The Cohesive Interactions in Phenylimidazoles

This work presents a comprehensive study exploring the thermodynamics of the solid phase of a series of phenylimidazoles, encompassing experimental measurements of heat capacity, volatility, and thermal behavior. The influence of successive phenyl group insertions on the imidazole ring on thermodynamic properties and supramolecular behavior was thoroughly examined through the evaluation of 2-phenylimidazole (2-PhI), 4-phenylimidazole (4-PhI), 4,5-diphenylimidazole (4,5-DPhI), and 2,4,5-triphenylimidazole (2,4,5-TPhI). Structural correlations between molecular structure and thermodynamic properties were established. Furthermore, the investigation employed UV–vis spectroscopy and quantum chemical calculations. Additive effects arising from the introduction of phenyl groups were found through the analysis of the solid–liquid and solid–gas equilibria, as well as heat capacities. A good correlation emerged between the thermodynamic properties of sublimation and the molar volume of the unit cell, evident across 2-PhI, 4,5-DPhI, and 2,4,5-TPhI. In contrast to its isomer 2-PhI, 4-PhI exhibited greater cohesive energy due to the stronger N–H···N intermolecular interactions, leading to the disruption of coplanar geometry in the 4-PhI molecules. The observed higher entropies of phase transition (fusion and sublimation) are consistent with the higher structural order observed in the crystalline lattice of 4-PhI.


Heat Capacity Measurements
The heat capacities at θ = 298.15K (solid phase), of all the studied compounds were measured using a high-precision heat capacity drop calorimeter. 1,2The calorimeter was calibrated with sapphire (α-Al2O3, NIST-RM 720), whose C p,m o (α-Al2O3, 298.15 K) = (79.03± 0.08) J•K −1 •mol −1 . 3Table S1 presents the standard molar heat capacities at 298.15 K for the phenylimidazoles: msample represents the mass of sample used in each independent experiment and Ndrop is the number of drop experiments.The reported uncertainty is twice the standard deviation of the mean and includes the calibration uncertainty.Ndrop = number of drop experiments; Tfurnace = average temperature of the furnace; Tcalorimeter = average temperature of the calorimeter; ε = calibration constant; the uncertainty reported is twice the standard deviation of the mean and the calibration uncertainty is included.

DSC Calibration:
Temperature correction :  Detailed results obtained for the study of the fusion equilibrium of 2-PhI (M = 114.17g•mol -1 ).
The reported uncertainties are twice the standard deviation of the mean and include the calibration uncertainty.

TABLE S3
. Detailed results obtained for the study of the fusion equilibrium of 4-PhI (M = 114.17g•mol -1 ).
The reported uncertainties are twice the standard deviation of the mean and include the calibration uncertainty.The reported uncertainties are twice the standard deviation of the mean and include the calibration uncertainty.
The reported uncertainties are twice the standard deviation of the mean and include the calibration uncertainty.
For each compound, the molar entropy of fusion at Tm was derived according to Equation S1.The molar enthalpies, entropies, and Gibbs energies of fusion at the reference temperature of θ = 298.15K were calculated according to Equations S2, S3, and S4.
In these equations, ∆fusCp o represents the difference between the molar heat capacity of the liquid and the molar heat capacity of the solid at the reference temperature θ.For this analysis, a typical value (recommended by Chickos) of ∆fusCp o = (54.4± 20) J•K −1 •mol −1 was considered. 8The value of ∆fusCp o was treated as a constant within the temperature range.The temperature θ = 298.15K was utilized as the reference temperature for this evaluation.The thermodynamic properties at Tm and derived at θ = 298.15K are presented in

Knudsen Effusion/Thermodynamic Properties of Sublimation
For each solid compound, the equilibrium vapor pressures (Table S6), denoted as p, were determined at various effusion temperatures using the Knudsen equation (Equation S5). 9,10 In Equation S5, Δm represents the mass of the sample effused during the experimental time Δt, M stands for the molar mass of the effusing vapor, R denotes the gas constant (8.314462618J•mol -1 •K -1 ), Ao signifies the area of the effusion orifice, and wo is the transmission probability factor.It was calculated using Equation S6, where l represents the length of the effusion orifice and r its radius. 9,10 this technique, only one effusion cell is used, with l = 0.0125 mm and r = 0.600 mm, yielding Ao = 1.1310 mm 2 and wo = 0.9922.The ln p = f (1 / T) results were fitted using the integrated form of the Clausius-Clapeyron equation (Equation S7). 9,10 where  = ∆ sub  m ⬚ (⟨⟩)/, and p* = 1 Pa.Table 3 of the manuscript details the results of fitting the sublimation data to the linear Clausius-Clapeyron equation.The mean temperature, <T>, was determined as the average temperature across all experimental data points; ( ) represents the pressure at that temperature, obtained from the ln p = f (1/T) linear regression.∆ sub  m ⬚ (⟨⟩, ⟨⟩) was subsequently calculated according to Equation S8. 9,10 ∆ sub  m ⬚ (⟨⟩, ⟨⟩) = ∆ sub  m ⬚ (⟨⟩) / ⟨⟩ (Equation S8) The standard molar entropy of sublimation, ∆ sub  m  , at θ = 298.15K, was calculated according to Equation S10, where p o = 10 5 Pa.
The standard molar Gibbs Energy of sublimation, ∆ sub  m  , at θ = 298.15K, was calculated according to Equation S11.
The thermodynamic properties of vaporization at θ = 298.15K (listed in Table 4 of the manuscript) were derived by combining the fusion and sublimation results according to Equations S12, S13, and S14.S12) S13)

Computational Results
Full geometry optimizations and frequency calculations, without symmetry restrictions, were conducted at the M06-2X/6-311++G(d,p) level of theory.No imaginary frequencies were found, confirming that the optimized structures correspond to true minima.These calculations were used to determine the gaseous phase heat capacities for all species at 298.15 K and evaluated gas phase molecular geometries and energetics.Comprehensive computational results are provided in Table S7.The optimized geometry of each molecule is presented in Figure S3 and Table S8.
Oscillator strength, the position of the transitions, and frontier orbitals are also depicted.

Figure S5 .
Figure S5.Comparison of the experimental (solid line) and theoretical (dashed line) UV-Vis spectra of 4-PhI, recorded in CH2Cl2, at T = 298.1 K. Oscillator strength, the position of relevant transitions, and frontier orbitals are also depicted.TD-DFT electronic absorption spectra were computed at the M06-2X/6-311++G(d,p) level of theory.

Figure S6 .
Figure S6.Comparison of the experimental (solid line) and theoretical (dashed line) UV-Vis spectra of 4,5-DPhI, recorded in CH2Cl2, at T = 298.1 K. Oscillator strength, the position of relevant transitions, and frontier orbitals are also depicted.TD-DFT electronic absorption spectra were computed at the M06-2X/6-311++G(d,p) level of theory.

TABLE S1 .
Standard molar heat capacity values of the studied compounds at θ = 298.15K.
Table 2 of the manuscript.