==== Front Sci Rep Sci Rep Scientific Reports 2045-2322 Nature Publishing Group UK London 78071 10.1038/s41598-020-78071-1 Article New approach to determine the surface and interface thermodynamic properties of H-β-zeolite/rhodium catalysts by inverse gas chromatography at infinite dilution Hamieh Tayssir tayssir.hamieh@ul.edu.lb 12 Ali Ahmad Ali 23 Roques-Carmes Thibault 4 Toufaily Joumana 3 1 SATIE-IFSTTAR, University Gustave Eiffel, Campus de Marne-La-Vallée, 25, allée des Marronniers, 78000 Versailles, France 2 grid.411324.10000 0001 2324 3572Laboratory of Materials, Catalysis, Environment and Analytical Methods (MCEMA), Faculty of Sciences and EDST, Lebanese University, Hariri Campus, Hadath, Beirut Lebanon 3 grid.411324.10000 0001 2324 3572LEADDER Laboratory, Faculty of Sciences and EDST, Lebanese University, Hariri Campus, Hadath, Beirut Lebanon 4 grid.29172.3f0000 0001 2194 6418Laboratoire Réactions Et Génie Des Procédés (LRGP), UMR CNRS, 7274, Université de Lorraine, 1 Rue Grandville, 54001 Nancy, France 1 12 2020 1 12 2020 2020 10 208948 4 2020 17 11 2020 © The Author(s) 2020Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.The thermodynamic surface properties and Lewis acid–base constants of H-β-zeolite supported rhodium catalysts were determined by using the inverse gas chromatography technique at infinite dilution. The effect of the temperature and the rhodium percentage supported by zeolite on the acid base properties in Lewis terms of the various catalysts were studied. The dispersive component of the surface energy of Rh/H-β-zeolite was calculated by using both the Dorris and Gray method and the straight-line method. We highlighted the role of the surface areas of n-alkanes on the determination of the surface energy of catalysts. To this aim various molecular models of n-alkanes were tested, namely Kiselev, cylindrical, Van der Waals, Redlich–Kwong, geometric and spherical models. An important deviation in the values of the dispersive component of the surface energy γsd determined by the classical and new methods was emphasized. A non-linear dependency of γsd with the specific surface area of catalysts was highlighted showing a local maximum at 1%Rh. The study of RTlnVn and the specific free energy ∆Gsp(T) of n-alkanes and polar solvents adsorbed on the various catalysts revealed the important change in the acid properties of catalysts with both the temperature and the rhodium percentage. The results proved strong amphoteric behavior of all catalysts of the rhodium supported by H-β-zeolite that actively react with the amphoteric solvents (methanol, acetone, tri-CE and tetra-CE), acid (chloroform) and base (ether) molecules. It was shown that the Guttmann method generally used to determine the acid base constants KA and KD revealed some irregularities with a linear regression coefficient not very satisfactory. The accurate determination of the acid–base constants KA, KD and K of the various catalysts was obtained by applying Hamieh’s model (linear regression coefficients approaching r2 ≈ 1.000). It was proved that all acid base constants determined by this model strongly depends on the rhodium percentage and the specific surface area of the catalysts. Subject terms ChemistryEngineeringMaterials scienceissue-copyright-statement© The Author(s) 2020 ==== Body Introduction The determination of the surface and interface properties of solid materials is of vital importance in many industrial domains including catalysis, biomedicine, chemical engineering, adsorption, adhesion, membrane fabrication, polymers and composites, clays1,2, nanomaterials and clay-polymer composites3,4, pharmaceutical and food products5–8. One of the most popular and interesting technique to determine the surface properties of solid materials is the inverse gas chromatography (IGC) at infinite dilution. The IGC technique can advantageously give access to the acid base properties in Lewis terms as well as to the thermodynamic parameters such as specific free energy, enthalpy and entropy of adsorption. In addition, Lewis acid–base character of the surface, surface nanoroughness parameter, can be also determined5–12. The IGC technique appears a real source of physiochemical data of surfaces and interfaces13 allowing the observation of the interactions between oxides, polymers or polymers adsorbed on oxides and organic solvent systems14. This is an important tool, precise, sensitive, and more competitive to determine the heterogeneous surfaces of textiles, their physicochemical properties15, and to determine surface energy and surface area of powdered materials16,17. In previous studies18,19, we used IGC technique to determine the surface characteristics of various oxides and polymers or polymers adsorbed on oxides, especially, their surface energies, their interactions with some organic molecules and the acid–base properties of solid materials or nanomaterials. The IGC technique was preferentially applied to characterize the surface properties of catalysts or metals containing catalysts that can be advantageously used in industrial applications20–22. It is well known that rhodium is used in automobile industries during the manufacturing of automobile catalytic converts20. It plays an important role in the oxidation of ammonia and carbon monoxide and also in the elimination of nitric oxide21,22. On the other hand, beta zeolite was proved to be an excellent catalyst due to the relatively high density of Brønsted acid sites and favorable pore structure23–25. Zeolite can be considered as an interesting support for metal catalysts. Moloy et al.26 studied the adsorption properties of zeolite and metal loaded zeolite. However, they did not provide details on the specific surface properties, the acid base constants in Lewis terms and the surface energy of H-β-zeolite supported rhodium catalysts. In this paper, a new approach for the determination of the surface and interfacial properties of H-β-zeolite and the rhodium impregnated in H-β-zeolite catalysts is developed. We used the inverse gas chromatography technique at infinite dilution, Papirer’s approach27–29 and Hamieh’s model18,19 to determine the specific free enthalpy and enthalpy of adsorption and the acid–base constants of the above materials. The dispersive component of the surface energy of such catalysts was also studied by using the various molecular models of n-alkanes. Theory and methods Inverse gas chromatography can be considered as powerful technique used to determine the superficial phenomena, the surface energy, the specific free energy enthalpy and entropy of adsorption and the acid–base properties of solid materials. IGC technique was applied in this study to determine the changes of the superficial properties of H-β-zeolite/rhodium catalysts as a function of the temperature. Probes of known properties were injected into the column containing the solid. The retention times of these probes, measured at infinite dilution, allowed us to determine the interactions between model organic molecules and the solid assuming that there was no interaction between the probe molecules. In parallel, the surface specific area of the various catalyst samples was determined by using Brunauer–Emmett–Teller (BET). The nitrogen adsorption–desorption experiments were carried out using BET gas adsorption method at 77 K, in an automatic Micromeritics ASAP 2420 apparatus. The samples were degassed under vacuum for 2 h at 100 °C followed by 300 °C for 10 h before the measurements. The specific surface SBET area was determined by using the classical BET method. The mesopore size distribution of the catalysts were calculated using the model of Barrett–Joyner–Halenda (BJH). Retention volume The net retention volume Vn was calculated from: 1 Vn=jDctR-t0 where tR is the retention time of the probe, t0 the zero retention reference time measured with a non adsorbing probe such as methane, Dc the corrected flow rate and j a correction factor taking into account the compression of the gas30. Dc and j are respectively given by the following expressions: 2 Dc=DmTcTaηTcηTa and 3 j=32ΔP+P0P02-1ΔP+P0P03-1 where Dm is the measured flow rate, Tc the column temperature, Ta the room temperature, η(T) the gas viscosity at temperature T, P0 the atmospheric pressure and ∆P the pressure variation. Determination of the dispersive component of the surface energy of a solid The free enthalpy of adsorption ∆G0 of n-alkanes on a solid is given by: 4 ΔG0=-RTlnVn+C where R is the ideal gas constant, T the absolute temperature and C a constant depending on the reference state of adsorption. In the case of n-alkanes, ∆G0 is equal to the free energy of adsorption corresponding to dispersive interactions ∆Gd only. The increment method Dorris and Gray31 proposed the increment method by applying the well-known relationship of Fowkes32 which gives at the same time the dispersive component of the surface energy of solids γsd by using the geometric mean of the dispersive components (exponent d) of the surface energy of the probe γld and the solid γsd: 5 Wa=2γldγsd where Wa is the work of adhesion between the probe and the solid. This energy of adhesion can be correlated to the free enthalpy of adsorption following 6 ΔG0=N,aWa=2Naγldγsd where N is Avogadro’s number and a the surface area of o adsorbed molecule on the solid. Dorris and Gray were the first who determined the dispersive component of the surface energy of a solid by considering the increment of ΔG-CH2-0 per methylene group in the n-alkanes series of general formula CnH2(n+1). They defined the increment ΔG-CH2-0 by: 7 ΔG-CH2-0=ΔG0Cn+1H2(n+2)-ΔG0CnH2(n+1) where CnH2(n+1) and CnH2(n+1) represent the general formula of two consecutive n-alkanes. By using the retention volumes VnCnH2(n+1) and VnCn+1H2(n+2) of two consecutive n-alkanes and the relation (4), the dispersive component of the surface energy γsd can be determined by the following equation: 8 γsd=RTlnVnCn+1H2(n+2)VnCnH2(n+1)24N2a-CH2-2γ-CH2- where a-CH2- is the surface area of methylene group (a-CH2- = 6 Å2) and γ-CH2- the surface energy of –CH2– group of a polyethylene polymer (with a finite molecular mass). The latter is given by: 9 γ-CH2-=52.603-0.058T(T in K;γ-CH2-in mJ/m2) By applying Dorris and Gray’s method, we determined the dispersive cponent of the surface energy γsd of H-β-zeolite for various temperatures. We only gave here the value determined at 480 K which was equal to γsd = 240.3 mJ/m2. The variation of γsd(T) of H-β-zeolite as a function of the temperature is given by the following straight-line equation: 10 γsd(T)=327-0.183 T.66 Note that the temperature T is in K while γsd is expressed in mJ/m2. The correlation coefficient was R2 = 0.9994. The n-alkane straight-line method This method, also based on Fowkes approach32, replaced the free enthalpy of adsorption by its value taken from relation (4). It leads to the following relationship: 11 RTlnVn+C=2Naγldγsd By plotting RTlnVn as a function of 2Naγld of n-alkanes, one obtains a typical straight line that allows to deduce, from its slope, the value of dispersive component γsd of the surface energy of the solid. The evolution of RTlnVn as a function of 2Naγld of n-alkanes adsorbed on H-β-zeolite is reproduced in the Fig. 1. The experimental relation can be extracted:Figure 1 Variations of the retention volume of n-alkanes (from n-pentane C5 to n-decane C10) versus 2Naγld of probes of H-β-zeolite. 12 RTlnVn=12.0982Naγld+9.962;r2=0.9992 The slope of the straight line is γsd=12.098 and then γsd = 146.36 mJ/m2. The same method was applied, at different temperatures, in order to obtain the values γsd(T) of H-β-zeolite at different temperatures. The results are displayed in the Fig. 2.Figure 2 Variations of the dispersive component γsd(T) of the surface energy of H-β-zeolite versus the temperature T. A non-linear variation of γsd(T) with the temperature can be noticed. This is certainly due to the presence of hydroxide layer on the zeolite surface which is likely to evolve with the heat treatment. In addition, a large deviation between the results obtained by this method compared to those of Dorris and Gray’s method is observed. This is because of the high temperatures reaching 560 K that can extremely affect the values of the surface tensions of n-alkanes depending on the temperature interval; whereas, the surface tension of ethylene group is given by the relation γ-CH2-= 52.603–0.058 T for all temperatures. Critics of the classic methods18 It is obvious, in the two previous methods based on Fowkes relation, that the determination of the dispersive surface energy component γsd of a solid, necessitates the precise knowledge of the surface areas, a, of n-alkanes adsorbed on the solid substrate. However, the surface area of a molecule adsorbed on a solid is not known with a good accuracy due to the large dependency on the temperature change. In a previous study, Hamieh and Schultz18 criticized the classical way and proposed to use various models giving the molecular areas of n-alkanes. The geometrical model, cylindrical molecular model, liquid density model, BET method, Kiselev results and the model using the two-dimensional Van der Waals (VDW) constant b that depends on the critical temperature and pressure of the liquid were considered. Redlich–Kwong (R–K) equation transposed from three-dimensional space to two-dimensional space was also used to calculate the areas of organic molecules. The value of γsd depends significantly on the chosen molecular models of the surface area of n-alkanes and on the temperature. The different molecular models for the different n-alkanes are listed in Table 1.Table 1 Surface areas of various molecules (in Å2) obtained from the various models of Van der Waals (VDW), Redlich–Kwong (R–K) and Kiselev models. Molecule VDW Kiselev Cylindrical R-K Spherical Geometrical C5H12 47.0 45 39.3 36.8 36.4 32.9 C6H14 52.7 51.5 45.5 41.3 39.6 40.7 C7H16 59.2 57 51.8 46.4 42.7 48.5 C8H18 64.9 63 58.1 50.8 45.7 56.2 C9H20 69.6 69 64.4 54.5 48.7 64.0 C10H22 74.4 75 70.7 58.2 51.7 71.8 The surface areas are also compared to those obtained by geometrical, cylindrical or spherical models. It appears relevant to strengthen our analysis and to show the effect of the method used and the molecular models chosen on γsd values. The variations of γsd as a function of the temperature for the various molecular models of n-alkane surface areas are displayed, respectively, in the case of the increment method (Fig. 3) and the straight methods (Fig. 4).Figure 3 Evolution of γsd(T) versus the temperature for the various molecular models: Van der Waals, Redlich–Kwong, Kiselev, geometrical, cylindrical and spherical models, by using the increment method for H-β-zeolite. Figure 4 Evolution of γsd(T) versus the temperature for the various molecular models: Van der Waals, Redlich–Kwong, Kiselev, geometrical, cylindrical and spherical models, by using the straight-line method for H-β-zeolite. Figures 3 and 4 clearly highlighted the extreme variation of γsd(T) depending on the chosen method and molecular model of the surface area of n-alkanes. The difference between the values of γsd reached about 100% from Kiselev, van der Waals or cylindrical models to Redlich–Kwong or spherical models for all used temperatures regardless of the method (increment method and straight-line method). However, for any method and molecular model used, some physico-chemical behavior can be deduced when comparing the surface energy of two solid materials. This can be useful to understand the change of catalyst properties when the temperature varies. Determination of the specific interactions The free energy of adsorption ∆G0 of a probe on a solid generally contains the two contributions relative to the dispersive and specific interactions. In the case of n-alkanes, ∆G0 is equal to the free energy of adsorption corresponding to the dispersive interactions ∆Gd only. When polar molecules are injected into the column, specific interactions are established between these probes and the solid surface and ∆G0 is now given by: 13 ΔG0=ΔGd+ΔGsp. where ∆Gsp refers to specific interactions of a polar molecule adsorbed on solid substrate. To calculate the specific interactions between the solid substrates and polar probes, several methods were used in the literature5–8,18,19,27–29. To avoid the use of the method based on the surface area of n-alkanes that cannot be known precisely as a function of the temperature, the method developed by Papier et al.29 is preferred. It allows to quantify more precisely the specific interactions. Saint Flour and Papirer's method Papirer method is employed to quantify the specific free energy of adsorption of polar molecules and obtain the acid–base constants of the different hydrocarbon materials. This method gives access to the specific enthalpy of interaction between a probe and a solid27–29 from the obtained straight line when plotting RTlnVn against the logarithm of the vapor pressure of the probes, i.e. lnP0. For a homologous series of n-alkanes, whatever the nature of the solid substrates: 14 RTlnVnn-alkane=AlnP0n-alkane+B where A and B are constants which depend on the nature of the solid substrate. Following Saint Flour and Papirer's approach26,27, RTlnVn values of the various solutes are first plotted versus the logarithm of their vapor pressure at saturation, Po. The points representative of n-alkanes define the so-called “n-alkane straight line” (see Fig. 5), and the distance between this line and the points corresponding to RTlnVn (polar molecule) value of polar probes are then taken as a measure of the specific interactions and it is defined as the specific free enthalpy of adsorption, ∆Gs, of polar molecule on the solid. It is given, for any temperature T, by the following equation: 15 ΔGsppolarmolecule=RTlnVnpolarmolecule-AlnP0polarmolecule-B Figure 5 Variations of RTlnVn of n-alkanes and polar probes as a function of lnP0 in the case of H-β-zeolite at 480 K. As example, the variations of RTlnVn of different n-alkanes and polar probes as a function of lnP0 in the case of H-β-zeolite at 480 K are reproduced in the Fig. 5. The equation of n-alkanes straight line is given with an excellent linearity: 16 RTlnVn=-21.812lnP0+111.4;R=0.9993 From this equation, the specific free enthalpy of adsorption of polar molecules can be deduced. For example, for trichloroethylene ∆Gsp is equal to 7.960 kJ/mol. In the following, the specific free enthalpy of adsorption of polar probes ∆Gs(T) can be determined by varying the temperature. The corresponding values of (∆Hsp) and entropy ∆Ssp of adsorption of polar molecules are obtained. Determination of acid–base constants of solid substrates By plotting ∆Gsp(T) of the polar molecules as a function of the temperature, the specific enthalpy. (∆Hsp) and entropy ∆Ssp of adsorption are calculated from: 17 ΔGsT=ΔHsp-TΔSsp The evolution of ∆Gsp(T) of the polar molecules as a function of the temperature in the case of H-β-zeolite is plotted in Fig. 6. In general, this relationship (17) is linear if ∆Hsp and ∆Ssp do not depend on the temperature. However, when the linear correlation coefficient is too small in front of 1, then the linearity is not verified; therefore, ∆Hsp(T) and ∆Ssp(T) strongly depend on the temperature. The curves representing the variations of ∆Gsp(T) versus the temperature give access to the thermodynamic calculations of specific enthalpy and entropy as a function of the temperature by using the classical thermodynamic equations. The specific enthalpy and entropy of adsorption determined from the linear relation between ∆Gsp and T are summarized in the Table 2.Figure 6 Curves of -∆Gsp(T) of polar molecules as a function of the temperature in the case of H-β-zeolite for a range temperature [480 K, 560 K]. Table 2 Values of the specific enthalpy − ∆Hsp and ∆Ssp entropy of adsorption of polar molecules adsorbed on H-β-zeolite substrate. Polar probes Specific enthalpy − ∆Hsp (kJ/mol) Specific entropy ∆Ssp (J/mol) Cyclohexane 9.713 − 10 Trichloroethylene 16.960 − 19 Tetrachloroethylene 32.242 − 30 Benzene 1.457 − 2 Chloroform 22.592 − 30 Ether 54.982 − 90 Methanol 71.74 − 127 Acetone 68.321 − 112 The Guttmann method Gutmann33 classified the polar molecules by assigning an electron donor (DN) and a number of electron acceptor (AN) which realize, respectively, the acidity and the basicity of the molecule. In analogy to the Guttmann approach, Papirer et al.27–29 proposed to characterize the solid by two parameters. The parameters KA and KD reflect the basic and the acidic character of the solid, respectively. These two constants measure the ability of the solid to develop, respectively, the acid and base interactions with basic, acidic or amphoteric probes. They are connected to the specific enthalpy ΔHaSP through the following equation: 18 (-DHsp)=KA.DN+KD.AN where KA and KD represent the acidic and the basic character of the solid, respectively, while AN and DN represent the donor number and the electron acceptor of the probe according to the scale of Gutmann33. Equation 11 can be rewritten as: 19 -ΔHSpAN=DNANKA+KD The representation of -ΔHSpAN as a function of DNAN gives, in general, a straight line of slope KA and intercept KD. The new model For several solid substrates, the Guttmann method cannot be applied because the linearity of Eq. 19 is not satisfied. This classical relationship was corrected. Then, a new equation was proposed18,19. A third parameter K was added. It reflected the amphoteric character of the oxide or polymer. The final expression becomes: 20 (-ΔHsp)=KADN+KDAN-KDNAN By dividing by AN, one can obtain: 21 -ΔHspAN=KADNAN+KD-K.DN The Eq. (21) can be symbolically written as: 22 X1=KD+KAX2-KX3 where X1=-ΔHspAN,X2=DNAN, X3=DN and K = K(KA,KD). Note that X1, X2 and X3 are known for every polar molecule, whereas KD, KA and K are unknown. By using N probes, relationship (22) leads to a set of linear system of three equations with three unknown variables: KD, KA and K. The matrix representing this linear application is a symmetrical one. It appears that Eq. (22) possesses a unique solution for N ≥ 3. This method can be applied to calculate the acid–base constants of solids if the Gutmann relation falls. Experimental results on rhodium supported by H-β-zeolite Materials and solvents The different catalysts analyzed in this study containing rhodium supported by H-β-zeolite were obtained by following the method developed by Navio et al.24 and Zhang et al.25 to have different percentages of rhodium. Classical organic probes, characterized by their donor and acceptor numbers, were used. Corrected acceptor number AN′ = AN–ANd were utilized. They were given by Riddle and Fowkes34. The idea was to subtract the contribution of Van der Waals interactions (or dispersion forces). The corrected acceptor number was then normalized by a dimensionless donor number DN′ according to the following relationship18,19: 23 DN′=2.5mol/kcalDNkcal/mol However, if one wants to use DN in kcal/mol, AN′ can be easily transformed to the kcal/mol unit using the following relationship: 24 AN′(kcal/mol)=40kcal/mol100AN′(unitless) The solvents used as probes for IGC measurements were selected based on their ability to interact with three different types of interaction forces, namely dispersive, polar, and hydrogen bonding. All probes were obtained from Aldrich. They were highly pure grade (i.e., 99%). The probes used were n-alkanes (pentane, hexane, heptane, octane, and nonane), amphoteric solvents (methanol, acetone, trichloroethylene (Tri-CE), tetrachloroethylene (Tetra-CE)), strong basic solvent (diethyl ether), very weak basic solvent (benzene), very acidic solvent (chloroform), and very weak acid (cyclohexane). The Table 3 gives the donor and acceptor numbers18,33 of polar probes used in this study.Table 3 Normalized donor and acceptor numbers of some polar molecules. Polar probe DN' AN' DN'/AN' Character Chloroform 18.600 0.000 0.000 Higher acidity Ether 4.900 48.000 9.796 Higher basicity Methanol 41.700 47.500 1.139 High amphoteric Acetone 8.700 42.500 4.885 High amphoteric Cyclohexane 0.141 3.520 24.965 Weaker acidity Trichloroethylene 3.313 2.500 0.755 Weak amphoteric Tetrachloroethylene 3.020 3.100 1.026 Weak amphoteric Benzene 0.600 0.250 0.417 Weak acidity and basicity GC conditions The IGC measurements were performed on a commercial Focus GC gas chromatograph equipped with a flame ionization detector. Dried nitrogen was the carrier gas. The gas flow rate was set at 20 mL/min. The injector and detector temperatures were maintained at 400 K during the experiments30. To achieve infinite dilution, 0.1 μL of each probe vapor was injected with 1 μL Hamilton syringes, in order to approach linear condition gas chromatography. All four columns used in this study were prepared using a stainless-steel column with a 2 mm inner diameter and with an approximate length of 20 cm. The column was packed with 1 g of solids in powder forms. In general, the surface properties of materials are studied by IGC at low temperatures. However, in certain case for lower temperatures, the retention times of organic molecules are very long due to the difficulties for the probes to find its path through the catalyst particles. For this reason, the experiments were conducted at higher temperatures in order to quantify the surface properties of catalysts by IGC at infinite dilution and deduce the acid base and dispersive surface energy of solid substrates. The column temperatures were 480–560 K, varied in 20 °C steps. Each probe injection was repeated three times, and the average retention time, tR, was used for the calculation. The standard deviation was less than 1% in all measurements. Results and discussion Variations of the net retention volume Experimental results obtained by IGC at infinite dilution with different percentages of rhodium (from 0 to 2%) supported by H-β-zeolite at various temperatures (from 480 to 560 K), are presented in Tables 4, 5, 6, 7, 8.Table 4 Values of RTlnVn (in kJ/mol) of n-alkanes and polar probes for different percentages of rhodium supported by H-β-zeolite substrate (%Rh = 0; 0.25; 0.50; 0.75; 1.00; 1.25; 1.50; 1.75; 2.00) at T = 480 K. %Rh/H-β-Z probes 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 C5 15.117 16.78 17.841 18.1 17.879 17.1 16.41 16.2 16.172 C6 20.481 22 23.35 24 23.342 22.2 21.7 21.6 21.523 C7 25.909 27.62 28.836 29.453 28.62 27.356 26.8 26.65 26.624 C8 31.294 32.854 34.337 35.116 34.021 32.511 32.012 31.912 31.892 C9 36.69 38.523 39.835 40.21 39.392 38.111 37.355 37.154 37.118 Cyclohexane 18.243 19.421 20.528 21.335 20.831 19.731 18.995 18.864 18.834 Tri-CE 16.082 18.025 19.560 22.613 24.406 21.311 18.461 18.370 18.331 Tetra-CE 18.046 20.147 24.566 31.244 33.136 29.277 20.608 19.997 19.947 Benzene 22.054 23.421 24.618 25.447 25.207 22.305 22.301 22.454 22.536 Chloroform 11.257 10.684 10.367 11.500 10.548 9.766 9.336 9.217 9.166 Ether 1.35 5.750 6.550 7.860 8.750 8.120 7.810 7.230 7.150 Methanol 2.000 7.240 8.450 9.850 10.560 9.870 8.450 8.012 7.980 Acetone 1.876 9.230 9.780 10.230 10.840 10.450 9.380 9.045 8.986 Table 5 Values of RTlnVn (in kJ/mol) of n-alkanes and polar probes for different percentages of rhodium supported by H-β-zeolite substrate (%Rh = 0; 0.25; 0.50; 0.75; 1.00; 1.25; 1.50; 1.75; 2.00) at T = 500 K. %Rh/H-β-Z probes 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 C5 13.701 15.275 16.619 17.254 16.541 15.740 15.102 14.900 14.814 C6 18.879 20.500 21.880 22.500 21.840 20.740 20.310 20.210 20.137 C7 24.145 25.777 27.192 28.078 26.956 25.612 25.178 25.100 25.006 C8 29.352 31.021 32.470 33.153 32.194 30.979 30.300 30.211 30.178 C9 34.574 36.200 37.757 38.362 37.402 36.000 35.530 35.370 35.274 Cyclohexane 4.169 16.821 18.000 19.070 19.688 19.327 18.711 18.182 17.930 Tri-CE 4.149 14.704 16.700 18.406 21.828 23.249 20.309 17.477 17.150 Tetra-CE 3.612 16.688 18.632 23.203 30.123 31.907 28.241 19.348 18.800 Benzene 4.200 20.419 21.761 23.104 24.123 23.645 20.740 20.830 21.024 Chloroform 4.343 10.293 9.945 9.755 10.612 10.291 9.522 8.780 8.601 Ether 4.603 1.570 6.250 6.780 7.230 7.789 7.641 7.230 7.100 Methanol 4.664 2.150 8.260 8.887 10.220 10.856 9.680 9.350 9.159 Acetone 4.480 2.150 8.740 9.210 10.230 10.840 10.450 10.184 9.879 Table 6 Values of RTlnVn (in kJ/mol) of n-alkanes and polar probes for different percentages of rhodium supported by H-β-zeolite substrate (%Rh = 0; 0.25; 0.50; 0.75; 1.00; 1.25; 1.50; 1.75; 2.00) at T = 520 K. %Rh/H-β-Z probes 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 C5 12.285 14.022 15.397 15.897 15.203 14.300 13.598 13.511 13.456 C6 17.277 19.000 20.410 21.023 20.338 19.526 19.000 18.878 18.751 C7 22.381 23.989 25.548 26.101 25.292 24.200 23.587 23.460 23.388 C8 27.410 29.230 30.603 31.489 30.367 29.454 28.845 28.566 28.464 C9 32.458 34.213 35.679 36.340 35.411 34.360 33.689 33.520 33.430 Cyclohexane 15.399 16.630 17.612 18.154 17.823 17.420 17.100 16.922 16.914 Tri-CE 13.326 15.378 17.252 20.797 22.094 19.469 16.120 15.900 15.819 Tetra-CE 15.330 17.200 21.843 28.994 30.697 27.588 18.237 17.612 17.463 Benzene 18.788 20.229 21.597 22.537 22.090 19.334 19.352 19.566 19.622 Chloroform 9.328 9.130 9.144 9.500 10.034 9.000 8.142 8.000 7.910 Ether 1.850 2.650 3.221 3.678 4.623 4.232 4.012 3.877 3.798 Methanol 2.350 3.456 4.412 4.941 5.714 5.245 5.014 4.886 7.778 Acetone 2.480 4.124 4.876 5.664 6.356 5.884 5.665 5.514 5.412 Table 7 Values of RTlnVn (in kJ/mol) of n-alkanes and polar probes for different percentages of rhodium supported by H-β-zeolite substrate (%Rh = 0; 0.25; 0.50; 0.75; 1.00; 1.25; 1.50; 1.75; 2.00) at T = 540 K. %Rh/H-β-Z probes 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 C5 10.869 12.700 14.175 15.211 13.865 12.890 12.350 12.200 12.098 C6 15.675 17.456 18.940 19.356 18.836 18.211 17.660 17.456 17.365 C7 20.617 22.411 23.904 24.789 23.628 22.678 22.100 21.900 21.770 C8 25.468 27.190 28.735 29.455 28.539 27.500 26.997 26.860 26.750 C9 30.342 32.233 33.600 34.300 33.421 32.300 31.800 31.677 31.586 Cyclohexane 13.977 15.200 16.154 16.520 16.319 16.080 16.050 16.000 15.954 Tri-CE 11.948 14.120 16.098 19.993 20.930 18.461 14.950 14.697 14.563 Tetra-CE 13.972 15.900 20.495 27.727 29.490 26.493 16.934 16.333 16.221 Benzene 17.154 18.665 20.081 21.212 20.530 17.737 17.923 18.120 18.166 Chloroform 8.364 8.400 8.532 9.078 9.778 8.700 7.543 7.355 7.282 Ether 2.115 2.733 3.420 3.876 3.456 3.117 2.977 2.778 2.612 Methanol 2.450 3.012 3.656 4.031 3.678 3.312 3.185 3.033 2.897 Acetone 2.851 3.654 4.221 4.785 4.335 3.915 3.687 3.421 3.334 Table 8 Values of RTlnVn (in kJ/mol) of n-alkanes and polar probes for different percentages of rhodium supported by H-β-zeolite substrate (%Rh = 0; 0.25; 0.50; 0.75; 1.00; 1.25; 1.50; 1.75; 2.00) at T = 560 K. %Rh/H-β-Z probes 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 C5 9.453 11.500 12.953 13.220 12.527 11.500 10.950 10.800 10.740 C6 14.073 15.870 17.470 18.000 17.334 16.600 16.166 16.070 15.979 C7 18.853 20.700 22.260 22.700 21.964 21.100 20.396 20.266 20.152 C8 23.526 25.417 26.868 27.700 26.712 25.800 25.290 25.159 25.036 C9 28.226 30.000 31.522 32.344 31.431 30.500 29.961 29.822 29.742 Cyclohexane 12.555 13.770 14.696 15.200 14.815 14.850 14.930 14.962 14.994 Tri-CE 10.570 12.900 14.944 18.779 19.770 17.450 13.700 13.412 13.307 Tetra-CE 12.614 14.581 19.149 26.725 28.295 25.647 15.700 15.025 14.979 Benzene 15.513 17.128 18.568 19.425 18.975 16.238 16.378 16.636 16.707 Chloroform 7.399 7.566 7.920 8.580 9.521 8.350 6.970 6.712 6.654 Ether 2.452 3.687 4.256 4.904 5.371 5.046 4.786 4.660 4.589 Methanol 2.750 3.425 3.879 4.623 4.970 4.572 4.364 4.271 4.182 Acetone 3.851 4.456 5.012 5.706 6.041 5.731 5.493 5.322 5.278 The Tables indicate substantial variations of RTlnVn between the probes adsorbed on the solid substrates. Consequently, significant variations of the surface free enthalpy of adsorption are expected. This aspect is emphasized in the Fig. 7.Figure 7 Variations of RTlnVn of n-alkanes and polar probes adsorbed on the solid substrates as a function of the percentage of rhodium supported by H-β-zeolite (%Rh). It is interesting to note a particular point represented by a maximum of RTlnVn. In the case of n-alkanes adsorbed on the catalysts it takes place for a percentage of rhodium %Rh/H-β-Z = 0.75% (Fig. 7). However, this maximum of the surface free enthalpy shifts to a percentage %Rh/H-β-Z of 1.00% in the case of polar solvents. This shift maybe attributed to the strong specific interaction of the polar molecules with rhodium. The evolution of RTlnVn as a function of the temperature for n-alkanes and polar molecules is given in the Fig. 8. In the case of H-β-zeolite substrate, a linear dependency for all alkane solvents is observed (Fig. 8a). Conversely, for all the polar molecules, a non-linear behavior occurs with a minimum for T = 500 K where the surface groups of the solid substrate are strongly affected by the temperature change. The same behavior takes place with all the polar molecules on H-β-zeolite at 500 K. At this temperature, they have identical resident or retention time due to a minimum polarity of the catalyst at this temperature leading to weak polar interactions between the probes and the H-β-zeolite.Figure 8 Variations of RTlnVn of n-alkanes and polar probes adsorbed on the solid substrates as a function of the temperature T (K) in the case of (a) H-β-zeolite and (b) rhodium supported by H-β-zeolite (2%Rh). The dispersive interactions can be considered similar for all polar molecules in this case. This gives similar values of RTlnVn at T = 500 K. It seems that, at this temperature, some surface groups of H-β-zeolite are inaccessible for polar probes that cross more quickly the chromatographic column. In addition, when the temperature increases, the acid base surface groups of the solid increases. Consequently, the values of RTlnVn also increase for all the polar molecules. In the presence of rhodium incorporated into H-β-zeolite catalyst, the minimum of RTlnVn with the temperature disappears for polar molecules (Fig. 8b). In addition, a global linear tendency is observed for polar and non-polar molecules. The presence of the rhodium particles on the surface of H-β-zeolite catalyst affects strongly the specific interactions between the polar molecules and the catalyst while whereas the dispersive interactions remain stable and constant. However, in order to better quantify the specific free enthalpy of interaction between the catalyst and the polar molecules, the classical thermodynamic equations are used in the “Determination of the specific interactions and acid–base properties”. The obtained specific free enthalpy of adsorption gives a real idea of the nature of acid base interactions at any temperature. Determination of the dispersive component of the surface energy of catalysts In this section, the dispersive component of the surface energy of the rhodium supported by H-β-zeolite are determined at different percentages of Rh and for various temperatures. The various surface areas of n-alkanes given by the different molecular models are also used knowing the values of the dispersive component of the surface energy γld(T) of n-alkanes as a function of the temperature. The calculations of γsd(T) are performed using the increment method of Dorris and Gray and the method of the straight-line for all solid substrates. They are calculated at different temperatures and molecular models. The results obtained by using Dorris and Gray method are listed on Table 9 while those calculated thanks to the straight-line method are given in Table 10. They are estimated for different temperatures, rhodium percentages and molecular models.Table 9 Values of γsd of different catalysts as a function of temperature, rhodium percentage and molecular model using the increment method. %Rh T (K) Van der Waals Kiselev Cylindrical Redlih–Kwong Spherical Geometric 0% (H-β-zeolite) 480 270 240 205 442 884 135 500 265 236 201 434 870 133 520 261 233 198 427 856 130 540 257 229 195 420 842 128 560 253 226 192 414 829 126 0.25% 480 283 250 213 462 919 140 500 265 237 202 433 870 133 520 262 233 199 429 861 131 540 263 233 198 430 855 130 560 250 225 192 409 826 126 0.50% 480 280 248 212 457 915 139 500 272 242 206 445 890 136 520 264 235 201 432 865 132 540 256 229 195 419 841 128 560 249 223 189 407 817 125 0.75% 480 266 241 205 436 886 135 500 269 243 206 439 888 136 520 263 237 202 431 873 133 540 263 241 203 429 875 134 560 259 231 198 425 854 130 1.00% 480 266 235 201 435 868 132 500 262 232 198 429 856 130 520 259 229 195 423 844 129 540 255 226 193 418 834 127 560 252 223 191 413 824 126 1.25% 480 266 231 198 435 854 130 500 252 223 191 413 825 126 520 252 221 190 412 821 125 540 241 210 180 394 780 119 560 247 217 185 404 802 122 1.50% 480 255 223 191 417 826 126 500 254 221 190 415 820 125 520 247 217 186 405 806 123 540 243 212 182 398 786 120 560 245 213 183 402 794 121 1.75% 480 251 221 189 410 815 124 500 251 220 188 410 813 124 520 247 215 185 404 800 122 540 246 214 184 402 796 121 560 244 211 182 400 789 120 2.00% 480 251 222 190 411 819 125 500 251 220 188 410 813 124 520 248 216 186 406 803 122 540 246 214 184 403 796 121 560 246 212 183 402 791 120 Table 10 Values of γsd of different catalysts as a function of temperature, rhodium percentage and molecular model using the straight-line method. %Rh T (K) Van der Waals Kiselev Cylindrical Redlih–Kwong Spherical Geometric 0% 480 134 145 156 237 329 141 500 96 98 110 156 202 108 520 113 110 122 184 241 114 540 49 50 58 80 99 59 560 119 116 130 195 247 125 0.25% 480 135 150 161 244 338 145 500 96 98 111 157 203 108 520 116 113 125 190 248 117 540 50 51 59 81 101 60 560 114 111 124 185 235 142 0.50% 480 139 150 162 245 341 146 500 98 100 113 160 207 111 520 114 111 123 186 244 115 540 49 50 58 79 99 59 560 117 114 128 191 242 123 0.75% 480 142 124 158 240 297 142 500 97 100 112 159 206 110 520 117 114 126 190 249 118 540 47 48 55 79 95 57 560 117 114 127 190 241 123 1.00% 480 131 143 154 233 323 139 500 94 97 109 154 199 106 520 114 111 123 186 243 115 540 50 51 59 79 101 60 560 120 117 131 196 249 126 1.25% 480 122 139 150 227 315 135 500 91 92 106 149 193 103 520 115 112 124 187 245 116 540 48 49 57 78 97 58 560 119 116 130 195 247 125 1.50% 480 122 136 146 221 307 132 500 90 92 104 147 190 102 520 113 111 122 185 243 115 540 49 50 58 79 99 59 560 118 115 129 192 244 124 1.75% 480 122 134 144 219 304 130 500 89 91 103 146 188 101 520 112 110 121 183 240 114 540 50 51 59 81 101 60 560 118 115 128 191 243 123 2.00% 480 135 135 145 220 306 131 500 89 91 103 146 188 101 520 112 109 121 183 240 113 540 50 51 59 81 102 61 560 120 117 131 195 248 126 Significant difference between the values of γsd obtained by the two applied methods can be noticed. This large difference in the values of γsd is due to the high temperatures neighboring 500 K. The surface tension of n-alkanes at such temperatures does not give an identical surface tension of the methylene group than that given by the classical relation (γ-CH2-= 52.603–0.058 T). The results also prove that γsd strongly depends on the molecular model chosen to estimate the surface areas of n-alkanes. Equation (11) can be also written as: γsd=RTlnVn+C24N2γlda2 This equation clearly shows an important variation of γsd of a solid substrate as a function of the surface area a of molecules. Table 1 gave the different molecular models for the different n-alkanes and showed a larger variation of the surface area of molecules depending on the chosen molecular model. The standard deviation can rich in many cases more than 50% from a molecular model to another model. This leads to larger difference between the obtained values of γsd of a solid substrate at fixed temperature for the various molecular models. The value of γsd can vary from the simple to the double when passing from geometric model to the spherical model. This problem was solved by another study showing the variation of the surface areas of polar and n-alkane molecules as a function of the temperature35. The variations of γsd with the temperature in the case of H-β-Zeolite (for 0%Rh) were given previously in “Critics of the classic methods”. The aim here is to study the effect of the methods (increment or the straight-line methods) on the values of γsd with the temperature increases for the different molecular models in the case of 2%Rh catalyst. To this aim, the curves of γsd(T) are plotted in in Fig. 9 for the case of where the increment method is used while the same curves with the straight-line method are given in Fig. 10. The same behavior is obtained with rhodium catalyst (supported by zeolite) as that of H-β-zeolite (without rhodium): linearity for the “increment method” and non-linearity for the “straight-line method”.Figure 9 Evolution of γsd(T) versus the temperature for the various molecular models: Van der Waals, Redlich–Kwong, Kiselev, geometrical, cylindrical and spherical models, by using the increment method in the case of 2%Rh catalyst. Figure 10 Evolution of γsd(T) versus the temperature for the various molecular models: Van der Waals, Redlich–Kwong, Kiselev, geometrical, cylindrical and spherical models, by using the straight-line method in the case of 2%Rh catalyst. The rhodium percentage deposited on the H-β-zeolite has a high impact on the dispersive component of the surface energy of catalysts whatever the used temperature (Fig. 11) and molecular model (Fig. 12).Figure 11 Evolution of γsd(T) versus the temperature at various rhodium percentages by using the Dorris and Gray method and Kiselev molecular model. Figure 12 Evolution of γsd(T) versus the rhodium percentage at various temperatures by using the Dorris and Gray method and Kiselev molecular model. It seems also relevant to evaluate the Variations of γsd as a function of the specific surface area of the catalysts. Experimental results obtained by the BET method are presented in Table 11 and Fig. 13.Table 11 Values of the specific surface area SBET (m2/g) and microporous volume Vm (cm3/g) of the various catalyst samples. %Rh SBET (m2/g) Vm (cm3/g) 0 687 0.198 0.25 640 0.185 0.5 603 0.175 0.75 610 0.177 1 622 0.182 1.25 591 0.172 1.5 568 0.165 1.75 563 0.164 2 561 0.163 Figure 13 Variations of the specific surface area SBET (m2/g) of catalysts versus the rhodium percentage %Rh. A non-linear decrease of the specific area with the amount of Rh occurs until 0.5%Rh. It is followed by a slight increase to reach a local maximum at 1%Rh. Then, the specific area decreases up to a plateau of specific surface area observed for %Rh larger than 1.50%. The same conclusion can be drawn for the microporous volume. However, the highest value of the specific surface area is obtained for H-β-zeolite. It can be deduced from the figure that when the rhodium is added to zeolite, more metal particles would block the micropores causing a decrease in the specific surface area and in the catalyst microporosity. However, the increase of the specific surface area, for the catalysts containing a rhodium percentage comprised between 0.5 and 1.0, can result from the smaller particle sizes that cannot block the zeolite micropores. For catalysts with a rhodium percentage larger than 1.50% Rh, the lower surface area and pore volume are certainly due to the larger nanoparticles blocking the micropores and, then, decreasing the surface area and the pore volume. The curves of Fig. 14 giving the variations of—dγsddT as a function of the impregnated rhodium percentage give a similar behavior as that reported in the Fig. 13 for all the molecular models of n-alkane surface area.Figure 14 Variations of -dγsddT of catalysts versus the impregnated rhodium percentage (%Rh) for different molecular models of the surface area of n-alkanes. Determination of the specific interactions and acid–base properties Variations of the specific free enthalpy The experimental results obtained by IGC technique at infinite dilution previously presented in the Tables 4, 5, 6, 7, 8 lead to the determination of the specific free enthalpy ∆Gsp(T) of polar molecules adsorbed on H-β-zeolite and rhodium supported by zeolite for various temperatures and impregnated rhodium percentages. The results are summarized in Table 12.Table 12 Values of the specific free enthalpy—∆Gsp(T) (in kJ/mol) of different polar molecules adsorbed on catalysts as a function of the temperature and impregnated rhodium percentage. %Rh T (K) polar probes 480 500 520 540 560 0% Cyclohexane 4.951 4.760 4.568 4.368 4.156 Tri-CE 7.690 7.309 6.928 6.541 6.145 Tetra-CE 17.975 17.366 16.772 16.183 15.592 Benzene 0.497 0.457 0.417 0.377 0.337 Chloroform 7.993 7.385 6.780 6.173 5.558 Ether 12.012 10.264 8.468 6.694 4.850 Methanol 10.896 8.317 5.738 3.307 0.720 Acetone 14.520 12.460 10.357 8.222 5.463 0.25% Cyclohexane 5.416 5.202 5.063 4.941 4.836 Tri-CE 7.395 6.935 6.602 6.165 5.709 Tetra-CE 17.606 17.080 16.658 16.064 15.407 Benzene 0.769 0.734 0.700 0.662 0.625 Chloroform 10.183 9.342 8.695 7.928 7.322 Ether 11.327 9.679 7.985 6.313 4.574 Methanol 9.262 7.070 4.877 2.811 0.612 Acetone 11.616 9.968 8.286 6.578 4.370 0.50% Cyclohexane 5.564 5.536 5.509 5.468 5.423 Tri-CE 7.121 6.638 6.158 5.668 5.179 Tetra-CE 14.595 14.030 13.480 12.915 12.351 Benzene 0.819 0.789 0.759 0.729 0.699 Chloroform 11.707 10.900 10.098 9.289 8.482 Ether 10.682 9.127 7.530 5.953 4.313 Methanol 7.872 6.009 4.146 2.389 0.520 Acetone 9.293 7.975 6.629 5.262 3.496 0.75% Cyclohexane 5.645 5.605 5.560 5.520 5.480 Tri-CE 4.583 3.903 3.223 2.543 1.863 Tetra-CE 8.521 7.801 7.081 6.361 5.641 Benzene 0.496 0.456 0.416 0.376 0.336 Chloroform 11.540 10.888 10.145 9.330 8.550 Ether 10.073 8.607 7.101 5.614 4.067 Methanol 6.692 5.108 3.524 2.031 0.442 Acetone 7.434 6.380 5.303 4.210 2.797 1.00% Cyclohexane 5.135 5.118 5.104 5.079 5.056 Tri-CE 2.136 1.628 1.120 0.612 0.104 Tetra-CE 5.597 4.977 4.357 3.737 3.117 Benzene 0.119 0.097 0.075 0.053 0.031 Chloroform 11.494 10.262 9.036 7.806 6.580 Ether 9.499 8.116 6.696 5.294 3.835 Methanol 5.688 4.342 2.995 1.726 0.376 Acetone 5.948 5.104 4.242 3.368 2.238 1.25% Cyclohexane 4.811 4.685 4.522 4.406 4.311 Tri-CE 4.087 3.447 2.807 2.167 1.527 Tetra-CE 7.996 7.216 6.436 5.656 4.876 Benzene 1.909 1.910 1.902 1.945 1.870 Chloroform 11.248 10.020 9.166 8.025 6.852 Ether 8.957 7.654 6.315 4.992 3.617 Methanol 4.835 3.690 2.546 1.467 0.319 Acetone 4.758 4.083 3.394 2.694 1.790 1.50% Cyclohexane 5.148 4.623 4.263 3.903 3.503 Tri-CE 6.342 5.849 5.528 5.145 4.737 Tetra-CE 16.044 15.600 15.156 14.712 14.268 Benzene 1.320 1.288 1.256 1.224 1.192 Chloroform 11.092 10.224 9.397 8.639 7.696 Ether 8.447 7.217 5.955 4.707 3.410 Methanol 4.109 3.137 2.164 1.247 0.272 Acetone 3.806 3.266 2.715 2.155 1.432 1.75% Cyclohexane 5.225 4.741 4.301 3.781 3.338 Tri-CE 6.278 5.943 5.606 5.227 4.892 Tetra-CE 16.506 16.039 15.562 15.169 14.807 Benzene 1.012 0.959 0.906 0.853 0.801 Chloroform 11.055 10.262 9.423 8.647 7.822 Ether 7.965 6.806 5.615 4.439 3.216 Methanol 3.493 2.666 1.840 1.060 0.231 Acetone 3.045 2.613 2.172 1.724 1.146 2.00% Cyclohexane 5.214 4.720 4.223 3.723 3.217 Tri-CE 6.277 5.941 5.601 5.257 4.907 Tetra-CE 16.525 16.062 15.614 15.177 14.745 Benzene 0.889 0.827 0.765 0.703 0.641 Chloroform 11.063 10.247 9.431 8.616 7.797 Ether 7.511 6.418 5.295 4.186 3.033 Methanol 2.969 2.266 1.564 0.901 0.196 Acetone 2.436 2.090 1.738 1.379 0.917 The results of the table gives a lot of information to understand the surface physicochemical properties of the various zeolite catalysts. Examples of the values of the specific free enthalpy ∆Gsp(T) of different polar molecules adsorbed on different catalysts are displayed in the Fig. 15. For H-β-zeolite at T = 480 K, the strong amphoteric behavior of this catalyst is emphasized (Fig. 15a). The catalyst actively reacts with the amphoteric solvents (methanol, acetone, tri-CE and tetra-CE), acid (chloroform) and base (ether) molecules.Figure 15 Comparison between the specific free enthalpy -∆Gsp(T) (in kJ/mol) of the different polar molecules adsorbed at T = 480 K on (a) H-β-zeolite, (b) 0.25% of rhodium supported by H-β-zeolite, and (c) 2% of rhodium supported by H-β-zeolite (c). For 0.25% of rhodium impregnated into H-β-zeolite, similar behavior take place. However, an evolution in the surface acid–base properties of catalyst is observed. The presence of 0.25% of rhodium produces a decrease of the amphoteric character of the catalyst. The magnitudes of methanol and acetone ∆Gsp decrease from 10.9 kJ/mol and 14.5 kJ/mol, respectively, to 9.3 kJ/mol and 11.6 kJ/mol. However, there is an increase in the acid character with a diminution of basic specific free enthalpy. It seems that the impregnation of the rhodium in H-β-zeolite causes a reduction in base character and an enhancement in the acid force. The tendency of the decrease of the basic character and the increase of acid character becomes more accentuated for greater percentage of impregnated rhodium (2%Rh, see Fig. 15c). The same behaviors are observed at all the temperatures (Table 12). Other thermodynamic measurements Some other thermodynamic parameters can be calcualted in this study. Experimental results led to determine the differential heat of adsorption ∆H 0 and the standard entropy change of adsorption ∆S 0 of the probe. These parameters can be obtained from relation (4) by using the two following Eqs. (25) and (26): 25 ΔH0=-R∂lnVn∂1T 26 ΔS0=-∂RTlnV∂T By plotting lnVn as a function of (1/T), one obtained the curves of Fig. 16. A linear dependency was proved and the following general Eq. (27) was obtained for all polar and n-alkanes adsorbed on the catalyst of 2% of rhodium supported by H-β-zeolite: 27 lnVn=A1T+B where A and B are constants depending on the probe nature.Figure 16 Variations of lnVn as a function of 1000/T of different polar and n-alkane molecules adsorbed on 2% of rhodium supported by H-β-zeolite. One deduced ∆H 0 and ∆S 0 from Eq. (27): 28 ΔH0=-RA;ΔS0=-RB By using relations (25–28) and Fig. 16, we obtained the values of the differential heat the standard entropy change of adsorption given by Table 13.Table 13 Values of ΔH0(kJ/mol), ΔS0(JK-1mol-1) and the expressions of ΔG0(T)(kJ/mol) of different polar and n-alkane molecules adsorbed on 2% of rhodium supported by H-β-zeolite. Molecules ΔH0(kJ/mol) ΔS0(JK-1mol-1) ΔG0(T)(kJ/mol) C5 − 48.764 − 67.9 − 48.764 + 6.79 × 10–2 T C6 − 54.787 − 69.3 − 54.787 + 6.93 × 10–2 T C7 − 65.456 − 80.9 − 65.456 + 8.09 × 10–2 T C8 − 73.026 − 85.7 − 73.026 + 8.57 × 10–2 T C9 − 81.372 − 92.2 − 81.372 + 9.22 × 10–2 T Cyclohexane − 42.018 − 48.3 − 42.018 + 4.83 × 10–2 T Tri-CE − 48.668 − 63.1 − 48.668 + 6.31 × 10–2 T Tetra-CE − 50.000 − 62.5 − 50.000 + 6.25 × 10–2 T Benzene − 57.363 − 72.6 57.363 + 7.26 × 10–2 T Chloroform − 24.401 − 31.7 − 24.401 + 3.17 × 10–2 T Ether − 31.179 − 50.3 − 31.179 + 5.03 × 10–2 T Methanol − 41.654 − 67.8 − 41.654 + 6.78 × 10–2 T Acetone − 43.644 − 71.3 − 43.644 + 7.13 × 10–2 T The values of -ΔH0 and -ΔS0 of the probe increase when the carbon atom number nC increases. Linear relations (29) and (30) were obtained as a function of nC for n-alkanes: 29 -ΔH0(kJ/mol)=-8.346nC-6.262 30 -ΔS0(JK-1mol-1)=-6.50nC-33.71 This increase is due to the increase in the boiling points of n-alkanes and to the stronger interaction between the solute and catalyst surface. Table 13 clearly showed that benzene exhibits more negative ΔH0 than the corresponding values for n-alkanes with the same carbon atom number (as for example n-hexane or cyclohexane where nC=6) The more negative the heat, the greater the interaction between the adsorbate and the adsorbent. This can be explained by the specific interactions between benzene’s electrons and the surfaces. The same results were previously observed by Bilgiç and Tümsek36. The -ΔH0 values of polar probes increase in the following order for the catalyst 2% of rhodium supported by H-β-zeolite: Chloroform < Ether < Methanol < Cyclohexane < Acetone < Tri-CE < Tetra-CE < benzene. This is conform to the relative polarities of polar molecules that decrease in the same order. Variations of the specific enthalpy and entropy of adsorption on different catalysts From the Table 12 it can be deduced that the curves of ∆Gsp(T) of the polar molecules as a function of the temperature follow linear dependency for all used catalysts in agreement with Eq. (17): 31 ΔGspT=ΔHsp-TΔSsp An example of straight lines obtained with the catalyst containing 1.75% of rhodium is shown in Fig. 17.Figure 17 Variations of − ∆Gsp(T) of polar molecules as a function of the temperature in the case of 1.75% impregnated rhodium in H-β-zeolite for a range temperature [480 K, 560 K]. The specific enthalpy ∆Hsp and entropy ∆Ssp of adsorption can be calculated by applying Eq. (17) to the data of Table 12. The results are reported in Tables 14 and 15. Note also that all linear regression coefficients, r2, are close to 1.Table 14 Values of the specific free enthalpy − ∆Hsp (kJ/mol) of adsorption of different polar molecules on catalysts as a function of the impregnated rhodium percentage. %Rh probes 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 Cyclohexane 9.713 8.788 6.408 5.562 5.614 7.871 14.712 16.581 17.196 Tri-CE 16.959 17.329 18.776 20.903 14.328 19.447 15.694 14.661 14.499 Tetra-CE 32.242 30.634 28.035 25.801 20.477 26.716 26.700 26.716 27.178 Benzene 1.457 1.633 1.539 1.456 0.647 1.907 2.088 2.279 2.377 Chloroform 22.592 27.243 31.054 29.689 4.974 37.111 31.191 30.452 30.654 Ether 54.982 51.848 48.892 46.106 43.478 40.999 38.662 36.459 34.38 Methanol 71.74 60.979 51.832 44.057 37.449 31.831 27.057 22.998 19.548 Acetone 68.321 54.657 43.726 34.981 27.984 22.388 17.91 14.328 11.462 Table 15 Values of the specific entropy − ∆Ssp (JK-1 mol-1) of adsorption of different polar molecules on catalysts as a function of the impregnated rhodium percentage. %Rh probes 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 Cyclohexane − 10 − 7 − 2 0 − 1 − 6 − 20 − 24 − 25 Tri-CE − 19 − 21 − 24 − 34 − 25 − 32 − 20 − 17 − 17 Tetra-CE − 30 − 27 − 28 − 36 − 31 − 39 − 22 − 21 − 22 Benzene − 2 − 2 − 2 − 2 − 1 0 − 1 − 3 3 Chloroform − 30 − 36 − 40 − 38 − 61 − 54 − 42 − 40 − 41 Ether − 90 − 84 − 80 − 75 − 71 − 67 − 63 − 59 − 56 Methanol − 127 − 108 − 92 − 78 − 66 − 56 − 48 − 41 − 35 Acetone − 112 − 89 − 72 − 57 − 46 − 37 − 29 − 23 − 19 The specific enthalpy of interaction between the catalysts and polar molecules is very large for the amphoteric probes as acetone and methanol and for base and acid solvents as ether and chloroform (Table 14). The negative value of the specific entropy of interaction proves the more ordered systems for basic and acidic interactions. This confirms the previous results concerning the acid–base properties of the catalysts. Lewis acid base constants of catalysts The acid–base constants KA and KD of the various catalysts can be obtained using the experimental data and applying the relation (19). To this aim, the evolution of − ∆Hsp/AN' as a function of DN'/AN' for H-β-zeolite is followed for various rhodium percentages. The Fig. 18 gives examples of these variations, for four amounts of Rh. The extracted acid and base constants obtained for the different solid substrates are presented in Table 16 with the corresponding linear regression coefficients used to fit the linear curves.Figure 18 Evolution of − ∆Hsp/AN' (kJ/mol) versus of DN'/AN' of polar molecules adsorbed on different percentages of rhodium impregnated surfaces: (a) H-β-Zeolite (0%Rh), (b) 0.5%Rh/HβZ, (c) 1.25%Rh/HβZ, and (d) 1.75%Rh/HβZ (d). Table 16 Values of KD, KA, of different catalysts as a function of the rhodium percentage impregnated in zeolite. %Rh/HβZ Acid constant KA (kJ/mol) Base constant KD (kJ/mol) Linear regression coefficient R2 0 2.522 0.088 0.9117 0.25 2.2691 0.908 0.9076 0.5 1.628 1.365 0.9089 0.75 1.4 1.581 0.9009 1 1.471 0.255 0.9219 1.25 2.007 0.143 0.8777 1.5 3.86 -4.031 0.8689 1.75 4.362 -5.192 0.8643 2 4.5223 -5.58 0.8608 It seems also interesting to follow the acid and base constants (KD and KA) as a function of the percentage of rhodium impregnated. The results are given in Fig. 19.Figure 19 Variations of the acid base constants KA and KD (in kJ/mol) of different catalysts as a function of the rhodium percentage impregnated in zeolite by using the classical model. The acid base properties of the zeolite surface are significantly affected by the impregnation of rhodium metal in H-β-zeolite. For a rhodium percentage less than 0.75%, the surface acidity of the catalysts decreases whereas the basicity increases. Conversely, for %Rh larger than 0.75%Rh, an opposite trend takes place since an increase of the acidity and decrease of the basicity are visible. For rhodium percentage larger than or equal to 1.5%Rh, KD and KA do not vary with the rhodium percentage. Note that, negative values of the basic constant for rhodium percentages larger than 1.25%Rh are observed. In this range of %Rh, the linear regression coefficients are not very satisfactory since r2 are comprised between 0.800 and 0.900. Actually, for all the rhodium percentages %Rh, no perfect straight line is obtained. This confirms that the model (Eq. (19)) does not satisfactorily apply to the results. One of reasons for obtaining bad linear regression coefficients r2 was the larger value of the ratio DN/AN equal to 25 for cyclohexane, the second reason was the insufficiency of the classical equation to describe with accuracy the experimental results. It becomes then pertinent to employ the Hamieh’s model in order to improve the accuracy of the acid–base constants. Discussion on the light of the new model Some similar irregularities when using Eq. (19) were observed by Hamieh et al.18,19. They proposed a new relationship by adding a third parameter K reflecting the amphoteric character of solid surfaces. This method is applied here and the Eq. (21) is used to calculate the three acid–base constants KA, KD and K of the various catalysts. These constants are obtained with an excellent three-dimension linear regression coefficients approaching r2 ≈ 1.000. The obtained results are presented in Table 17 and Fig. 20 where the acid–base constants KD, KA, K and the ratio KA/KD of different substrates are expressed for various rhodium percentages %Rh.Table 17 Acid–base constants KD, KA, K and of the ratio KA/KD of different catalysts as a function of the rhodium percentage %Rh. %Rh/HbZ KD KA K KA/KD 0 1.215 2.652 0.464 2.18 0.25 1.465 2.387 0.528 1.63 0.5 1.670 1.690 0.552 1.01 0.75 1.596 1.446 0.619 0.91 1 0.267 1.524 0.519 5.70 1.25 1.995 2.112 0.335 1.06 1.5 1.677 4.130 0.026 2.46 1.75 1.637 4.684 0.068 2.86 2 1.648 4.866 0.056 2.95 Figure 20 Variations of the acid base constants KA, KD and K (in kJ/mol) and KA/KD of different catalysts as a function of the impregnated rhodium percentage by using Hamieh’s model. The H-β-zeolite is more acidic than basic. In the presence of rhodium, the acidity constant KA decreases from 2.7 to 1.5 kJ/mol when the percentage %Rh increases from 0 to 0.75%. On the opposite, the basicity constant KD, increases from 1.2 to 1.7 kJ/mol and dramatically decreases until 0.3 kJ/mol at rhodium percentage equal to 1%. For Rh percentages larger than 1%Rh, the acid base constants increase until %Rh reaches 1.5% and then stabilize. On the other hand, the amphoteric constant K remains constant up to 1%Rh. It then decreases to reach a plateau above 1.5%Rh. The ratio KA/KD showing a maximum at 1%Rh confirms the previous results on the incorporation of rhodium into the channels of H-β-zeolite observed when discussing the variations of RTlnVn, ∆Gsp and the dispersive component of the surface energy γsd of the different catalysts. It seems interesting to compare the order of magnitudes of the constants with those reported in the literature. Bilgiç and Tümsek determined the surface acid base properties of MgY and NH4Y using inverse gas chromatography36. According to results obtained by the above authors for KA and KD, the surface of MgY exhibits predominantly basic character with the ratio of KD/KA = 3.50, while surface of NH4Y shows a less basic character with the ratio of KD/KA = 2.61. These results showed basic than acidic character of the zeolite materials. However, when comparing these data with those obtained in our study, it appears that our catalysts are rather acidic than basic since the ratios KA/KD are comprised between 0.9 and 5.7. The difference between the two materials results from the presence of framework oxygens adjacent to alkali cations which are the Lewis basic sites in zeolites. This was previously proved by Bilgic and Tumsek36, Barr and Lishka37, Okamoto et al.38 and Vinek et al.39. Other catalysts exhibit acidic surface similar to the catalysts of the present study. As an example, the sepiolite surface characterized by Morales et al.40 for which the ratio of acid base constants KA/KD was equal to 3. It seems also relevant to evaluate the error committed on the values of acid base constants. To this aim, the following approach is employed. The error committed on the net retention time is: 10-3min≤Δtnprobe≤3×10-3min The relative standard deviation on the retention time is given by the following inequalities: 5×10-5≤Δtnprobetnprobe≤10-4 This gives a relative standard deviation on the net retention volume: 5×10-5≤ΔVnprobeVnprobe≤10-4 And therefore, we obtain for free enthalpy of adsorption the following error: 5×10-4kJ/mol≤ΔΔGa0≤3×10-3kJ/mol Moreover, the relative deviation is given by: 3×10-4≤ΔΔGa0ΔGa0≤5×10-4 And the error on the specific free enthalpy reads as: 10-3kJ/mol≤ΔΔGasp≤6×10-3kJ/mol Finally, the relative error committed on the acid–base constants KA, KB and K are: 1×10-3≤ΔKA,BKA,B≤2×10-3 Therefore, the error committed on the values of acid base constants is equal to 5×10-3. Conclusion In this paper, new thermodynamic methods and models were developed to study the surface energy and acid base properties of H-β-zeolite impregnated with rhodium metal at different percentages %Rh. The effect of the temperature and the rhodium content on the acid base properties in Lewis terms of the various catalysts were analyzed by inverse gas chromatography at infinite dilution. The variation of RTlnVn of n-alkanes adsorbed on the catalysts with the rhodium percentage revealed the presence of a maximum at %Rh = 0.75%. Conversely, for polar solvents the maximum occurs at 1.00%Rh. This is due to the variation of some surface groups because of the change in the acid base properties of the catalysts when adding rhodium in zeolite. The specific surface area SBET of different catalysts decreases with the rhodium percentage until 0.5%Rh, followed by a slight increase to reach a local maximum at 1%Rh. Finally, SBET decreases up to a plateau observed for %Rh larger than 1.50%. The same conclusion was observed for the microporous volume. The highest value of the specific surface area was obtained for H-β-zeolite. In the presence of rhodium, the metal particles block the micropores causing a decrease in the specific surface area and in the catalyst microporosity. However, the increase of the specific surface area, for the catalysts containing a rhodium percentage comprised between 0.5 and 1.0, can result from the smaller particle sizes that cannot block the zeolite micropores. For catalyst with a rhodium percentage larger than 1.50% Rh, the much lower observed surface area and pore volume is certainly due to the larger nanoparticles which block the micropores. The same behavior was observed when studying the variations dγsddT of catalysts as a function of the impregnated rhodium percentage regardless of the molecular model of n-alkane surface areas used. The results relative to the specific free enthalpy ∆Gsp(T) of different polar molecules adsorbed on H-β-zeolite clearly demonstrated the strong amphoteric behavior of all supported Rh catalysts. The rhodium supported by H-β-zeolite actively react with the amphoteric solvents (methanol, acetone, tri-CE and tetra-CE), acid (chloroform) and base (ether) molecules. A decrease of the amphoteric character of the catalyst with 0.25% of rhodium is reported. The magnitudes of methanol and acetone ∆Gsp decrease from 10.9 kJ/mol and 14.5 kJ/mol, respectively, to 9.3 kJ/mol and 11.6 kJ/mol. Whereas, an increase in the acid character with a decrease of basic specific free enthalpy were highlighted. It seems that the impregnation of the rhodium in H-β-zeolite causes a decrease in base character and an increase in the acid magnitude. The tendency of the decrease of basic character and the increase of acid character became more accentuated for greater percentage of impregnated rhodium (2%Rh) for all temperatures. The classic Gutmann relationship was not well suited for an accurate determination of the acid base constants. Negative values of the basic constant for rhodium percentage more than 1.25%Rh coupled to weak linear regression coefficients of the order of 0.8 and 0.9 are obtained. The previous results were corrected by applying the Hamieh’s model. In this case, the acid–base constants KA, KD and K of the various catalysts were determined with an excellent accuracy. The H-β-zeolite is more acidic than basic with more important specific interactions. The acidity constant KA decreases with the Rh content while the basicity constant KD, increases up to 1%Rh. At the same time, the amphoteric constant K remains constant until 1%Rh and then decreases to reach its plateau from 1.5%Rh. An interesting correlation was highlighted between the surface specific area of the various catalysts, the rhodium percentage in zeolites and the specific acid base interactions between the catalysts and the polar organic molecules. Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Author contributions T.H.: conceptualization; formal analysis; investigation; methodology; supervision; validation; writing—review & editing. A.A.A.: investigation; methodology. T.R.C.: conceptualization; supervision; validation; writing—review & editing. J.T.: conceptualization; formal analysis; investigation. Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Competing interests The authors declare no competing interests. ==== Refs References 1. Chow AHL Tong HHY Shekunov BY York P Letter to editor: Use of inverse gas chromatography (IGC) to determine the surface energy and surface area of powdered materials Pharma. Res. 2004 21 9 1718 1720 10.1023/B:PHAM.0000041470.92345.e8 2. 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