Expanding the Equilibrium Solubility and Dissolution Thermodynamics of Benzoic Acid in Aqueous Alcoholic Mixtures

Abstract

The equilibrium solubility of benzoic acid in water and ethanol, as well as in nine {ethanol (1) + water (2)} mixtures, was determined from T = (293.15 to 323.15) K. Benzoic acid mole fraction solubility in these aqueous-ethanolic mixtures was adequately correlated with some well-known correlation/prediction models, obtaining mean percentage deviations of 2.2 to 7.6%. Apparent thermodynamic quantities, namely, Gibbs energy, enthalpy, and entropy, for the dissolution, mixing and solvation processes, were computed by means of the van ’t Hoff and Gibbs equations. The enthalpy–entropy compensation plot of apparent enthalpy vs. apparent Gibbs energy of dissolution was not linear, indicating enthalpy and entropy mechanisms for transfer. Ultimately, by using the inverse Kirkwood–Buff integrals, it is observed that benzoic acid is preferentially solvated by water molecules in water-rich mixtures but preferentially solvated by ethanol molecules in those {ethanol (1) + water (2)} mixtures of 0.24 < x₁ < 1.00.

1. Introduction

The equilibrium solubility of compounds in chemical, food, pharmaceutical and material industries is of critical importance in selecting the most suitable solvent and operating temperature in separation techniques such as extraction, pollution prevention, purification and crystallization [1]. In addition, the dependence of the solubility on temperature allows the evaluation of the preferential dissolution analysis of the solute by types of co-solvents in the mixture, as well as the related thermodynamic analysis, which provides insight into the molecular mechanisms involved in the dissolution process [2,3,4].

Benzoic acid (Systematic IUPAC name: Benzenecarboxylic acid, molecular structure shown in Figure 1) is a colorless crystalline solid that is generally used as a significant intermediate in the production of a variety of chemical compounds [5,6,7,8,9,10]. Benzoic acid and its salts have a wide range of applications in different industries, especially in the chemical industry and pharmaceutical industry, such as resins, dyes, polymers, preservatives, cosmetics and pharmaceuticals [11,12]. In this regard, benzoic acid is widely used as an intermediate in the synthesis of phenol and caprolactam [8] as well as in the preparation of resins, cosmetics, dyes, inks, plasticizers, pharmaceutical products and rust inhibitor [7,8,10,13,14,15]. Its salts are mainly used as intermediates in the preservation of foods, fruit juices, pickles and fats as well as in the production of synthetic flavors, perfumes and for the flavoring of tobacco. In addition, they are known to have very good antifungal activity [8,10,16]. Furthermore, the other end products of benzoic acid are also used as corrosion inhibitors [8,17].

Benzoic acid is sparingly soluble in water (0.3 g/100 g of water at 293 K) and the limited solubility of benzoic acid in water makes it difficult to use its application in products containing large amounts of water [10,18]. For this reason, it is necessary to increase its solubility in the aqueous phase. In this context, many researchers have reported temperature dependent solubility data of benzoic acid in both monosolvents and binary mixtures and have tried to understand the solubility and dissolution behavior of this excipient agent [6,7,8,9,10,12,13,14,15,16,17,19,20]. Thus, Chertkoff and Martin [21] measured the solubility of benzoic acid in binary mixtures of hexane, ethyl acetate, ethanol and water. Sahay et al. [22] examined the solubility of benzoic acid in distilled water and aqueous solutions of poly (vinyl alcohol) and poly (vinylpyrroildone) containing 0.5–6.0 wt% polymer at temperatures from 303.15 to 328.15 K. Li et al. [14] determined the solubility of benzoic acid in N,N-dimethylformamide as a function of temperature in the temperature range of 294.75–370.45 K using a laser monitoring observation technique. Oliveira et al. [19] performed benzoic acid solubility determinations in pure water and in mixtures of an organic solvent (DMF, ethanol, and 1,4 dioxane) with water within the temperature range of 303.65 to 355.65 K. Pires and Franco [20] investigated the solubility of benzoic acid in water-ethanol and water-n-propanol mixtures at temperatures between 303.5 and 333.7 K. Long et al. [6] carried out the solubility determination of benzoic acid in acetone, 2-propanol, acetic acid and cyclohexane over the temperature range of 277–346 K by using a gravimetric method. Thati et al. [16] conducted the solubility analysis of benzoic acid in monosolvents (ethanol, toluene, heptane, cyclohexane, pentane and chloroform) and binary solvent mixtures (ethanol + heptane and ethanol + toluene) at temperatures ranging from 278.15 to 323.15 K by the gravimetric method. Cheng et al. [15] measured the solubility values of benzoic acid in ethanol, benzene, acetic acid and ethyl acetate over the temperature range of 291.69–356.27 K. Wang et al. [12] investigated the solubility of benzoic acid in binary (benzyl alcohol + benzaldehyde) solvent mixtures at temperatures from 298.35 to 355.65 K under atmospheric pressure by a synthetic method. Kumari et al. [17] investigated the solubility of benzoic acid in (acetic acid + water) and (acetic acid + toluene) binary mixtures at different mass fractions of acetic acid ranging from 0.1 to 0.8 and at 299.15 K by the gravimetric method. Xue et al. [7] examined the solubility of benzoic acid in six alcohols (isobutanol, n-pentanol, isoamylol, 2-pentanol, tert-amyl alcohol and n-hexanol) at different temperatures from 288.15 K to 336.15 K by the dynamic method. Sandeepa et al. [8] performed the solubility study of benzoic acid in six monosolvents (tributyl phosphate, diacetone alcohol, methyl-n-propyl ketone, methyl acetate, amyl acetate and isooctane) at temperatures from (283.15 to 328.15) K and five binary systems, i.e., ethanol + hexane (288.15–328.15 K), isopropyl alcohol + hexane and chloroform + hexane (288.15–323.15 K), acetone + hexane (288.15–318.15 K), and acetone + water (288.15–318.15 K) at atmospheric pressure. AbouEllef et al. [9] determined the solubility of benzoic acid in binary mixtures of ethanol–water solvents at different temperatures by using the gravimetric method. Muhammad et al. [10] conducted the benzoic acid solubility study in the aqueous phase and various aqueous mixtures of methanol, ethanol and 2-propanol at temperatures ranging from 303 to 333 K by an analytical technique. Wang et al. [13] measured the solubility of benzoic acid in twelve organic solvents (1,2-ethanediol, 3,6-dioxa-1-decanol, octane, 3-oxa-1,5-pentanediol, N,N-dimethylacetamide, 1,4-dimethylbenzene, 1,3-dimethylbenzene, 1,2-dimethylbenzene, propanoic acid, cyclohexanone, decahydronaphthalene, thiacyclopentane dioxide) at temperature ranging from 286.15 to 354.15 K using the laser dynamic method. Mendez-Santiago and Teja [23] measured the solubility of benzoic acid in mixtures of CO₂ + hexane at temperatures ranging from (308 to 338) K, pressures ranging from 10 to 35 MPa, and cosolvent concentrations containing 0–7.0 mol% hexane. Sandeepa et al. [24] evaluated the solubility of benzoic acid in monosolvents (benzyl acetate and benzyl benzoate), in binary solvent systems (benzyl alcohol + toluene, benzyl acetate + toluene and benzyl benzoate + toluene) in the temperature range of 288.15–328.15 K, and in ternary solvent systems (benzoic acid + phthalic acid + benzyl alcohol/methanol) at two different temperatures (298.15 and 308.15) K. More recently, Zhang et al. [25] have carried out the solubility analysis of benzoic acid in seven pure solvents (water, methanol, ethanol, acetonitrile, dichloromethane, toluene, and ethyl acetate) within the temperature range of 273.15 to 323.15 K. Despite of all the benzoic acid solubility studies reported, there are limited experimental or modeling information about determination of the solubility of benzoic acid in binary solvent mixtures of {ethanol (1) + water (2)} at different temperatures and compositions.

Because of all of these reasons, the purposes of this study were to (1) measure the solubility of benzoic acid in solvent mixtures of {ethanol (1) + water (2)} in the temperature range of 293.15–323.15 K at constant pressure; (2) describe the determined experimental solubilities with various mathematical models; (3) evaluate the apparent thermodynamic parameters of dissolution, mixing and solvation of benzoic acid in binary solvent mixtures of {ethanol (1) + water (2)}; and (4) calculate the preferential solvation parameters of benzoic acid by ethanol in binary solvent combinations using the inverse Kirkwood–Buff integrals.

2. Materials and Methods

2.1. Materials

Benzoic acid was obtained from Sigma-Aldrich Chemical (Steinheim, Germany). Acetonitrile, ethanol and formic acid were of chromatographic grade and provided by Merck Chemical (Istanbul, Turkey). Ultra-pure water (18.2 MΩ·cm) was prepared by the MilliPore Milli-Q-Gradient water purification system (Billerica, MA, USA). More detailed information about these chemicals is listed in

Table 1. Source and purities of the compounds used in this research.

Compound	CAS	Formula	Molar Mass/g·mol−1	Source	Purity in Mass Fraction	Analytic Technique a
Benzoic acid	65-85-0	C7H6O2	122.12	Sigma-Aldrich	≥0.995 b	HPLC
Ethanol	64-17-5	C2H6O	46.07	Merck	>0.995 b	GC
Acetonitrile	75-05-8	C2H3N	41.05	Merck	≥0.999 b	GC
Formic acid	64-18-6	CH2O2	46.03	Merck	≥0.999 b	GC
Water	7732-18-5	H2O	18.02	Obtained by Millipore Milli-Q-Gradient water purification system	>0.999	-

^a HPLC is high performance liquid chromatography, GC is gas chromatography. ^b As indicated by the suppliers.

.

2.2. Solubility Determinations

All {ethanol (1) + water (2)} binary solvent mixtures in mass fractions from 0.10 to 0.90, varying by 0.10, were prepared in 50.0 g dark glass flasks. All mass fractions were adjusted by using an analytical balance with an accuracy of ±0.1 mg (Kern ABJ 220-4NM, Balingen, Germany).

Solubility measurements of benzoic acid against the mass fraction of ethanol in {ethanol (1) + water (2)} binary solvent mixtures at temperatures ranging from 293.15 K to 323.15 K under atmospheric pressure were determined using the traditional shake-flask method [26] and are described in the following. Briefly, an excess amount of benzoic acid was added to the flask containing pure solvents and solvent mixtures in known mass ratios and capped tightly. Then, the suspensions were shaken for 18 h to saturation in a shaking thermostatic water bath (±0.1 K) at selected temperatures. After the 18 h equilibration period, all the saturated mixtures were centrifuged to remove fine solid particles and suitably diluted with the water−acetonitrile mixture (50:50% v/v) without observing solute precipitation. The solubility samples were then analyzed by high performance liquid chromatography (HPLC) for quantitative determination, described as follows.

The quantification of benzoic acid in solubility samples was carried out in triplicate by using an Agilent 1200 HPLC (Santa Clara, CA, USA) coupled with an Agilent Zorbax SB Phenyl column (4.6 × 150 mm, 5 μm-Santa Clara, CA, USA). The chromatographic determination was conducted by using the mixture of acetonitrile:water:formic acid (20:79:1% v/v/v) as the mobile phase at a flow rate of 1.0 mL·min⁻¹ and detected at 240 nm with a diode array detector. The volume of the injected sample was 10 μL.

Besides the solubility data, thermal and spectroscopic analyses such as differential scanning calorimetry (DSC) and powder X-ray diffraction (XRD) were used to investigate the crystalline structure of crude and equilibrated benzoic acid samples before and after the solubility determination.

2.3. Solid Phase Characterization

2.3.1. Differential Scanning Calorimetry Analysis

Solid samples obtained after solubility determination were analyzed using a differential scanning calorimetric analyzer to determine the nature of the crude and equilibrated forms of benzoic acid. DSC analysis of samples was carried out using a METTLER TOLEDO STARe System DSC 3 series (Ohio, ABD). The instrument was calibrated by Indium as standard. About 5.0 mg of benzoic acid was placed in a crimped, sealed aluminum pan. Then, the sample was heated from 25 to 440 °C at a heating rate of 10 °C·min⁻¹ under a dynamic nitrogen atmosphere (40 mL·min⁻¹).

2.3.2. X-ray Diffraction Analysis

The crystal form of benzoic acid, both crude and after equilibration in neat water, the mixture w₁ = 0.50, and neat ethanol were identified using XRD analyses. The powder XRD patterns of the solid benzoic acid were performed on Rigaku Smart Lab system (Tokyo, Japan) using CuKα radiation (1.5418 Å). The samples were scanned from ~10° to 90° (2θ°) to provide a degree of crystallinity and structural information.

3. Results and Discussion

3.1. Mole Fraction Solubility of Benzoic Acid

The mole fraction equilibrium solubilities of benzoic acid in aqueous–ethanol mixtures at seven temperatures from T = (293.15 to 323.15) K and an atmospheric pressure of 90 kPa as a function of the mass fraction of ethanol in the solvent mixtures are summarized in

Table 2. Experimental and ideal mole fraction solubility of benzoic acid (x₃) in {ethanol (1) + water (2)} mixtures at several temperatures and p = 90 kPa ^a,c.

w1 b,c	x1 b,c	T/K c
		293.15	298.15	303.15	308.15	313.15	318.15	323.15
0.00	0.0000	4.69 × 10−4	5.61 × 10−4	6.64 × 10−4	7.87 × 10−4	9.34 × 10−4	1.12 × 10−3	1.32 × 10−3
0.10	0.0417	8.05 × 10−4	9.51 × 10−4	1.12 × 10−3	1.26 × 10−3	1.42 × 10−3	1.58 × 10−3	1.77 × 10−3
0.20	0.0891	1.57 × 10−3	1.75 × 10−3	1.98 × 10−3	2.22 × 10−3	2.48 × 10−3	2.81 × 10−3	3.14 × 10−3
0.30	0.1436	2.84 × 10−3	3.19 × 10−3	3.56 × 10−3	4.02 × 10−3	4.54 × 10−3	5.07 × 10−3	5.69 × 10−3
0.40	0.2068	4.92 × 10−3	5.61 × 10−3	6.34 × 10−3	7.09 × 10−3	7.94 × 10−3	8.98 × 10−3	1.00 × 10−2
0.50	0.2812	8.93 × 10−3	1.04 × 10−2	1.16 × 10−2	1.31 × 10−2	1.48 × 10−2	1.65 × 10−2	1.87 × 10−2
0.60	0.3698	1.60 × 10−2	1.88 × 10−2	2.11 × 10−2	2.38 × 10−2	2.67 × 10−2	3.01 × 10−2	3.38 × 10−2
0.70	0.4772	2.88 × 10−2	3.37 × 10−2	3.80 × 10−2	4.30 × 10−2	4.86 × 10−2	5.44 × 10−2	6.00 × 10−2
0.80	0.6101	5.03 × 10−2	5.67 × 10−2	6.43 × 10−2	7.16 × 10−2	8.04 × 10−2	9.03 × 10−2	0.102
0.90	0.7788	8.56 × 10−2	9.93 × 10−2	0.112	0.126	0.139	0.153	0.167
1.00	1.0000	0.163	0.184	0.206	0.228	0.254	0.286	0.321
	Ideal	0.194	0.213	0.233	0.255	0.278	0.304	0.331

^a p is the atmospheric pressure in Aksaray, Turkey. ^b w₁ and x₁ are the mass and mole fractions of ethanol (1) in the {ethanol (1) + water (2)} mixtures free of benzoic acid (3), respectively. ^c Standard uncertainty in p is u(p) = 3.0 kPa. Mean relative uncertainties in w₁ and x₁ were u(w₁) = 0.0008 and u(x₁) = 0.0008. Standard uncertainty in T is u(T) = 0.10 K. Average relative uncertainty in x₃, u_r(x₃) = 0.025.

. Minimum and maximum benzoic acid solubilities are observed in neat water and in neat ethanol, respectively, at all temperatures studied. Benzoic acid solubility increases with temperature-arising, implying endothermic dissolution processes in all cases. Solubility in neat water is in agreement with data reported by Belhachemi et al. [27]. It is noteworthy that our solubility values in these binary mixtures at T = 298.15 K are in good agreement with those reported graphically by Chertkoff and Martin [21]. Although several authors have studied the solubility of benzoic acid in different solvent systems, no more direct comparisons are possible. For instance, Pal and Lahiri reported benzoic acid solubility values in {ethanol (1) + water (2)} mixtures determined at (288, 293 and 298) K but expressed in molarity scale [28].

On the other hand, Figure 2 depicts the benzoic acid solubility as a function of the Hildebrand solubility parameters of the aqueous-ethanol mixtures (δ₁₊₂). The Hildebrand solubility parameter is a polarity index widely used in pharmaceutical studies regarding cosolvency effects on the physical and chemical stabilities of drugs. Mixtures δ₁₊₂ values were calculated from the corresponding δ values, i.e., δ₁ = 26.5 MPa^1/2 for ethanol and δ₂ = 47.8 MPa^1/2 for water [29,30] and volumetric solvent fractions as described in Equation (1) [31,32]. Volume fractions (f_i) were calculated assuming additive behavior.

$${\delta }_{1+2}={\displaystyle \sum _{i=1}^2{f}_i{\delta }_i}$$

(1)

Accordingly, it is expected that organic compounds reach their maximum solubilities in solvent systems exhibiting the same or similar Hildebrand solubility parameters, as them [33,34]. By keeping this in mind, the δ₃ value of benzoic acid would be lower than δ₁ value (i.e., neat ethanol δ value, 26.5 MPa^1/2) at T = 298.2 K, where the maximum benzoic acid solubilities are observed at all temperatures. Effectively, the calculated Fedors δ₃ value of benzoic acid is 24.4 MPa^1/2, as shown in

Table 3. Internal energy, molar volume, and Hildebrand solubility parameter of benzoic acid calculated according to the Fedors method.

Group	Number	ΔU°/kJ·mol−1	V°/cm3·mol−1
Phenyl ring	1	31.9	71.4
–COOH	1	27.6	28.5
	Σ	59.5	99.9
	δ3 = (59,500/99.9)1/2 = 24.4 MPa1/2

[35]. From this point, it would be very important to study the benzoic acid solubility in mixed solvent systems exhibiting lower polarities than that of neat ethanol. A very good option would be {ethyl acetate (1) + ethanol (2)} mixtures, with an interval of 19.8 ≤ (δ₁₊₂/MPa^1/2) ≤ 26.5, where most likely, a maximum benzoic acid solubility peak could be observed.

3.2. The Cosolvency Models

Among all the reported cosolvency models used to calculate drug solubilities in mixed solvents at ambient or any other temperature [36,37], the log-linear model proposed by Yalkowsky is the simplest one [38], which requires only two experimental solubility determinations (in neat solvents) to predict the solubility at other solvent compositions. The Yalkowsky model is commonly represented as:

ln x₃₍₁₊₂₎ = w₁ ln x₃₍₁₎ + w₂ ln x₃₍₂₎

(2)

where x₃₍₁₊₂₎ is the mole fraction solubility of benzoic acid in the solvent mixtures, x₃₍₁₎ is the mole fraction solubility in neat ethanol (component 1), x₃₍₂₎ is the mole fraction solubility in neat water (component 2), and w₁ and w₂ are the mass fractions of ethanol (1) and water (2) in the solvent mixtures in the absence of benzoic acid (3). Thus, the obtained mean percentage deviation (MPD) values for calculating the solubility of benzoic acid in {ethanol (1) + water (2)} mixtures at T = (293.15, 298.15, 303.15, 308.15, 313.15, 318.15 and 323.15) K by using this model are (2.4, 2.2, 2.7, 4.2, 6.1, 9.0 and 5.5) %, respectively, with an overall MPD of 5.5%. It is important to note that the MPD is computed using:

$$MPD=\frac{100}{N}{\displaystyle \sum \frac{\left|x_3^{cal}-x_3^{}\right|}{x_3^{}}}$$

(3)

where N is the number of experimental data points.

As mentioned above, Equation (2) is capable of estimating drug solubilities using only solubility data in the mono-solvents. However, it could be extended as:

$$\ln x_{3(1+2),T}=w_1\left(A_1+\frac{B_1}{T}\right)+w_2\left(A_2+\frac{B_2}{T}\right)$$

(4)

to be applied at various temperatures (x_3(1+2),T) using a single equation. In Equation (4), A and B terms are the model constants [39]. In particular, the obtained constants for equilibrium solubility of benzoic acid in {ethanol (1) + water (2)} mixtures are A₁ = 5.392, B₁ = −2113.628, A₂ = 3.466 and B₂ = −3266.275, which resulted in the MPD of 5.7%.

In addition to above mentioned simple and extended models, the Jouyban–Acree model is the most accurate model to describe the drug solubility in binary solvent mixtures at various temperatures and it is expressed as [36]:

$$\ln x_{3(1+2),T}=w_1\ln x_{3(1),T}+w_2\ln x_{3(2),T}+\left(\frac{w_1{w}_2}{T}\right){\displaystyle \sum _{i=0}^2{J}_i{\left(w_1-w_2\right)}^i}$$

(5)

where J_i terms are the model constants that are computed using a no intercept least square analysis [34]. Accordingly, the generated solubility values of benzoic acid in {ethanol (1) + water (2)} was fitted to Equation (5) and the trained model is:

$$\ln x_{3(1+2),T}=w_1\ln x_{3(1),T}+w_2\ln x_{3(2),T}+\left(\frac{w_1{w}_2}{T}\right)\left[-65.289+144.997\left(w_1-w_2\right)\right]$$

(6)

The F value of Equation (6) was 23, and the correlation and the model constants were significant with p < 0.0005. Equation (6) is valid for calculating the solubility of benzoic acid in {ethanol (1) + water (2)} mixtures at various temperatures by employing the solubility data of benzoic acid in ethanol and water at each temperature or mixture-composition of interest. The obtained MPD for the back-calculated solubility data of benzoic acid using Equation (6) was 4.6%.

Although Equation (6) provided accurate correlation for solubility of benzoic acid in {ethanol (1) + water (2)} mixtures, it requires the experimental solubility data in neat ethanol and water at any temperature of interest (i.e., x_3(1),T and x_3(2),T) to calculate the solubility of benzoic acid in binary mixtures. However, one may combine the trained version of Equation (4) with Equation (6) to provide a full predictive model as:

$$\begin{array}{ll}\ln x_{3(1+2),T}& =w_1\left(5.392-\frac{2113.628}{T}\right)+w_2\left(3.466-\frac{3266.275}{T}\right)\\ & +\left(\frac{w_1{w}_2}{T}\right)\left[-65.882+144.819\left(w_1-w_2\right)\right]\end{array}$$

(7)

Equation (7) calculates the solubilities of benzoic acid in these binary mixtures at various temperatures with an MPD of 4.6% and does not require any experimental input data. It should be noted that the J₂ term in both Equations (6) and (7) was not statistically significant (p > 0.05) and was excluded from the equation. In practical applications of Equation (7), one may train the model using a minimum number of seven experimental data points and then predict the rest of the required data in any solvent composition and temperature of interest, as has been exemplified in a previous communication [40]. When the model was trained with the solubility data in ethanol and water at T = (293.15 and 323.15) K (the lowest and highest temperatures) and in w₁ = 0.3, 0.5 and 0.7 at T = 298.15 (totally, 7 data points), the rest of data points were predicted with the MPD of 7.6% (N = 70). All produced MPD values rely on experimental relative standard deviation values (usually < 10.0%) and could be considered as an acceptable error level.

3.3. Solid Phases’ Analyses

The DSC thermograms of benzoic acid as the original untreated sample and after saturating it in neat water, in the aqueous mixture of w₁ = 0.50, and in neat ethanol, are depicted in Figure A1, Figure A2, Figure A3 and Figure A4 (Appendix A). There are two endothermic peaks corresponding to the melting and thermal degradation of benzoic acid. Observed on-set temperatures of melting varied from (394.8 to 395.1) K; whereas, enthalpies of melting varied from (20.49 to 21.04) kJ·mol⁻¹, which are in agreement with those reported earlier [41]. The X-ray diffraction spectra for benzoic acid without any treatment and after dissolving it in neat water, in the aqueous mixture of w₁ = 0.50, and in neat ethanol, are shown in Figure A5, Figure A6, Figure A7 and Figure A8 (Appendix B). As observed, the positions of the characteristic peaks are comparable in each of the samples analyzed and also are in agreement with those reported earlier [42]. Thus, it is observable that benzoic acid did not suffer crystal polymorphic transitions or solvate formation after dissolution and saturation in these solvent systems.

3.4. Ideal Solubility and Activity Coefficients of Benzoic Acid in Mixed Solvents

Ideal solubilities of benzoic acid ($x_3^{\mathrm{id}}$) at the temperatures of interest were calculated by means of:

$$\ln x_3^{\mathrm{id}}=-\frac{{\Delta }_{\mathrm{fus}}H(T_{\mathrm{fus}}-T)}{R{T}_{\mathrm{fus}}T}+\left(\frac{\Delta C_p}{R}\right)\left[\frac{(T_{\mathrm{fus}}-T)}{T}+\ln \left(\frac{T}{T_{\mathrm{fus}}}\right)\right]$$

(8)

Here, Δ_fusH is the molar enthalpy of melting of the pure benzoic acid (obtained at the melting point, i.e., 18.01 kJ·mol⁻¹ [39]), T_fus is the absolute melting point (i.e., 395.2 K [39]), T is the absolute dissolution temperature, R is the universal gas constant (8.3145 J·mol⁻¹·K⁻¹), and ΔC_p is the difference between the molar heat capacities of benzoic acid as crystalline form and its hypothetical super-cooled liquid form at the dissolution temperature [43]. However, due to the difficulty in ΔC_p determination, this value has been considered in this research as the same as the one of the entropy of fusion (Δ_fusS = Δ_fusH/T_fus, i.e., 45.7 J·mol⁻¹·K⁻¹). Table 2 shows that the ideal solubilities of benzoic acid are higher than the experimental solubilities at all the temperatures studied.

On the other hand,

Table 4. Activity coefficients of benzoic acid (γ₃) in {ethanol (1) + water (2)} mixtures at several temperatures and p = 90 kPa. ^a,c.

w1 b,c	x1 b,c	T/K c
		293.15	298.15	303.15	308.15	313.15	318.15	323.15
0.00	0.0000	413	379	351	324	298	272	251
0.10	0.0417	241	223	209	202	197	192	186
0.20	0.0891	123	121	117	115	112	108	106
0.30	0.1436	68.2	66.5	65.4	63.3	61.2	59.9	58.2
0.40	0.2068	39.4	37.9	36.7	36.0	35.1	33.8	32.9
0.50	0.2812	21.7	20.4	20.1	19.4	18.9	18.4	17.6
0.60	0.3698	12.1	11.3	11.0	10.7	10.4	10.1	9.78
0.70	0.4772	6.73	6.31	6.12	5.92	5.72	5.58	5.52
0.80	0.6101	3.85	3.75	3.62	3.56	3.46	3.36	3.24
0.90	0.7788	2.26	2.14	2.07	2.02	2.00	1.98	1.98
1.00	1.0000	1.18	1.15	1.13	1.12	1.10	1.06	1.03

^a p is the atmospheric pressure in Aksaray, Turkey. ^b w₁ and x₁ are the mass and mole fractions of ethanol (1) in the {ethanol (1) + water (2)} mixtures free of benzoic acid (3), respectively. ^c Standard uncertainty in T is u(T_hm) = 0.10 K. Standard uncertainty in p is u(p) = 3.0 kPa. Mean relative uncertainties in w₁ and x₁ were u(w₁) = 0.0008 and u(x₁) = 0.0008. Standard uncertainty in T is u(T) = 0.10 K. Average relative uncertainty in γ₃ is u_r(γ₃) = 0.028.

summarizes the asymmetrical activity coefficients of benzoic acid (γ₃) in neat solvents and in {ethanol (1) + water (2)} mixtures at all temperatures. The γ₃ values were calculated as the quotient: $x_3^{\mathrm{id}}/x_3^{}$ from the experimental and ideal benzoic acid solubilities summarized in Table 2. As observed, at T = 298.15 K γ₃ values vary from 379 in neat water (where the minimum benzoic acid solubility is obtained) to 1.15 in neat ethanol (where the maximum benzoic acid solubility is achieved). In all solvent systems, the γ₃ values decrease with the temperature-arising. All the obtained γ₃ values are higher than the unit because the experimental solubilities in all the solvent systems are lower than $x_3^{\mathrm{id}}$ at all temperatures tested. However, in neat ethanol, the γ₃ values are near unit, indicating nearly-ideal behavior of benzoic acid in this solvent. Furthermore, a rough estimate of the respective solute-solvent intermolecular interactions was performed from the γ₃ values, based on [44]:

$$\ln {\gamma }_3=(e_{ss}+e_{33}-2e_{s3})\frac{V_3{\phi }_s^2}{RT}$$

(9)

Here, subscript s stands for the solvent system (which corresponds to neat solvents or aqueous–ethanol binary mixtures), e_ss, e₃₃ and e_s3 represent the solvent–solvent, solute–solute and solvent–solute interaction energies, respectively. However, it is important to note that in multicomponent systems like ethanol–water–benzoic acid, some water–cosolvent interactions are present, which could also play an important role in the magnitudes of equilibrium solubility and dissolution rate of drugs. V₃ is the molar volume of the super-cooled liquid benzoic acid and φ_s is the volume fraction of the solvent system in every saturated solution. It is noteworthy that when low x₃ values are obtained, the term (V₃φ_s²/RT) can be considered as almost constant regardless of the solvent system composition because φ_s is almost invariant and near to 1.0. Thus, the γ₃ values would depend mainly on the terms e_ss, e₃₃ and e_s3 [44]. As well-known, e_ss and e₃₃ are unfavorable for drug dissolution and equilibrium solubility, but e_s3 favors the respective drug dissolution processes. The contribution of e₃₃ towards the equilibrium solubility and dissolution rate of benzoic acid was considered to be almost constant regardless of the solvent system studied.

As indicated above, a qualitative approach to intermolecular interactions was made based on the e_ss, e₃₃ and e_s3 energetic terms of Equation (9). Hence, based on polarities, it follows that e_ss is highest in neat water (δ = 47.8 MPa^1/2) and lowest in neat ethanol (δ = 26.5 MPa^1/2) [27,28]. Neat water and water-rich mixtures, exhibiting γ₃ values higher than 250, would imply high e_ss and low e_s3 values. Otherwise, in ethanol-rich mixtures exhibiting γ₃ values lower than 2.5, the e_ss values are relatively low and the e_s3 values would be comparatively high regarding water-rich mixtures.

3.5. Apparent Thermodynamic Functions of Dissolution

All the apparent thermodynamic quantities of dissolution of benzoic acid in neat solvents and in {ethanol (1) + water (2)} mixtures were estimated at T = 298.15 K. In this way, the apparent standard enthalpic changes for benzoic acid dissolution processes (Δ_solnH°) were obtained by means of the modified van ’t Hoff equation, as [45,46]:

$${\left(\frac{\partial \ln x_3}{\partial \left(1/(T/\mathrm{K})-1/298.15 \mathrm{K}\right)}\right)}_P=-\frac{({\Delta }_{\mathrm{soln}}H°/\mathrm{J}\cdot {\mathrm{mol}}^{-1})}{R}$$

(10)

The apparent standard Gibbs energy changes for the benzoic acid dissolution processes (Δ_solnG°) were calculated by means of:

$${\Delta }_{\mathrm{soln}}G°=-R\cdot 298.15 \mathrm{K}\cdot \mathrm{intercept}$$

(11)

Here, the intercepts used are those obtained in the respective linear regressions of ln x₃ vs. (1/T − 1/298.15 K). As visual help, Figure 3 depicts the solubility van ’t Hoff plots obtained in the neat solvents, water and ethanol, and in the nine {ethanol (1) + water (2)} mixtures. Linear regressions with determination coefficients higher than 0.99 were obtained in all cases [47,48,49]. Standard apparent entropic changes for benzoic acid dissolution processes (Δ_solnS°) were obtained from the respective Δ_solnH° and Δ_solnG° values by using Equation (12) [50]:

$${\Delta }_{\mathrm{soln}}{S}_{}^{\mathrm{o}}/\mathrm{J}\cdot {\mathrm{mol}}^{-1}\cdot {\mathrm{K}}^{-1}=\frac{\left({\Delta }_{\mathrm{soln}}H°/\mathrm{J}\cdot {\mathrm{mol}}^{-1}-{\Delta }_{\mathrm{soln}}G°/\mathrm{J}\cdot {\mathrm{mol}}^{-1}\right)}{298.15\mathrm{K}}$$

(12)

Table 5. Apparent thermodynamic functions relative to dissolution processes of benzoic acid (3) in {ethanol (1) + water (2)} mixtures at T = 298.15 K and p = 90 kPa. ^a,c.

w1 b,c	x1 b,c	ΔsolnG°/kJ·mol−1 c	ΔsolnH°/kJ·mol−1 c	ΔsolnS°/J·mol−1·K−1 c	TΔsolnS°/kJ·mol−1 c	ζH d	ζTS d
0.00	0.0000	18.57	27.15	28.79	8.58	0.760	0.240
0.10	0.0417	17.25	20.45	10.72	3.20	0.865	0.135
0.20	0.0891	15.72	18.19	8.30	2.48	0.880	0.120
0.30	0.1436	14.25	18.29	13.57	4.04	0.819	0.181
0.40	0.2068	12.86	18.60	19.26	5.74	0.764	0.236
0.50	0.2812	11.35	19.07	25.89	7.72	0.712	0.288
0.60	0.3698	9.89	19.30	31.55	9.41	0.672	0.328
0.70	0.4772	8.43	19.19	36.09	10.76	0.641	0.359
0.80	0.6101	7.11	18.46	38.06	11.35	0.619	0.381
0.90	0.7788	5.74	17.41	39.15	11.67	0.599	0.401
1.00	1.0000	4.21	17.53	44.70	13.33	0.568	0.432
	Ideal	3.84	14.06	34.28	10.22	0.579	0.421

^a p is the atmospheric pressure in Aksaray, Turkey. ^b w₁ and x₁ are the mass and mole fractions of ethanol (1) in the {ethanol (1) + water (2)} mixtures free of benzoic acid (3), respectively. ^c Standard uncertainty in T is u(T) = 0.1 K. Standard uncertainty in p is u(p) = 3.0 kPa. Mean relative uncertainties in w₁ and x₁ were u(w₁) = 0.0008 and u(x₁) = 0.0008. Average relative standard uncertainty in apparent thermodynamic quantities of real dissolution processes are u_r(Δ_solnG°) = 0.027, u_r(Δ_solnH°) = 0.035, u_r(Δ_solnS°) = 0.044, u_r(TΔ_solnS°) = 0.044. ^d ζ_H and ζ_TS are the relative contributions by enthalpy and entropy toward apparent Gibbs energy of dissolution.

summarizes the standard apparent molar thermodynamic functions for dissolution processes of benzoic acid (3) in neat solvents and in all the {ethanol (1) + water (2)} mixtures at T = 298.15 K.

Apparent standard Gibbs energies, enthalpies and entropies relative to the benzoic acid dissolution processes are positive in all cases, as shown in Table 5. This implies endothermic and entropy-driven dissolution processes. Moreover, Δ_solnG° values decrease continuously from neat water to neat ethanol indicating more affinity of benzoic acid by semipolar solvent media like the one corresponding to neat ethanol or even by less polar solvents if they were analyzed. Relative contributions by enthalpy (ζ_H) and entropy (ζ_TS) toward the benzoic acid dissolution processes were calculated by means of [51]:

$${\zeta }_H=\frac{\left|{\Delta }_{\mathrm{soln}}H°\right|}{\left|{\Delta }_{\mathrm{soln}}H°\right|+\left|T{\Delta }_{\mathrm{soln}}S°\right|}$$

(13)

$${\zeta }_{TS}=\frac{\left|T{\Delta }_{\mathrm{soln}}S°\right|}{\left|{\Delta }_{\mathrm{soln}}H°\right|+\left|T{\Delta }_{\mathrm{soln}}S°\right|}$$

(14)

As shown in Table 5, the main contributor to the positive standard apparent molar Gibbs energies of dissolution of benzoic acid was the positive enthalpy, with ζ_H values higher than 0.56, which demonstrates the energetic predominance in all these benzoic acid dissolution processes.

3.6. Apparent Thermodynamic Quantities of Mixing

Global dissolution processes of benzoic acid in {ethanol (1) + water (2)} mixtures may be represented by means of the following hypothetical process:

Solute_(Solid) at T → Solute_(Solid) at T_fus → Solute_(Liquid) at T_fus → Solute_(Liquid) at T → Solute_(Solution) at T

Here, the hypothetical stages are as follows: (i) the heating and melting of benzoic acid at T_fus = 395.2 K [41], (ii) the cooling of the liquid benzoic acid to the considered temperature (T = 298.15 K), and (iii) the subsequent mixing of both the hypothetical benzoic acid super-cooled liquid and the solvent system under consideration at T = 298.15 K [52]. This allowed the calculation of the individual thermodynamic contributions by fusion and mixing toward the overall benzoic acid dissolution processes, by means of:

$${\Delta }_{\mathrm{soln}}H°={\Delta }_{\mathrm{fus}}{H}^{298.15}+{\Delta }_{\mathrm{mix}}H°$$

(15)

$${\Delta }_{\mathrm{soln}}S°={\Delta }_{\mathrm{fus}}{S}^{298.15}+{\Delta }_{\mathrm{mix}}S°$$

(16)

where ${\Delta }_{\mathrm{fus}}{H}^{298.15}$ and ${\Delta }_{\mathrm{fus}}{S}^{298.15}$ indicate the thermodynamic quantities of benzoic acid fusion and its cooling at T = 298.15 K. In turn, these two functions were calculated by means of Equations (17) and (18), respectively [53]:

$${\Delta }_{\mathrm{fus}}{H}^{298.15}={\Delta }_{\mathrm{fus}}{H}^{T_{\mathrm{fus}}}-\Delta C_p\left(T_{\mathrm{fus}}-298.15 \mathrm{K}\right)$$

(17)

$${\Delta }_{\mathrm{fus}}{H}^{298.15}={\Delta }_{\mathrm{fus}}{H}^{T_{\mathrm{fus}}}-\Delta C_p\left(T_{\mathrm{fus}}-298.15 \mathrm{K}\right)$$

(18)

Table 6. Apparent thermodynamic functions relative to mixing processes of benzoic acid (3) in {ethanol (1) + water (2)} mixtures at T = 298.15 K and p = 90 kPa. ^a,c.

w1 b,c	x1 b,c	ΔmixG°/kJ·mol−1 c	ΔmixH°/kJ·mol−1 c	ΔmixS°/J·mol−1·K−1 c	TΔmixS°/kJ·mol−1 c	ζH d	ζTS d
0.00	0.0000	14.73	13.09	−5.49	−1.64	0.889	0.111
0.10	0.0417	13.41	6.39	−23.56	−7.02	0.476	0.524
0.20	0.0891	11.88	4.14	−25.98	−7.74	0.348	0.652
0.30	0.1436	10.41	4.23	−20.71	−6.18	0.407	0.593
0.40	0.2068	9.02	4.54	−15.01	−4.48	0.504	0.496
0.50	0.2812	7.51	5.01	−8.39	−2.50	0.667	0.333
0.60	0.3698	6.05	5.24	−2.73	−0.81	0.866	0.134
0.70	0.4772	4.59	5.13	1.82	0.54	0.905	0.095
0.80	0.6101	3.27	4.40	3.78	1.13	0.796	0.204
0.90	0.7788	1.90	3.35	4.87	1.45	0.698	0.302
1.00	1.0000	0.37	3.47	10.42	3.11	0.528	0.472

^a p is the atmospheric pressure in Aksaray, Turkey. ^b w₁ and x₁ are the mass and mole fraction of ethanol (1) in the {ethanol (1) + water (2)} mixtures free of benzoic acid (3). ^c Standard uncertainty in T is u(T) = 0.1 K. Standard uncertainty in p is u(p) = 3.0 kPa. Mean relative uncertainties in w₁ and x₁ were u(w₁) = 0.0008 and u(x₁) = 0.0008. Average relative standard uncertainty in apparent thermodynamic quantities of mixing processes are u_r(Δ_mixG°) = 0.029, u_r(Δ_mixH°) = 0.038, u_r(Δ_mixS°) = 0.048, u_r(TΔ_mixS°) = 0.048. ^d ζ_H and ζ_TS are the relative contributions by enthalpy and entropy toward apparent Gibbs energy of mixing.

summarizes the apparent thermodynamic quantities of mixing of the hypothetical benzoic acid as a super-cooled liquid with all the aqueous–ethanol mixtures and the neat solvents at T = 298.15 K. The apparent Gibbs energies of mixing are positive because the experimental solubilities of benzoic acid are lower than the ideal ones at all temperatures, as indicated above. Otherwise, Δ_mixH° are positive in all cases, but Δ_mixS° are negative from neat water to the mixture of w₁ = 0.60, but positive from the mixture of w₁ = 0.70 to neat ethanol. In this way, non-enthalpy-nor entropy driving is observed for benzoic acid mixing processes in the composition interval 0.0 ≤ w₁ ≤ 0.6 (where Δ_mixH° > 0 and Δ_mixS° < 0); whereas, in the interval 0.7 ≤ w₁ ≤ 1.0, the mixing processes are entropy-driven (where Δ_mixH° > 0 and Δ_mixS° > 0). Moreover, to compare the relative contributions of enthalpy (ζ_H) and entropy (ζ_TS) to the mixing processes, two equations analogous to Equations (13) and (14) were also employed. Thus, in almost all cases the main contributor to apparent Gibbs energy of mixing is the apparent mixing enthalpy, except in the interval 0.1 ≤ w₁ ≤ 0.3, where the apparent mixing entropy is the dominant thermodynamic function.

3.7. Thermodynamic Quantities of Solvation

Furthermore, in addition to the previous fusion-mixing subprocesses, the overall dissolution process may also be represented by the hypothetic stages: Solute_(Solid) → Solute_(Vapor) → Solute_(Solution); where, the respective partial processes toward the dissolution process are sublimation and solvation of benzoic acid [54]. This treatment allowed the calculation of the partial thermodynamic contributions to the dissolution process by means of Equations (19) and (20), respectively, while the Gibbs energy of solvation was calculated by means of Equation (21):

Δ_solnH° = Δ_sublH° + Δ_solvH°

(19)

Δ_solnS° = Δ_sublS° + Δ_solvS°

(20)

Δ_solnG° = Δ_sublG° + Δ_solvG°

(21)

where, Δ_sublG° = 5.58 kJ·mol⁻¹ and Δ_sublH° = 92.11 kJ·mol⁻¹ at 298.15 K as reported by Kruif and Blok [55]. Thermodynamic functions of solvation are summarized in

Table 7. Apparent thermodynamic functions relative to solvation processes of benzoic acid (3) in {ethanol (1) + water (2)} mixtures at T = 298.15 K and p = 90 kPa. ^a,c.

w1 b,c	x1 b,c	ΔsolvG°/kJ·mol−1 c	ΔsolvH°/kJ·mol−1 c	ΔsolvS°/J·mol−1·K−1 c	TΔsolvS°/kJ·mol−1 c	ζH d	ζTS d	εH e	εS e
0.00	0.0000	12.99	−64.96	−261.43	−77.94	0.455	0.545	0.666	0.795
0.10	0.0417	11.67	−71.66	−279.49	−83.33	0.462	0.538	0.838	0.924
0.20	0.0891	10.14	−73.91	−281.91	−84.05	0.468	0.532	0.896	0.941
0.30	0.1436	8.67	−73.81	−276.65	−82.48	0.472	0.528	0.893	0.903
0.40	0.2068	7.28	−73.50	−270.95	−80.78	0.476	0.524	0.885	0.863
0.50	0.2812	5.77	−73.03	−264.32	−78.81	0.481	0.519	0.873	0.816
0.60	0.3698	4.31	−72.81	−258.66	−77.12	0.486	0.514	0.868	0.775
0.70	0.4772	2.85	−72.91	−254.12	−75.77	0.490	0.510	0.870	0.743
0.80	0.6101	1.53	−73.65	−252.15	−75.18	0.495	0.505	0.889	0.729
0.90	0.7788	0.16	−74.69	−251.06	−74.85	0.499	0.501	0.916	0.721
1.00	1.0000	−1.37	−74.57	−245.51	−73.20	0.505	0.495	0.913	0.682

^a p is the atmospheric pressure in Aksaray, Turkey. ^b w₁ and x₁ are the mass and mole fraction of ethanol (1) in the {ethanol (1) + water (2)} mixtures free of benzoic acid (3). ^c Standard uncertainty in T is u(T) = 0.1 K. Standard uncertainty in p is u(p) = 3.0 kPa. Mean relative uncertainties in w₁ and x₁ were u(w₁) = 0.0008 and u(x₁) = 0.0008. Average relative standard uncertainty in apparent thermodynamic quantities of mixing processes are u_r(Δ_solvG°) = 0.032, u_r(Δ_solvH°) = 0.042, u_r(Δ_solvS°) = 0.053, u_r(TΔ_solvS°) = 0.053. ^d ζ_H and ζ_TS are the relative contributions of enthalpy and entropy toward the apparent Gibbs energy of mixing. ^e ε_H and ε_S are the relative ratios of specific and non-specific solute–solvent interactions expressed in terms of enthalpy and entropy.

, where negative enthalpies and entropies of solvation were obtained. On the other hand, with the aim of comparing the relative contributions of enthalpy (ζ_H) and entropy (ζ_TS) toward the solvation process, two equations analogous to Equations (13) and (14) were employed again.

From the ζ_H and ζ_TS values presented in Table 7, it follows that enthalpy is the main contributing force to the standard Gibbs energy of the solvation process of benzoic acid in every solvent mixture (ζ_H values are higher than 0.66 in all solvent systems). Moreover, because the balance between specific and non-specific solute–solvent interactions are also important, some parameters describing the relative ratio of specific and non-specific solute-solvent interaction in terms of enthalpies (ε_H) and entropies (ε_S) were also calculated, according to the following definitions [51]:

$${\epsilon }_H=\left|\frac{{\Delta }_{\mathrm{spec}}H°}{{\Delta }_{\mathrm{non}\text{-}\mathrm{spec}}H°}\right|$$

(22)

$${\epsilon }_S=\left|\frac{{\Delta }_{\mathrm{spec}}S°}{{\Delta }_{\mathrm{non}\text{-}\mathrm{spec}}S°}\right|$$

(23)

where, Δ_specH° = Δ_{soln(solvent-i)}H° − Δ_soln(CH)H° = Δ_{soln(CH → solvent-i)}H°, Δ_non-specH° = Δ_soln(CH)H° − Δ_sublH° = Δ_solv(CH)H°, Δ_specS° = Δ_{soln(solvent-i)}S° − Δ_soln(CH)S° = Δ_{soln(CH → solvent-i)}S°, and Δ_non-specS° = Δ_soln(CH)S°.

Cyclohexane (CH) was chosen as an “inert” solvent because it interacts with benzoic acid molecules mainly by non-specific interactions like London dispersion forces, while the cosolvent mixtures studied here interact with benzoic acid by specific interactions such as hydrogen bonding. Dissolution thermodynamics quantities of benzoic acid in CH were calculated by means of the modified van ’t Hoff and Gibbs equations from solubility values taken from the literature [6]. The respective values for apparent dissolution thermodynamic functions are: Δ_soln(CH)G° = 11.25 kJ·mol⁻¹, Δ_soln(CH)H° = 53.12 kJ·mol⁻¹, and Δ_soln(CH)S° = 140.43 J·mol⁻¹·K⁻¹. In this way, the ε_H and ε_S values for benzoic acid solvation are also presented in Table 7. These values indicate that during the dissolution process of this chemical in all solvent systems, the specific solute-solvent interactions (hydrogen bonding, mainly) effectively affect the entropic term of Gibbs energy with respect to non-specific interactions.

3.8. Enthalpy-Entropy Compensation Analysis

Extra-thermodynamic studies, which include enthalpy–entropy compensation analysis, provide powerful tools to inquire into the main molecular mechanisms implicated in several physical and chemical processes that involve organic compounds like a variety of drugs [56,57]. Hence, non-enthalpy-entropy compensation effects have been associated to the dissolution processes of several organic compounds in {ethanol (1) + water (2)} mixtures as has been summarized earlier [58]. These physicochemical studies were performed in order to identify the main mechanisms involved in the cosolvent action of ethanol for increasing or decreasing the solubility depending on the solvent mixtures composition. Normally, weighted plots of Δ_solnH° vs. Δ_solnG° have been used for performing these analyses [59,60,61]. In particular, Figure 4 clearly shows that benzoic acid exhibits a non-linear Δ_solnH° vs. Δ_solnG° trend defined by the quotient polynomial shown as Equation (24), where the obtained statistical parameters were as follows: adjusted r² = 0.982, typical error = 0.335, and F = 161.3. Thus, variant-positive slopes are observed from neat water to the mixture of w₁ = 0.20 and the composition interval of 0.60 < w₁ < 0.90, whereas variant-negative slopes are observed in the composition interval of 0.20 < w₁ < 0.60 and from the mixture of w₁ = 0.90 to neat ethanol. Accordingly, in the first cases, the benzoic acid transfer processes are entropy-driven, whereas, in the other cases, the driving mechanism for the transfer processes of benzoic acid from the more polar to the less polar solvent systems is entropy.

Δ_solnH° = 37.40 − 10.69·(Δ_solnG°) + 1.96·(Δ_solnG°)² − 0.142·(Δ_solnG°)³ + 3.56·10⁻³·(Δ_solnG°)⁴

(24)

3.9. Preferential Solvation of Benzoic Acid

The preferential solvation parameters of benzoic acid (identified as compound 3) by ethanol (identified as compound 1) molecules in the different {ethanol (1) + water (2)} mixtures (δx_1,3), are defined as [2,3,4]:

$$\delta x_{1,3}=x_{1,3}^L-x_1=-\delta x_{2,3}$$

(25)

where $x_{1,3}^L$ is the local mole fraction of ethanol in the molecular environment of benzoic acid and x₁ is the bulk mole fraction of ethanol in the initial {ethanol (1) + water (2)} binary solvent mixture free of benzoic acid. Thus, if the δx_1,3 value is positive, benzoic acid molecules are preferentially solvated by ethanol molecules in the solutions. In contrast, benzoic acid molecules are preferentially solvated by water molecules if this δx_1,3 parameter is negative. The values of δx_1,3 were obtained from the inverse Kirkwood-Buff integrals (IKBI) as described earlier [2,3,4]:

$$\delta x_{1,3}=\frac{x_1{x}_2\left(G_{1,3}-G_{2,3}\right)}{x_1{G}_{1,3}+x_2{G}_{2,3}+V_{\mathrm{cor}}}$$

(26)

with,

$$G_{1,3}=RT{\kappa }_T-{\overline{V}}_3+x_2{\overline{V}}_{2}D/Q$$

(27)

$$G_{2,3}=RT{\kappa }_T-{\overline{V}}_3+x_1{\overline{V}}_{1}D/Q$$

(28)

$$V_{\mathrm{cor}}=2522.5{\left(r_3+0.1363{\left(x_{1,3}^L{\overline{V}}_1+x_{2,3}^L{\overline{V}}_2\right)}^{1/3}-0.085\right)}^3$$

(29)

Here, κ_T denotes the isothermal compressibility of the aqueous-ethanol mixtures. ${\overline{V}}_1$, ${\overline{V}}_2$ and ${\overline{V}}_3$ are respectively the partial molar volumes of ethanol, water, and benzoic acid in the dissolutions. The function D, defined in Equation (30), corresponds to the first derivative of the standard molar Gibbs energies of transfer of benzoic acid from neat water to every aqueous-ethanol mixture regarding the mole fraction of ethanol. The function Q, defined in Equation (31), involves the second derivative of the excess molar Gibbs energy of mixing of ethanol and water ($G_{1+2}^{Exc}$) regarding the mole fraction of water [2,3,4]. V_cor and r₃ are, respectively, the correlation volume and the molecular radius of benzoic acid. Here, r₃ was roughly calculated by means of Equation (32), where N_Av is the number of Avogadro.

$$D={\left(\frac{\partial {\Delta }_{\mathrm{tr}}{G}_{3,2\to 1+2}^{\mathrm{o}}}{\partial x_1}\right)}_{T,p}$$

(30)

$$Q=RT+x_1{x}_2{\left(\frac{{\partial }^2{G}_{1+2}^{Exc}}{\partial x_2^2}\right)}_{T,p}$$

(31)

$$r_3={\left(\frac{3\cdot {10}^{21}{V}_3}{4\pi N_{\mathrm{Av}}}\right)}^{1/3}$$

(32)

To obtain definitive V_cor values, iteration processes are required because they depend on the local mole fractions of ethanol and water around the benzoic acid molecules in the respective solutions. Thus, these iteration processes were performed by replacing δx_1,3 and V_cor in the Equations (25), (26) and (29) to recalculate the $x_{1,3}^L$ values until obtaining non-variant values of V_cor.

Figure 5 shows the apparent Gibbs energies of transfer of benzoic acid from neat water to all aqueous-ethanol mixtures (${\Delta }_{\mathrm{tr}}{G}_{3,2\to 1+2}^{\mathrm{o}}$) at T = 298.15 K. These ${\Delta }_{\mathrm{tr}}{G}_{3,2\to 1+2}^{\mathrm{o}}$ values were calculated from the experimental mole fraction solubility values reported in Table 2 by using:

$${\Delta }_{\mathrm{tr}}{G}_{3,2\to 1+2}^{\mathrm{o}}=RT\ln \left(\frac{x_{3,2}}{x_{3,1+2}}\right)$$

(33)

Obtained ${\Delta }_{\mathrm{tr}}{G}_{3,2\to 1+2}^{\mathrm{o}}$ values were correlated by means of the regular polynomial shown as Equation (34), where the obtained statistical parameters were as follows: adjusted r² = 0.9999, typical error = 0.0421, and F = 32,256.

$${\Delta }_{\mathrm{tr}}{G}_{3,2\to 1+2}^{\mathrm{o}}=0.02-34.92x_1^{}+38.11x_1^2-21.95x_1^3+4.37x_1^4$$

(34)

The D values summarized in

Table 8. Some properties associated to preferential solvation of benzoic acid (3) in {ethanol (1) + water (2)} mixtures at T = 298.15 K.

x1 a	D/kJ·mol−1	G1,3/cm3·mol−1	G2,3/cm3·mol−1	Vcor/cm3·mol−1	100 δx1,3
0.00	−34.92	−353.4	−98.8	583	0.00
0.05	−31.28	−340.2	−136.4	601	−2.13
0.10	−27.94	−320.5	−173.1	628	−3.01
0.15	−24.91	−297.0	−206.3	663	−2.61
0.20	−22.17	−272.3	−234.6	704	−1.31
0.25	−19.71	−248.6	−258.1	748	0.36
0.30	−17.51	−227.2	−277.8	791	2.01
0.35	−15.56	−208.7	−295.1	832	3.46
0.40	−13.85	−193.3	−311.9	873	4.68
0.45	−12.36	−180.6	−329.7	911	5.69
0.50	−11.09	−170.3	−350.3	949	6.54
0.55	−10.01	−161.8	−375.1	985	7.26
0.60	−9.12	−154.4	−404.3	1021	7.83
0.65	−8.40	−147.5	−436.3	1055	8.15
0.70	−7.83	−140.0	−464.3	1085	8.03
0.75	−7.42	−131.2	−476.7	1112	7.24
0.80	−7.13	−121.4	−461.8	1134	5.76
0.85	−6.97	−112.2	−419.6	1154	3.94
0.90	−6.91	−104.9	−363.8	1174	2.23
0.95	−6.94	−100.0	−309.4	1197	0.92
1.00	−7.05	−97.0	−264.0	1222	0.00

^a x₁ is the mole fraction of ethanol (1) in the {ethanol (1) + water (2)} mixtures free of benzoic acid (3).

were calculated as the first derivative of Equation (34) solved in mixture composition steps of x₁ = 0.05. For the studied aqueous-ethanol mixtures, the Q, RTκ_T, ${\overline{V}}_1$ and ${\overline{V}}_2$ values at T = 298.15 K were taken from the literature [62,63]. In this research, the ${\overline{V}}_3$ value was considered the same as the one calculated by using the Fedors method (i.e., 99.9 cm³·mol⁻¹, Table 3). Table 8 also shows that the G_1,3 and G_2,3 values are negative in all the mixed-solvent systems. The benzoic acid r₃ value was calculated as 0.341 nm. As indicated above, the V_cor values shown in Table 8 were obtained after three iterations. V_cor values increase with the ethanol-proportion in the mixtures because the ${\overline{V}}_1$ values are higher than the ${\overline{V}}_2$ values in all cases. Moreover, Table 8 also summarizes the preferential solvation parameters of benzoic acid by ethanol molecules (δx_1,3) in all the aqueous-ethanol mixtures at T = 298.15 K.

Figure 6 shows a non-linear variation of benzoic acid δx_1,3 values regarding the ethanol proportion in the solvent mixtures as expressed by their mole fractions before solute adding. Initially, the addition of ethanol to neat water as solvent makes negative the δx_1,3 values of benzoic acid in the composition interval of 0.00 < x₁ < 0.24. The maximum negative δx_1,3 value is obtained in the mixture of x₁ = 0.10 (i.e., δx_1,3 = –3.01 × 10⁻²), which is higher than |1.0 × 10⁻²|. Hence, this result could be considered as a consequence of real preferential solvation effects of benzoic acid by water molecules, rather than a consequence of merely the uncertainties propagation associated to IKBI calculations [64,65]. Possibly the structuring of water molecules around the phenyl ring of this compound by hydrophobic hydration contributes to the lowering of the net δx_1,3 to negative values in these water-rich mixtures.

In the mixtures’ composition interval of 0.24 < x₁ < 1.00 the local mole fractions of ethanol around benzoic acid molecules are higher than those in the bulk aqueous-ethanol mixtures free of this drug. The maximum positive δx_1,3 value is obtained in the mixture of x₁ = 0.65 (i.e., δx_1,3 = 8.15 × 10⁻²), which is also higher than |1.0 × 10⁻²|. Thus, it could be considered as a consequence of preferential solvation effects of this drug by ethanol molecules [64,65]. In this composition interval, benzoic acid could be acting as a Lewis acid with ethanol molecules by means of its carboxylic hydroxyl group that would be interacting with the unshared electron pairs of the oxygen atoms of ethanol. It is noteworthy that ethanol exhibits higher Lewis base behavior compared with water [30].

4. Conclusions

Dissolution physicochemical properties of benzoic acid in aqueous-ethanol binary mixtures depend strongly on the ethanol proportion in the cosolvent mixtures. The experimental mole fraction solubility values of benzoic acid in these mixtures were adequately correlated with the classical Jouyban–Acree model and other well-known correlation models. The apparent thermodynamic quantities of dissolution, mixing and solvation of benzoic acid in aqueous-ethanol mixtures were calculated based on the van ’t Hoff and Gibbs equations. Non-linear enthalpy-entropy compensation was found for benzoic acid in these aqueous-ethanol mixtures, indicating different mechanisms for the drug transfer depending on the solvent mixture composition. Moreover, based on IKBI calculations, it was stated that benzoic acid is preferentially solvated by water molecules in water-rich mixtures but preferentially solvated by ethanol molecules in the composition interval of 0.24 < x₁ < 1.00. Finally, the thermodynamic results presented in this communication could be useful in optimizing different physical and chemical processes involving benzoic acid in mixed aqueous–ethanol media at the industrial level.