chemical methods can contribute to a reliable thermodynamic analysis and optimisation of hydrogenation reactions. The following steps leading to the solution of the Gibbs–Helmholtz equation for each of the fifteen reactions of the carvone hydrogenation reactions were performed in this work.

-

Step I: the high-level QC calculations are first performed in order to obtain ${∆}_{\mathrm{f}}{H}_{\mathrm{m}}^{\mathrm{o}}$(g) and $S_{\mathrm{m}}^{\mathrm{o}}$(g) values.

-

Step II: vapour pressure–temperature dependencies for the molecules involved in the reaction network are measured or taken from the literature. All possible experimental and empirical data and methods are used to evaluate and validate the ${∆}_{\mathrm{l}}^{\mathrm{g}}{H}_{\mathrm{m}}^{\mathrm{o}}$ and ${∆}_{\mathrm{l}}^{\mathrm{g}}{S}_{\mathrm{m}}^{\mathrm{o}}$ values.

-

Step III: the ${∆}_{\mathrm{f}}{H}_{\mathrm{m}}^{\mathrm{o}}$(liq) and $S_{\mathrm{m}}^{\mathrm{o}}$(liq) values in the liquid phase are derived from the results of the first and second steps. The enthalpies of formation of the liquid phase are validated (e.g., with the help of combustion experiments in this work). The ${∆}_r{H}_{\mathrm{m}}^{\mathrm{o}}$(liq) and ${∆}_r{S}_{\mathrm{m}}^{\mathrm{o}}$(liq) values are derived according to Hess’s Law and inserted into Equation (1) to derive the ${∆}_r{G}_{\mathrm{m}}^{\mathrm{o}}$(liq) of a desired reaction, and an analysis of all three quantities is performed.

This workflow helps to follow the combination of experimental, empirical and quantum chemical methods used in this work to evaluate and compare the feasibility and energetics of the reaction network depicted in Figure 1 and Figure 2.

3. Theoretical and Experimental Methods

The Gaussian 16 series software [6] was used for quantum chemical calculations under the “rigid rotator-harmonic oscillator” assumption. The total H₂₉₈ enthalpies of the most stable conformers were calculated using the G4 method [7]. The H₂₉₈ values were finally converted to the theoretical ${∆}_{\mathrm{f}}{H}_{\mathrm{m}}^{\mathrm{o}}$(g, 298 K) values and discussed. The $S_{\mathrm{m}}^{\mathrm{o}}$(g, 298 K)_QC values were calculated according to (Equation (1)) using H₂₉₈ and G₂₉₈ from the output file.

The transpiration method was used to measure the absolute vapour pressures of (-)-carvone. The vapour pressures temperature dependence was used to derive the enthalpy and entropy of vaporisation of (-)-carvone. The necessary details can be found in ESI.

The standard molar energy of combustion of (-)-carvone was measured with a home-made isoperibolic calorimeter with a static bomb and a stirred water bath. The standard molar enthalpy of formation of (-)-carvone in the liquid phase, ${∆}_{\mathrm{f}}{H}_{\mathrm{m}}^{\mathrm{o}}$(liq), was obtained from the specific energy of combustion. The sample of (-)-carvone used in this work was of commercial origin (see Table S1).

4. Results and Discussion

4.1. Step I: Theoretical Gas-Phase Enthalpies of Formation from Quantum Chemistry

The most stable conformer for the reactants shown in Figure 1 and Figure 2 was localised using a computer code CREST (conformer-rotamer ensemble sampling tool) [8] and optimised with the B3LYP/6-31g(d,p) method [9]. The structures of the most stable conformers of carvone are shown in Figure 3 as an example.

The structures of the most stable conformers of carvone derivatives are shown in Figure S1. The general atomisation (AT) reaction

C_mH_nO_k = m × C + n × H + k × O

(9)

was used to convert the total H₂₉₈ values available directly from the output file to the standard molar enthalpies of formation ${∆}_{\mathrm{f}}{H}_{\mathrm{m}}^{\mathrm{o}}$(g, 298 K)_AT. However, the AT method must be corrected using a linear correlation between the ${∆}_{\mathrm{f}}{H}_{\mathrm{m}}^{\mathrm{o}}$(g, 298 K)_AT values and the experimental enthalpies of formation, ${∆}_{\mathrm{f}}{H}_{\mathrm{m}}^{\mathrm{o}}$(g, 298 K)_exp, of a number of molecules containing similar functional groups (as the reactants in Figure 1 and Figure 2) with reliable experimental data (see

Table 1. Correlation of the G4 calculated and experimental gas-phase enthalpies of formation ${∆}_{\mathrm{f}}{H}_{\mathrm{m}}^{\mathrm{o}}$(g) at T = 298 K and p° = 0.1 MPa (in kJ·mol⁻¹).

Compound	AT a	Exp b	ATcorr c	Δ d
cyclohexanol	−289.8	−290.0 ± 2.1 [10]	−288.8	−1.2
butanol	−274.4	−275.0 ± 0.4 [10]	−273.9	−1.1
pentanol	−295.7	−294.7 ± 0.5 [10]	−294.9	0.2
hexanol	−317.0	−315.8 ± 0.6 [10]	−315.8	0.0
heptanol	−338.4	−336.4 ± 1.0 [10]	−336.7	0.3
octanol	−359.7	−356.5 ± 1.5 [10]	−357.7	1.2
cylopentanone	−192.9	−192.1 ± 1.8 [10]	−193.9	1.8
cyclohexanone	−228.4	−226.1 ± 2.1 [10]	−228.7	2.6
phenol	−90.8	−92.5 ± 1.2 [11]	−93.6	1.1
2-methylphenol	−127.0	−128.4 ± 0.9 [10]	−129.1	0.7
4-methylphenol	−122.0	−125.8 ± 1.5 [10]	−124.2	−1.6
2,3-dimethylphenol	−156.6	−157.2 ± 1.1 [10]	−158.2	1.0
2,5-dimethylphenol	−159.5	−161.6 ± 1.0 [10]	−161.1	−0.5
2,6-dimethylphenol	−161.1	−161.8 ± 0.5 [10]	−162.6	0.8
3,4-dimethylphenol	−156.2	−156.6 ± 0.6 [10]	−157.8	1.2
3,5-dimethylphenol	−157.7	−161.6 ± 0.7 [10]	−159.3	−2.2
2,3,6-trimethylphenol	−190.3	−192.9 ± 1.4 [10]	−191.3	−1.6
2,4,6-trimethylphenol	−192.0	−192.4 ± 1.3 [12]	−193.0	0.6
carvacrol (5-isopropyl-2-methylphenol)	−210.8		−211.4

^a Calculated according to atomisation reaction in Equation (9). ^b Experimental values from the literature. ^c Calculated according to Equation (10). ^d Difference between columns 3 and 4.

, column 3).

We deliberately included the aliphatic and aromatic alcohols (phenols) as well as the cyclic aliphatic alcohols and ketones in the correlation. However, all these structures lie on the same line with a very high correlation coefficient R² = 0.9997 as follows:

$${{∆}_{\mathrm{f}}{H}_{\mathrm{m}}^{\mathrm{o}}(\mathrm{g},\mathrm{G}4)}_{\mathrm{corr}}/\mathrm{kJ}·{\mathrm{mol}}^{-1}=0.9818\times {∆}_{\mathrm{f}}{H}_{\mathrm{m}}^{\mathrm{o}}(\mathrm{g},\mathrm{AT})-4.3\mathrm{with}{R}^2=0.9997$$

(10)

This equation was used to obtain the “corrected” G4 results for the carvone derivatives (see Figure 1 and Figure 2). The summary of the QC calculations is given in Table 2.

Table 2. Compilation of the G4 calculated gas-phase enthalpies of formation for the carvone derivatives (at T = 298 K, p° = 0.1 MPa, in kJ·mol⁻¹).

CAS	Compound	${∆}_{\mathbf{f}}{\mathit{H}}_{\mathbf{m}}^{\mathbf{o}}$(g, AT) a	${∆}_{\mathbf{f}}{\mathit{H}}_{\mathbf{m}}^{\mathbf{o}}$(g, G4)corr b	${∆}_{\mathbf{l}}^{\mathbf{g}}{\mathit{H}}_{\mathbf{m}}^{\mathbf{o}}$c	${∆}_{\mathbf{f}}{\mathit{H}}_{\mathbf{m}}^{\mathbf{o}}$(liq)theor d
6485-40-1	(-)-carvone	−122.1	−124.2	61.7	−185.9
99-48-9	trans-carveol	−168.7	−169.9	72.6	−242.5
5524-05-0	trans-(±)-dihydrocarvone	−221.1	−221.4	57.2	−278.6
3792-53-8	cis-(±)-dihydrocarvone	−220.0	−220.3	56.8	−277.1
499-71-8	carvotanacetone	−238.4	−238.4	59.3	−297.7
499-70-7	(±)-carvomenthone	−337.1	−335.3	58	−393.3
	cis-(±)-tetrahydrocarvone	−335.1	−333.3	58.4	−391.7
499-75-2	carvacrol	−210.8	−211.3	72.5	−283.8
499-69-4	carvomenthol	−397.0	−394.1	69.9	−464.0
619-01-2	trans-dihydrocarveol	−281.8	−281.0	71.9	−352.9
22567-21-1	cis-dihydrocarveol	−274.0	−273.3	70.1	−343.4
536-30-1	carvotanacetol	−280.5	−279.7	73.1	−352.8

^a Calculated using the atomisation reactions Equation (9) with the expanded uncertainties of ±3.5 kJ·mol⁻¹ [7]. ^b The result obtained from the atomisation reaction Equation (9) was corrected with Equation (10). ^c From Table 3. ^d Difference columns 4 and 5 in this table.

Table 2. Compilation of the G4 calculated gas-phase enthalpies of formation for the carvone derivatives (at T = 298 K, p° = 0.1 MPa, in kJ·mol⁻¹).

CAS	Compound	${∆}_{\mathbf{f}}{\mathit{H}}_{\mathbf{m}}^{\mathbf{o}}$(g, AT) a	${∆}_{\mathbf{f}}{\mathit{H}}_{\mathbf{m}}^{\mathbf{o}}$(g, G4)corr b	${∆}_{\mathbf{l}}^{\mathbf{g}}{\mathit{H}}_{\mathbf{m}}^{\mathbf{o}}$c	${∆}_{\mathbf{f}}{\mathit{H}}_{\mathbf{m}}^{\mathbf{o}}$(liq)theor d
6485-40-1	(-)-carvone	−122.1	−124.2	61.7	−185.9
99-48-9	trans-carveol	−168.7	−169.9	72.6	−242.5
5524-05-0	trans-(±)-dihydrocarvone	−221.1	−221.4	57.2	−278.6
3792-53-8	cis-(±)-dihydrocarvone	−220.0	−220.3	56.8	−277.1
499-71-8	carvotanacetone	−238.4	−238.4	59.3	−297.7
499-70-7	(±)-carvomenthone	−337.1	−335.3	58	−393.3
	cis-(±)-tetrahydrocarvone	−335.1	−333.3	58.4	−391.7
499-75-2	carvacrol	−210.8	−211.3	72.5	−283.8
499-69-4	carvomenthol	−397.0	−394.1	69.9	−464.0
619-01-2	trans-dihydrocarveol	−281.8	−281.0	71.9	−352.9
22567-21-1	cis-dihydrocarveol	−274.0	−273.3	70.1	−343.4
536-30-1	carvotanacetol	−280.5	−279.7	73.1	−352.8

^a Calculated using the atomisation reactions Equation (9) with the expanded uncertainties of ±3.5 kJ·mol⁻¹ [7]. ^b The result obtained from the atomisation reaction Equation (9) was corrected with Equation (10). ^c From Table 3. ^d Difference columns 4 and 5 in this table.

Table 3. Compilation of the standard molar enthalpies of vaporisation, ${∆}_{\mathrm{l}}^{\mathrm{g}}{H}_{\mathrm{m}}^{\mathrm{o}}$, of the carvone derivatives (in kJ·mol⁻¹).

Compounds, CAS	T-Range/ K	${∆}_{\mathbf{l}}^{\mathbf{g}}{\mathit{H}}_{\mathbf{m}}^{\mathbf{o}}$/ Tav	${∆}_{\mathbf{l}}^{\mathbf{g}}{\mathit{H}}_{\mathbf{m}}^{\mathbf{o}}$/ a 298 K
(˗)-carvone, 6485-40-1	279.6–333.0	61.4 ± 0.2	61.7 ± 0.3 b
	320–504	52.2 ± 0.5	60.2 ± 1.7
cis-(±)-dihydrocarvone, 3792-53-8	322–495	49.7 ± 0.8	56.8 ± 1.6
trans-(±)-dihydrocarvone, 5524-05-0	343–495	49.4 ± 0.6	57.2 ± 1.7
cis-(±)-tetrahydrocarvone	335–496	50.6 ± 0.5	58.4 ± 1.6
trans-(±)-tetrahydrocarvone (carvomenthone), 499-70-7	335–494	50.0 ± 0.5	58.0 ± 1.7
carvotanacetone, 499-71-8	317–501	52.4 ± 0.7	59.3 ± 1.5
carvomenthol, 499-69-4	354–491	53.7 ± 0.5	69.9 ± 3.3
(±)-cis-carveol, 1197-06-4	338–504	57.0 ± 1.2	71.0 ± 3.0
(±)-trans-carveol, 99-48-9	314–501	58.5 ± 0.8	72.6 ± 2.9
trans-dihydrocarveol, 619-01-2	350–498	55.0 ± 0.6	71.9 ± 3.4
cis-dihydrocarveol, 22567-21-1	353–498	55.4 ± 1.3	70.1 ± 3.6
carvotanacetol, 536-30-1	337–495	59.6 ± 1.3	73.1 ± 3.0
carvacrol, 499-75-2			72.5 ± 0.1 c
2-methyl-2-cyclohexen-1-one, 1121-18-2	325–452	44.3 ± 0.3	48.2 ± 0.8 d
2-methylcyclohex-2-en-1-ol, 20461-30-7	298–441	54.0 ± 1.0	60.2 ± 1.3 d

^a Vapour pressures from Table S2 were treated using Equations (11) and (12) using the heat capacity differences from Table S4 to calculate the enthalpies of vaporisation at 298 K. Uncertainties of the vaporisation enthalpies U(${∆}_{\mathrm{l}}^{\mathrm{g}}{H}_{\mathrm{m}}^{\mathrm{o}}$) are the expanded uncertainties (0.95 level of confidence). They include uncertainties from the fitting equation and uncertainties from temperature adjustment to T = 298 K. Uncertainties in the temperature adjustment of vaporisation enthalpies to the reference temperature T = 298 K are estimated to account with 20% to the total adjustment. ^b From Table S3. ^c From Ref. [13]. ^d Used as model compounds for the calculations according to the “centrepiece” approach (see Section 4.2.3).

Table 3. Compilation of the standard molar enthalpies of vaporisation, ${∆}_{\mathrm{l}}^{\mathrm{g}}{H}_{\mathrm{m}}^{\mathrm{o}}$, of the carvone derivatives (in kJ·mol⁻¹).

Compounds, CAS	T-Range/ K	${∆}_{\mathbf{l}}^{\mathbf{g}}{\mathit{H}}_{\mathbf{m}}^{\mathbf{o}}$/ Tav	${∆}_{\mathbf{l}}^{\mathbf{g}}{\mathit{H}}_{\mathbf{m}}^{\mathbf{o}}$/ a 298 K
(˗)-carvone, 6485-40-1	279.6–333.0	61.4 ± 0.2	61.7 ± 0.3 b
	320–504	52.2 ± 0.5	60.2 ± 1.7
cis-(±)-dihydrocarvone, 3792-53-8	322–495	49.7 ± 0.8	56.8 ± 1.6
trans-(±)-dihydrocarvone, 5524-05-0	343–495	49.4 ± 0.6	57.2 ± 1.7
cis-(±)-tetrahydrocarvone	335–496	50.6 ± 0.5	58.4 ± 1.6
trans-(±)-tetrahydrocarvone (carvomenthone), 499-70-7	335–494	50.0 ± 0.5	58.0 ± 1.7
carvotanacetone, 499-71-8	317–501	52.4 ± 0.7	59.3 ± 1.5
carvomenthol, 499-69-4	354–491	53.7 ± 0.5	69.9 ± 3.3
(±)-cis-carveol, 1197-06-4	338–504	57.0 ± 1.2	71.0 ± 3.0
(±)-trans-carveol, 99-48-9	314–501	58.5 ± 0.8	72.6 ± 2.9
trans-dihydrocarveol, 619-01-2	350–498	55.0 ± 0.6	71.9 ± 3.4
cis-dihydrocarveol, 22567-21-1	353–498	55.4 ± 1.3	70.1 ± 3.6
carvotanacetol, 536-30-1	337–495	59.6 ± 1.3	73.1 ± 3.0
carvacrol, 499-75-2			72.5 ± 0.1 c
2-methyl-2-cyclohexen-1-one, 1121-18-2	325–452	44.3 ± 0.3	48.2 ± 0.8 d
2-methylcyclohex-2-en-1-ol, 20461-30-7	298–441	54.0 ± 1.0	60.2 ± 1.3 d

^a Vapour pressures from Table S2 were treated using Equations (11) and (12) using the heat capacity differences from Table S4 to calculate the enthalpies of vaporisation at 298 K. Uncertainties of the vaporisation enthalpies U(${∆}_{\mathrm{l}}^{\mathrm{g}}{H}_{\mathrm{m}}^{\mathrm{o}}$) are the expanded uncertainties (0.95 level of confidence). They include uncertainties from the fitting equation and uncertainties from temperature adjustment to T = 298 K. Uncertainties in the temperature adjustment of vaporisation enthalpies to the reference temperature T = 298 K are estimated to account with 20% to the total adjustment. ^b From Table S3. ^c From Ref. [13]. ^d Used as model compounds for the calculations according to the “centrepiece” approach (see Section 4.2.3).

The “corrected” ${∆}_{\mathrm{f}}{H}_{\mathrm{m}}^{\mathrm{o}}$(g, G4)_corr values for carvone derivatives (Table 2, column 4) obtained in this step are now ready for thermochemical calculations in further steps.

4.2. Step II: Vaporisation Thermodynamics

4.2.1. Absolute Vapour Pressures

The vapour pressures of carvone derivatives are of interest for the appropriate design of the technology, the separation of the products and for safety reasons. Unfortunately, the systematic vapour pressure–temperature dependencies for carvone derivatives are practically non-existent in the literature (except for carvacrol [13]). However, the boiling points (BPs) of these compounds at various reduced pressures were collected from the literature (see a compilation in Table S2), and these data were evaluated to obtain vapour pressure–temperature dependencies for each compound and derive the corresponding enthalpies of vaporisation. It is important to note that these boiling points were recorded during the distillation of the reaction mixtures after synthesis. Therefore, the temperatures were given in the range of a few degrees, and the pressures were measured with uncalibrated manometers. In our experience, even such rough data generally show a reasonable trend of vapour pressure–temperature dependencies [14]. To prove this general observation for the carvone derivatives, we collected the available BP for (-)-carvone in the range 320–504 K (see Table S2), and to verify these data, the vapour pressure–temperature dependence in the range 279.6–333.0 K for (-)-carvone was measured by the transpiration method (see Table S3).

In Figure S2, the experimental vapour pressures for (-)-carvone were compared with the boiling points found in the literature for this compound at reduced pressures (see Table S2). As can be seen from this figure, the results from both data sets generally show the same trend. Once the applicability of the BP method for carvone derivatives was established, the boiling points at reduced pressure for each compound were collected (see Table S2) and involved in the thermodynamic analysis of the carvone hydrogenation reaction network.

The vapour pressures collected in Tables S2 and S3 were fit by the following equation:

$$R·\ln (p_{\mathrm{i}}{/p}_{\mathrm{r}\mathrm{e}\mathrm{f}})=\mathrm{a}+{\displaystyle \frac{\mathrm{b}}{T}}+{∆}_{\mathrm{l}}^{\mathrm{g}}{C}_{p,\mathrm{m}}^{\mathrm{o}}·\mathrm{l}\mathrm{n}\left({\displaystyle \frac{T}{T_0}}\right)$$

(11)

where a and b are adjustable parameters, T₀ applied in Equation (11) is the arbitrary temperature (was chosen to be T₀ = 298 K in this work), R = 8.314462 J·K⁻¹·mol⁻¹ is the molar gas constant, and the reference pressure $p_{\mathrm{r}\mathrm{e}\mathrm{f}}=1\mathrm{Pa},{∆}_{\mathrm{c}\mathrm{r},\mathrm{l}}^{\mathrm{g}}{C}_{p,\mathrm{m}}^{\mathrm{o}}$ = $C_{p,\mathrm{m}}^{\mathrm{o}}$(g) − $C_{p,\mathrm{m}}^{\mathrm{o}}$(liq) is the difference between the molar heat capacities of the gaseous $C_{p,\mathrm{m}}^{\mathrm{o}}$(g) and the liquid phase $C_{p,\mathrm{m}}^{\mathrm{o}}$(liq), respectively. The ${∆}_{\mathrm{l}}^{\mathrm{g}}{C}_{p,\mathrm{m}}^{\mathrm{o}}$ values used in Equation (11) are given in Table S4. The vapour pressures and fitting parameters of Equation (11) are given in Tables S2 and S3.

4.2.2. Vaporisation Enthalpies

The standard molar enthalpies of vaporisation of carvone derivatives at temperatures T were derived from the temperature dependence of the vapour pressures, which was approximated by Equation (11) using the following equation:

$${∆}_{\mathrm{l}}^{\mathrm{g}}{H}_{\mathrm{m}}^{\mathrm{o}}\left(T\right)=-b+{∆}_{\mathrm{l}}^{\mathrm{g}}{C}_{p,\mathrm{m}}^{\mathrm{o}}\times T$$

(12)

where b is one of the adjustable parameters of Equation (11). The vaporisation enthalpies at the reference temperature T = 298 K, ${∆}_{\mathrm{l}}^{\mathrm{g}}{H}_{\mathrm{m}}^{\mathrm{o}}$(298 K), of (carvone derivatives are summarised in Table 3.

As can be seen from this table, for (-)-carvone, our transpiration result ${∆}_{\mathrm{l}}^{\mathrm{g}}{H}_{\mathrm{m}}^{\mathrm{o}}$(298 K) = (61.7 ± 0.3) kJ·mol⁻¹ agrees very well with the vaporisation enthalpy, ${∆}_{\mathrm{l}}^{\mathrm{g}}{H}_{\mathrm{m}}^{\mathrm{o}}$(298 K) = (60.2 ± 1.7) kJ·mol⁻¹ derived in this work from the boiling points collected for this compound in Table S2.

4.2.3. Validation of Enthalpies of Vaporisation Using the “Centrepiece” Approach

Before the enthalpies of vaporisation of the carvone derivatives summarised in Table 3 are used for further thermochemical calculations, it is useful to validate these results with empirical methods based on structure–property correlations. The crucial advantage of empirical methods is that they are developed from reliable experimental data and have internal consistency. Therefore, the new results should be integrated into the network of already known data. If the new results are consistent with the already known trends, they can be validated for the thermochemical calculations. If this is not the case, the reason for the specific behaviour should be uncovered and justified.

A group addition (GA) method is one of the best manifestation of structure–property correlation. The conventional methodology of GA methods is to divide the experimental enthalpies of the molecules into the smallest possible groups and calculate numerical contributions for them from the matrix of well-established data. The prediction is then made by building a framework of the desired model molecule from the corresponding number and type of these contributions. Admittedly, the conventional GA methods have difficulties with the cyclic molecules. To overcome these difficulties, a series of unique “ring” corrections are proposed to improve the prediction. To address this shortcoming, the GA method can be modified by taking the specific ring as the centrepiece as the starting point for constructing the frame of the desired model molecule when the small additional contributions are attached to the “centrepiece”.

The general idea of this approach is to first select a potentially large “centrepiece” molecule that has a reliable enthalpy and can generally mimic the structure of other molecules of interest [15]. Then, the required substituents (or groups) are added to the “centrepiece” molecule, leading to the construction of the desired molecule. In this work, cyclohexane derivatives (cyclohexanone and cyclohexanol derivatives) can serve as a suitable “centrepiece” for the prediction of the enthalpy of vaporisation of carvone derivatives. Two types of substituents are specific to the carvone derivatives: the iso-propenyl and the iso-propyl substituents for the exchange of the H atom in the cyclohexane ring, as shown in Figure 4.

The idea of the “centrepiece” approach for carvone derivatives is illustrated in Figure 5.

To construct the carvone, the contribution for the substituent H→iPr(en) = 10.8 kJ·mol⁻¹ (see Figure 4) should be added to the enthalpy of vaporisation of 2-methyl-2-cyclohexen-1-one ${∆}_{\mathrm{l}}^{\mathrm{g}}{H}_{\mathrm{m}}^{\mathrm{o}}$(298 K) = 48.2 ± 0.8 kJ·mol⁻¹ (see Table S2), which is used as the “centrepiece”. The resulting value ${∆}_{\mathrm{l}}^{\mathrm{g}}{H}_{\mathrm{m}}^{\mathrm{o}}$(298 K) = 59.0 ± 1.4 kJ·mol⁻¹ (see Figure 5) has acceptable agreement with the value ${∆}_{\mathrm{l}}^{\mathrm{g}}{H}_{\mathrm{m}}^{\mathrm{o}}$(298 K) = 61.7 ± 0.3 kJ·mol⁻¹ (see Table 3) measured with transpiration, as well as with the value ${∆}_{\mathrm{l}}^{\mathrm{g}}{H}_{\mathrm{m}}^{\mathrm{o}}$(298 K) = 60.2 ± 1.7 kJ·mol⁻¹ (see Table S2) derived from BP in this work.

To construct the trans-tetrahydrocarvone, the contribution for the substituent H→iPr = 10.9 kJ·mol⁻¹ (see Figure 4) should be added to the enthalpy of vaporisation of 2-methyl-cyclohexanone ${∆}_{\mathrm{l}}^{\mathrm{g}}{H}_{\mathrm{m}}^{\mathrm{o}}$(298 K) = 46.5 ± 0.2 kJ·mol⁻¹ (see Table S5), which is used as the “centrepiece”. The resulting value ${∆}_{\mathrm{l}}^{\mathrm{g}}{H}_{\mathrm{m}}^{\mathrm{o}}$(298 K) = 57.4 ± 1.0 kJ·mol⁻¹ (see Figure 5) is in agreement with the value ${∆}_{\mathrm{l}}^{\mathrm{g}}{H}_{\mathrm{m}}^{\mathrm{o}}$(298 K) = 58.0 ± 1.7 kJ·mol⁻¹ (see

Table 4. Compilation of the standard molar entropies of vaporisation, ${∆}_{\mathrm{l}}^{\mathrm{g}}{S}_{\mathrm{m}}^{\mathrm{o}}$, and the absolute standard molar entropies, $S_{\mathrm{m}}^{\mathrm{o}}$(g or liq), of carvone derivatives (all values at T = 298 K in J·mol⁻¹·K⁻¹).

CAS	Compound	${∆}_{\mathbf{l}}^{\mathbf{g}}{\mathit{S}}_{\mathbf{m}}^{\mathbf{o}}$ a	${\mathit{S}}_{\mathbf{m}}^{\mathbf{o}}$(g) b	${\mathit{S}}_{\mathbf{m}}^{\mathbf{o}}$(liq) c
6485-40-1	(-)-carvone	130.8 ± 0.7	441	310.2 d
99-48-9	trans-carveol	159.5 ± 2.8	434	274.5
3792-53-8	cis-(±)-dihydrocarvone	122.8 ± 2.8	439	316.2
5524-05-0	trans-(±)-dihydrocarvone	123.0 ± 2.0	448	325.0
499-71-8	carvotanacetone	126.6 ± 2.5	445	318.4
499-70-7	trans-(±)-tetrahydrocarvone (carvomenthone)	124.8 ± 1.8	450	325.2
	cis-(±)-tetrahydrocarvone	125.9 ± 1.9	453	327.1
619-01-2	trans-dihydrocarveol	159.8 ± 2.0	451	291.2
22567-21-1	cis-dihydrocarveol	155.4 ± 4.5	449	293.6
536-30-1	carvotanacetol	163.2 ± 4.5	450	286.8
499-75-2	carvacrol	157.1 ± 0.5 [13]	455 [13]	297.9

^a From the vapour pressure measurements compiled in Tables S2 and S3. ^b Calculated with the G4 method [7]. ^c Calculated according to Equation (3) using entries from columns 3 and 4 from this table. ^d For comparison, the experimental value $S_{\mathrm{m}}^{\mathrm{o}}$(liq) =314.3 ± 0.6 J·mol⁻¹·K⁻¹ [16] was measured for (-)-carvone with adiabatic calorimetry.

) derived from BP in this work.

Similarly, the enthalpies of vaporisation for other carvone derivatives were calculated, as shown in Figures S3–S5. The “empirical” enthalpies of vaporisation derived using the “centrepiece” approach are compared in Table S6 and agree well with results obtained by other methods. This validation using the structure–property correlations makes it possible to recommend the ${∆}_{\mathrm{l}}^{\mathrm{g}}{H}_{\mathrm{m}}^{\mathrm{o}}$(298 K) values (Table 3) for further thermochemical calculations.

4.2.4. Entropies of Vaporisation

The change in the liquid-phase standard molar entropy of a chemical reaction, ${{∆}_{r}S}_{\mathrm{m}}^{\mathrm{o}}$(liq), as the entropic contribution to the Gibbs–Helmholtz equation is required for further thermochemical calculations. The standard molar entropies, $S_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K), of reactants (see Figure 1 and Figure 2) in the liquid phase are necessary as input for the calculation of the ${{∆}_{r}S}_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K) values according to Equation (4). The standard molar entropies, $S_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K), of the carvone derivatives were obtained using Equation (5). The gas-phase entropies were calculated using the G4 method. The standard molar entropies of vaporisation, ${∆}_{\mathrm{l}}^{\mathrm{g}}{S}_{\mathrm{m}}^{\mathrm{o}}$(298 K), were calculated according to Equation (6). The compilation of the standard molar entropies of vaporisation, ${∆}_{\mathrm{l}}^{\mathrm{g}}{S}_{\mathrm{m}}^{\mathrm{o}}$(298 K), and the absolute standard molar entropies, $S_{\mathrm{m}}^{\mathrm{o}}$(g or liq, 298 K), of carvone derivatives is shown in Table 4.

The resulting liquid-phase standard molar entropies, $S_{\mathrm{m}}^{\mathrm{o}}$(liq), required to estimate the entropic contribution to the Gibbs–Helmholtz equation (see Equation (1)) are given in Table 4, column 5.

4.3. Step III: Thermodynamic Analysis of the Reaction Network for the Hydrogenation of Carvone

4.3.1. Liquid-Phase Enthalpies of Formation and Liquid-Phase Reaction Enthalpies

The “corrected” quantum–chemical gas-phase enthalpies of formation, ${∆}_{\mathrm{f}}{H}_{\mathrm{m}}^{\mathrm{o}}$(g, 298 K)_G4, of the carvone derivatives were calculated using the G4 method (Table 2, column 4). They were recalculated to the liquid-phase enthalpies of formation, ${∆}_{\mathrm{f}}{H}_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K), via the enthalpies of vaporisation, ${∆}_{\mathrm{l}}^{\mathrm{g}}{H}_{\mathrm{m}}^{\mathrm{o}}$(298 K), which are shown in Table 3. The ${∆}_{\mathrm{f}}{H}_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K) values were calculated according to Equation (3), and the results are listed in Table 2 (last column).

The enthalpies of formation of the carvone derivatives in the liquid phase determined in this way must also be validated before they can be used for the thermodynamic analysis of the reaction network shown in Figure 1 and Figure 2. The theoretical result for (-)-carvone ${∆}_{\mathrm{f}}{H}_{\mathrm{m}}^{\mathrm{o}}$(liq)_theor = −185.9 ± 3.5 kJ·mol⁻¹ (Table 2) was compared with the experimental result, which was directly measured in this work using high-precision combustion calorimetry as follows.

The standard specific energy of combustion ${∆}_{\mathrm{c}}{u}^{\mathrm{o}}$(liq) = −38222.5 ± 4.3 J·g⁻¹ of the (-)-carvone was determined from five experiments. The results of the combustion experiments are summarised in Table S8. The ${∆}_{\mathrm{c}}{u}^{\mathrm{o}}$(liq) value was used to calculate the experimental standard molar enthalpy of combustion, ${∆}_{\mathrm{c}}{H}_{\mathrm{m}}^{\mathrm{o}}$ (liq) = −5479.1 ± 1.9 kJ·mol⁻¹, which refers to the reaction

C₁₀H₁₄O (liq) + 13 O₂ (g) = 10 CO₂ (g) + 7 H₂O (liq)

(13)

The experimental standard molar enthalpy of formation in the liquid state ${∆}_{\mathrm{f}}{H}_{\mathrm{m}}^{\mathrm{o}}$(liq) = −186.8 ± 2.3 kJ·mol⁻¹ for (-)-carvone is in very good agreement with the theoretical result ${∆}_{\mathrm{f}}{H}_{\mathrm{m}}^{\mathrm{o}}$(liq)_theor = −185.9 ± 3.5 kJ·mol⁻¹ (Table 2). Such a good agreement supports the results derived for the carvone derivatives in Steps I and II.

The enthalpies of formation in the liquid phase have now been validated, and the reliable ${∆}_r{H}_{\mathrm{m}}^{\mathrm{o}}$(liq) values for reactions of carvone hydrogenation (see Figure 1 and Figure 2) were derived according to Equation (2). The resulting reaction enthalpies in the liquid phase at the reference temperature, ${∆}_{\mathrm{r}}{H}_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K), are shown in

Table 5. Calculation of the liquid-phase thermodynamic properties for reactions R1 to R20 (see Figure 1, Figure 2 and Figure S6).

R	${∆}_{\mathbf{r}}{\mathit{H}}_{\mathbf{m}}^{\mathbf{o}}$ a (298 K)	${∆}_{\mathbf{r}}{\mathit{S}}_{\mathbf{m}}^{\mathbf{o}}$ b (298 K)	${∆}_{\mathbf{r}}{\mathit{C}}_{\mathit{p},\mathbf{m}}^{\mathbf{o}}$ c (298 K)	${∆}_{\mathbf{r}}{\mathit{G}}_{\mathbf{m}}^{\mathbf{o}}$ d (298 K)	${∆}_{\mathbf{r}}{\mathit{G}}_{\mathbf{m}}^{\mathbf{o}}$ e (400 K)	${∆}_{\mathbf{r}}{\mathit{G}}_{\mathbf{m}}^{\mathbf{o}}$ f (500 K)	lnKa g (298 K)	lnKa g (400 K)	lnKa g (500 K)
	kJ·mol−1	J·mol−1·K−1	J·mol−1·K−1	kJ·mol−1	kJ·mol−1	kJ·mol−1
				partial hydrogenation
1	−92.7	−115.9	−29.0	−58.2	−45.9	−33.1	23.5	18.5	13.4
2	−114.7	−130.5	−19.0	−75.8	−62.2	−48.4	30.6	25.1	19.5
3	−70.7	−158.6	−0.7	−23.4	−7.3	8.6	9.5	2.9	−3.5
4	−111.8	−122.5	−21.6	−75.3	−62.5	−49.3	30.4	25.2	19.9
5	−95.6	−123.9	−26.4	−58.7	−45.7	−32.1	23.7	18.4	13.0
6	−56.9	−166.4	−3.3	−7.0	10.0	26.8	2.8	−4.0	−10.8
7	−110.4	−114.0	−3.9	−76.4	−64.8	−53.2	30.8	26.1	21.5
8	−111.1	−124.6	−41.5	−74.0	−60.7	−46.4	29.9	24.5	18.7
9	−110.3	−118.4	−19.0	−75.0	−62.7	−50.0	30.3	25.3	20.2
10	−111.2	−120.2	−26.4	−75.4	−62.8	−49.6	30.4	25.3	20.0
				dehydrogenation
11	13.9	119.2	67.2	−21.6	−34.7	−49.6	8.7	14.0	20.0
				isomerisation
12	−97.9	−3.3	45.6	−96.9	−97.2	−98.9	39.1	39.2	39.9
13	−1.5	8.8	-	−4.1	−5.0	−5.9	1.7	2.0	2.4
14	−1.6	−1.9	-	−1.0	−0.8	−0.7	0.4	0.3	0.3
15	−9.5	−2.4	-	−8.8	−8.5	−8.3	3.5	3.4	3.4
				full hydrogenation
16	−207.4	−246.4	−48.0	−133.9	−108.2	−81.4	54.1	43.7	32.9
17	−278.1	−404.9	−48.7	−157.4	−115.4	−72.8	63.5	46.6	29.4
18	−180.2	−401.6	−94.3	−60.5	−18.2	26.1	24.4	7.4	−10.5
				simple ketones h
19	−70.0	−150.5	6.9	−25.1	−9.9	4.8	10.1	4.0	−2.0
20	−77.0	−155.8	7.6	−30.6	−14.8	0.4	12.3	6.0	−0.2

^a Calculated according to Equation (2) from the standard molar enthalpies of formation of the reactants from Table 2. ^b Calculated according to Equation (4) from the standard molar entropies of the reactants from Table 4. ^c Difference in the standard molar heat capacities of reactants calculated from the values in Table S4. ^d Calculated according to Equation (1) from the results given in columns 2 and 3 and referenced to 298 K. ^e Calculated according to Equation (1) from the results given in columns 2 and 3 and adjusted to 400 K according to Kirchhoff’s Law (see text). ^f Calculated according to Equation (1) from the results given in column 2 and 3 and adjusted to 500 K according to Kirchhoff´s Law (see text). ^g The logarithm of the thermodynamic equilibrium constant K_a, calculated according to Equation (1) for the corresponding temperature. ^h The reactions are shown in Figure S6.

, column 2.

It is interesting to compare the energetics of the hydrogenation of the double bonds (DB): the endo-, the exo-, and the carbonyl C=O, all three of which are present in carvone.

The enthalpy of Reaction 1, ${∆}_{\mathrm{r}}{H}_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K) = −92.7 kJ·mol⁻¹ (Table 5), represents the energetics of endo-DB hydrogenation. For comparison, the hydrogenation enthalpy of cyclohexene (−117.9 kJ·mol⁻¹, calculated from data given in Table S5) which also has the endo-DB in the six-member ring, is significantly more exothermic, reflecting the effect of the conjugation of double bonds in the carvone.

In contrast, the enthalpy of Reaction 4, ${∆}_{\mathrm{r}}{H}_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K) = −111.8 kJ·mol⁻¹ (Table 5), which represents the energetics of exo-DB hydrogenation, is indistinguishable from the hydrogenation of exo-DB of α-methyl-styrene (−111.6 kJ·mol⁻¹, calculated from data given in Table S5), demonstrating the absence of conjugation of exo-DB with the double bonds on the carvone ring.

The enthalpy of Reaction 6, ${∆}_{\mathrm{r}}{H}_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K) = −56.9 kJ·mol⁻¹ (Table 5), represents the energetics of the carbonyl DB hydrogenation. For comparison, the hydrogenation enthalpy of cyclohexanone (−77.0 kJ·mol⁻¹, Table 5) which has a similar shape, is more exothermic. It is likely that the energetics of Reaction 6 are influenced by the conjugation of all three double bonds in carvone and by steric interactions of the substituents on the cyclohexane ring. This conclusion is supported by the fact that the enthalpy of hydrogenation of tetrahydrocarvone in carvomenthol, ${∆}_{\mathrm{r}}{H}_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K) = −70.7 kJ·mol⁻¹ (R3, Table 5), lies energetically between the values of R3 and cyclohexanone (R20). Moreover, the specific effect of the carbonyl double bond in the carvone on the hydrogenation energetics becomes clear when one considers the quite comparable (at the level of −110 kJ·mol⁻¹) hydrogenation enthalpies of reactions R7, R8, R9 and R10, in which the hydroxyl is present on the cyclohexane ring instead of the carbonyl group.

The only endothermic reaction in the network shown in Figure 1 is the dehydrogenation of carvotanacetone to carvacrol, ${∆}_{\mathrm{r}}{H}_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K) = 13.9 kJ·mol⁻¹ (R11, Table 5). However, this very moderate energetics indicates that this dehydrogenation reaction is not far from equilibrium conditions even at low temperatures.

The most surprising aspect of the network shown in Figure 1 is that the isomerisation of carvone to carvacrol, ${∆}_{\mathrm{r}}{H}_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K) = −97.9 kJ·mol⁻¹ (R12, Table 5), is strongly exothermic, which is not typical for this type of reaction. The very strong energetics indicates that the isomerisation reaction competes with the hydrogenation reactions even at low temperatures.

The reaction enthalpies of the cis-trans isomerisation of the carvone derivatives (see Figure 2) as a side reaction of carvone hydrogenation reactions are generally not exceeding 10 kJ·mol⁻¹ (see R13, R14 and R15 in Table 5). The low energetics is also typical for this type of isomerisation reaction.

The network of hydrogenation reactions shown in Figure 1 illustrates the gradual stepwise hydrogenation of carvone. However, it is also of interest to compare the energetics of the partial hydrogenation of endo-DB and exo-DB in carvone (see Figure 6), ${∆}_{\mathrm{r}}{H}_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K) = −207.4 kJ·mol⁻¹ (R16, Table 5) and the complete hydrogenation of all three bonds in carvone, ${∆}_{\mathrm{r}}{H}_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K) = −278.1 kJ·mol⁻¹ (R17, Table 5).

It is obvious that the reaction enthalpy of R17 is greater than the reaction enthalpy of R16 exactly by the contribution to the energetics of C=O bond hydrogenation (see R3 in Table 5). Since different numbers of hydrogen molecules are involved R16 and R17, the reaction enthalpies are usually related to the amount of hydrogen consumed (as kJ·mol⁻¹/_H2 unit) in order to be able to correctly compare the energetics of hydrogenation reactions, taking the stoichiometry into account. In these new units, the hydrogenation enthalpy for R16, ${∆}_{\mathrm{r}}{H}_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K) = −103.7 kJ·mol⁻¹/_H2 and ${∆}_{\mathrm{r}}{H}_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K) = −92.4 kJ·mol⁻¹/_H2 for R17 were recalculated for comparison. Indeed, the two results do not differ significantly, but both hydrogenation enthalpies of the aliphatic carvone are significantly higher than those of its aromatic isomer carvacrol: ${∆}_{\mathrm{r}}{H}_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K) = −180.2/3 = −60.1 kJ·mol⁻¹/_H2 for R18 (see Table 5). This significant difference between the energetics of the hydrogenation of aromatic and aliphatic unsaturated systems indicates that the reaction is facilitated for the aliphatic unsaturated molecules.

4.3.2. Liquid-Phase Reaction Entropies

The reaction entropies, ${∆}_r{S}_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K), which were calculated for R1 to R20 according to Equation (4), are summarised in Table 5, column 3. Admittedly, the reaction entropies (in relation to the amount of hydrogen consumed) do not differ significantly. As can be seen from Table 5, the ${∆}_r{S}_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K) values are quite similar for all hydrogenation reactions of endo- or exo-DB (see R1, R2, R4, R5, R7 to R10) at the 120 J·mol⁻¹·K⁻¹ level. The reactions R3, R4 as well as R19 (hydrogenation of acetone) and R20 (hydrogenation of cyclohexanone), in which a C=O bond is involved, are somewhat higher, but they are also very close to the 150 J·mol⁻¹·K⁻¹ level. As expected, the reaction entropies, ${∆}_r{S}_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K), for the isomerisation reactions R12 to R15 are very small and do not exceed 10 J·mol⁻¹·K⁻¹ (see Table 5).

In this way, both the enthalpic (Section 4.3.1) and entropic (Section 4.3.2) contributions to the Gibbs–Helmholtz equation for the network reactions of carvone hydrogenation were derived (see Table 5) according to Equation (1), and the feasibility of these chemical reactions in terms of the standard molar Gibbs energies can be performed.

4.3.3. Gibbs Energies and Thermodynamic Analysis of the Hydrogenation of Carvone

From the experimental studies on the hydrogenation of carvone in the presence of various catalysts, the following order of decreasing reactivity was determined empirically: exo-DB > endo DB > C=O [17]. This sequence corresponds to the preferred formation of unsaturated and saturated ketones (see R1, R2 and R4, R5 in Figure 1) as the main products in the hydrogenation of carvone and the low possibility of forming the hydroxyl derivatives of carvone (see R3 and R6 to R10). The ${∆}_r{G}_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K) values calculated for reactions 1 to 15 fundamentally support and quantify (see Table 5) these empirical observations. Indeed, the ${∆}_r{G}_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K) = −75.3 kJ·mol⁻¹ for the hydrogenation of the exo-DB of carvone to carvotaneacetone (R4) is noticeably greater than the ${∆}_r{G}_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K) = −58.2 kJ·mol⁻¹ for the hydrogenation of the endo-DB of carvone to dihydrocarvone (R1). At the same time, the ${∆}_r{G}_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K) = −7.0 kJ·mol⁻¹ for the hydrogenation of the C=O from carvone to carveol (R4) is drastically lower than that for endo- and exo-DB. Thus, the results of the order of thermodynamic stability correspond exactly to the empirically determined order of decreasing reactivity.

The same conclusion can be drawn for the hydrogenation of the remaining hydrogen bonds in the second step: the ${∆}_r{G}_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K) = −76.8 kJ·mol⁻¹ for the hydrogenation of the exo-DB of dihydrocarvone to tetrahydrocarvone (R2) is greater compared to ${∆}_r{G}_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K) = −58.7 kJ·mol⁻¹ for the hydrogenation of the endo-DB of carvotanacetone to tetrahydrocarvone (R5). Also, the ${∆}_r{G}_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K) = −23.4 kJ·mol⁻¹ for the hydrogenation of the C=O from tetrahydrocarvone to carvomenthol (R3).

Interestingly, although the formation of the hydroxyl derivatives of carvone is unfavourable, their thermodynamic stabilities are quite similar: the ${∆}_r{G}_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K) values of ≈ −75 kJ·mol⁻¹ are the same for the hydrogenation of the exo-DB in dihydrocarveol (R8) and carveol (R9) and for the hydrogenation of the endo-DB in carvotanacetol (R10). However, the same trend has already been discussed for the reaction enthalpies of the hydroxyl derivatives of carvone (see Section 4.3.1).

The ${∆}_r{G}_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K) values of the cis-trans isomerisation reactions R13, R14, and R15 are only slightly negative (see Table 5, column 5), which indicates their general feasibility at 298 K. Most strikingly, the ${∆}_r{G}_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K) = −96.9 kJ·mol⁻¹ for the isomerisation of carvone to carvacrol (R12) is by far the largest in the reaction network shown in Figure 1. This proves that the highly selective direct conversion of carvone to carvacrol is possible provided a suitable catalyst is found that does not promote the partial hydrogenation reactions. The additional formation of carvacrol, as the end product of the network reactions in Figure 1, can be achieved by the dehydrogenation of carvotanacetone to carvacrol according to R11. The ${∆}_r{G}_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K) = −21.6 kJ·mol⁻¹ for this reaction is large enough to make a sufficient contribution.

As part of the thermodynamic analysis of chemical reactions, it is usually discussed whether the process is enthalpically or entropically driven. This assessment is usually made by comparing the size of the two contributions to the Gibbs energy according to Equation (1) as shown in Table S9. Interestingly, the entropy contributions ($T\times {∆}_r{S}_{\mathrm{m}}^{\mathrm{o}}$ ≈ 35 kJ·mol⁻¹) to the Gibbs energy of all hydrogenation reactions in Figure 1 are approximately constant and amount to only 1/3 compared to the enthalpic term (see Table S9). Consequently, from a thermodynamic point of view, these reactions equilibrium are mainly determined by the enthalpic term.

Basically, the thermodynamic calculations are initially carried out at the reference temperature T = 298 K. However, the typical temperature range for the reaction network is between 293 and 500 K. Using the standard molar isobaric heat capacities $C_{p,\mathrm{m}}^{\mathrm{o}}$ of the reactions’ participants (see Table S4), the Gibbs energies and equilibrium constants of reactions 1 to 15 at 400 K and 500 K were calculated using Equations (7) and (8). The results are shown in Table 5 and help to understand whether increasing the temperature shifts the equilibrium toward an increase or decrease in yield. As the temperature increases, the equilibrium constants for R1 decrease considerably. In contrast, as the temperature increases, the equilibrium constants for R11 increase. The equilibrium of R12, R14 and R15 is practically independent of the temperature. Only in the cis-trans isomerisation of dihydrocarvone does the equilibrium constant increase with increasing temperature.

4.3.4. Hydrogenation of Carvone: Catalytic Aspects

Quantitative thermodynamic analysis has shown that negative Gibbs energies are recorded for all reactions that may occur during the hydrogenation of carvone (Figure 1 and Figure 2), although these differ significantly in magnitude. This means that all these reactions are thermodynamically feasible in a wide range from ambient temperature to elevated temperatures. Therefore, all these reactions definitely take place under kinetic and not thermodynamic control. Nevertheless, the numerical ${∆}_r{G}_{\mathrm{m}}^{\mathrm{o}}$(liq, 298 K) values can help to establish the chemoselectivity of catalysts used to convert carvone to either carvacarol or to dihydro- and terahydrocarvone, either in carvotanacetone or carveol.

For example, Benavente et al. [5] demonstrated the exclusive formation of carvacrol with the complete carvone conversion via Pd/Al₂O₃ and Pd/C. In the same study, it was shown that Pd/CeO₂ promotes the formation of carveol due to the activation of C=O bonds at certain oxygen vacancies of the catalyst.

The selective hydrogenation of carvone to dihydrocarvone was carried out at 373 K with an Au/TiO₂ catalyst [4]. The gold catalyst showed high activity as well as stereo- and chemoselectivity in the conjugated endo-DB hydrogenation with a predominant formation of trans-dihydrocarvone. The overall selectivity to dihydrocarvone 62% was achieved with an almost complete carvone conversion (90%) after 13 h, whereby the ratio of trans- to cis-dihydrocarvone was about 1.8 (see reaction R13 in Figure 1). According to the calculations with the Gibbs energy of R13 given in Table 5, a ratio of trans- to cis-dihydrocarvone of 3.7 is expected when the system reaches equilibrium. This means that extending the reaction time or doubling the amount of catalyst could increase the carvone conversion and the yield of dihydrocarvones.

5. Conclusions

The thermodynamic analysis of the network of carvone hydrogenation reactions shows that even if the hydrogenations reactions run under kinetic control, the quantitative knowledge of thermodynamic characteristics of reactions is essential to assess the level of equilibrium constants at a given temperature in order to manipulate the amount of catalyst, time, and temperature and achieve sufficient yields of industrially important products of carvone hydrogenation.