Thermodynamic Analysis of the Solubility of Propylparaben in Acetonitrile–Water Cosolvent Mixtures

Abstract

Parabens are substances used in the food, pharmaceutical and cosmetic industries. Recent studies have indicated that these substances have toxic potential, cause endocrine disruption and can easily bioaccumulate; therefore, their physicochemical properties are of industrial, biological and environmental interest. Due to their potential use in the development of more efficient and cleaner processes, the design of environmental recovery strategies and more reasonable designs for solubility in cosolvent mixtures, studies of thermodynamic analysis and mathematical modeling are of great interest. This research studies the solubility of propylparaben in acetonitrile + water cosolvent mixtures at nine temperatures by UV/Vis spectrophotometry, analyzing the solid phase by differential scanning calorimetry to evaluate possible polymorphic changes. The solubility of propylparaben is an endothermic process, where phase separation occurs at intermediate mixtures, reaching its minimum solubility in pure water at 278.15 K and the maximum solubility in pure acetonitrile at 315.15 K. The experimental data are well-correlated with the va not Hoff, Apelblat and Buchowski–Ksiazaczak models. The results revealed that possible microheterogeneity of the MeCN + W mixture can generate phase separation in intermediate mixtures, possibly due to the formation of solvates or hydrates.

1. Introduction

Propylparaben (PP), (C${}_{10}$H${}_{12}$O${}_3$, CAS Number: 94-13-3, Figure 1) is a propyl ester of parahydroxybenzoic acid that is widely used in the pharmaceutical, cosmetic and food industries due to its low toxicity and broad antibacterial and antifungal spectrum over a wide pH range [1,2,3,4].

Due to the low solubility of PP in water, solubility studies in organic solvents and cosolvent mixtures are crucial lines of research. These studies aim to devise strategies to improve PP solubility, considering the importance of this physicochemical property in processes, such as recrystallization, purification, quality analysis, quantification, preformulation studies and pharmaceutical dosage design [6,7,8,9,10].

Regarding environmental factors, solubility data allow for the design of more efficient processes, which means a reduction in waste and, therefore, a reduction in environmental impact, which is an issue that has become highly relevant and where many researchers have focused their attention [11,12]. In addition, some environmental parameters are calculated from the solubility data, such as Abraham parameters, which are used for the calculation of the lethal median molar concentration, bioaccumulation analysis and partitioning processes [13,14].

Cosolvency is one of the main strategies used to increase the solubility of poorly water-soluble drugs since the addition of an organic solvent miscible with water can favor water destructuration. This reduces the “squeezing out” effect, in addition to reducing the polarity of the cosolvent system, thereby, favoring non-polar interactions [15,16,17,18].

One of the most widely used cosolvent mixtures in the pharmaceutical industry is {acetonitrile (MeCN) + water (W)} [19,20]. MeCN is an aprotic polar solvent, miscible with water in all proportions and used to dissolve semipolar substances, which theoretically makes it a suitable choice to develop aqueous mixtures, which are widely used in HPLC. On the other hand, water is the green solvent par excellence, and water solubility data are highly relevant in the development of pharmaceutical products [7].

Broadly speaking, solubility data are the baseline for many industrial processes and the key to understanding the design of pharmaceuticals and cosmetics. Therefore, solubility studies are essential to the industry for routine process development, quantification, quality control and for research and development purposes. The aim of this research was to thermodynamically analyze the solubility of PP in cosolvent mixtures {MeCN (1) + W (2)} at nine temperatures, where one of the most relevant results was the separation of liquid phases in intermediate cosolvent mixtures.

2. Materials and Methods

2.1. Materials

For the development of the research, propylparaben (PP) (purity > 0.990, CAS: 57-83-0, Sigma-Aldrich, Burlington, VT, USA) was used as the study drug and defined as compound 3; two solvents were also used, acetonitrile (MeCN) (purity > 0.998 CAS: 75-05-8, Merck Millipore, Burlington, VT, USA) defined as compound 1 and water (W) defined as compound 2.

2.2. Preparation of Solvent Mixtures

Nineteen cosolvent mixtures {MeCN (1) + W (2)} from 0.05 up to 0.95 in MeCN mass fraction were prepared in 15 mL capacity amber bottles using an analytical balance (RADWAG AS 220.R2, Torun, Poland) of four decimal places with a sensitivity of $\pm 0.0001$ g. Three samples of approximately 10.00 ± 0.00 g were prepared for each of the concentrations.

2.3. Solubility Determination

The determination of PP solubility in mixtures {MeCN (1) + W (2)} was performed following the protocol of the shake-flask method proposed by Higuchi and Connors [21,22,23]. The method was developed in three steps:

• Sample saturation: PP was added to each of the cosolvent mixtures {MeCN (1) + W (2)} until a saturated solution in equilibrium with the solid phase was obtained. The samples were subjected to ultrasound for 30 min and then placed in a recirculating water bath (thermostat) at each of the study temperatures (278.15, 283.15, 288.15, 293.15, 298.15, 303.15, 308.15, 313.15 and 318.15 K) for a period of 72 h (a constant concentration was verified).
• Phase separation: The filtration method was employed using membranes with a pore diameter of 0.45 $\mathsf{\mu }$m (Millipore Corp. Swinnex, Burlington, VT, USA). Using a syringe previously thermostatted to the study temperature, an aliquot of the supernatant was taken and then poured through the membrane. In order to reduce the possible effects of solute sorption on the filter membrane, the first drops of the filtrate were discarded.
• Saturated solution and solid phase analysis: The concentration of PP in each of the samples was determined by UV/Vis spectrophotometry. From the filtered aliquot, gravimetric dilutions were performed with NaOH 0.1 N (NaOH 0.1 N was used since mixing the saturated solution with the NaOH solution forms the sodium salt of PP, and as the salt is very soluble in aqueous media, the probability of PP precipitation decreases). After dilution, the samples were analyzed in a UV/Vis spectrophotometer UV/Vis (UV/VIS EMC-11- UV spectrophotometer, Dresden, Germany) at 256 nm (the wavelength of maximum absorbance) for the prior construction of the calibration curve. In order to identify possible polymorphic changes, solid samples in equilibrium were analyzed using differential scanning calorimetry (DSC) and X-ray powder diffraction (XRPD).

3. Results and Discussion

3.1. Solubility ($x_3$) of Propylparaben (3) in 10 Cosolvent Mixtures {Acetonitrile (1) + Water (2)}

Table 1 shows the solubility of PP (3) in cosolvent mixtures {MeCN (1) + W (2)} (Figure 2). In each case, the solubility of PP increases with increasing temperature suggesting an endothermic process. Furthermore, as the concentration of MeCN in the system increases from pure water to $w_1=0.20$, the solubility of PP increases, and from $w_1=0.20$ up to $w_1=0.80$, a separation of liquid phases is observed. The latter is possibly due to a higher affinity of PP for one of the solvents in the mixture (MeCN or W), which would disfavor the solvent–solvent interaction generating phase separation. Moreover, the mixture {MeCN (1) + W (2)} exhibits microheterogeneity in intermediate mixtures [24,25], which could be an additional factor contributing to phase separation.

This phenomenon has also been reported by other authors in solubility studies of parabens in the same cosolvent system. Romero-Nieto et al. reports phase separation in solubility studies of methylparabens [19] and ethylparabens [20] in cosolvent mixtures {MeCN (1) + W (2)}. Paruta et al. and Yang and Rasmuson also reported phase separation in solubility studies of parabens in cosolvent mixtures {dioxane (1) + water (2)} [26] and {ethanol (1) + water (2)} [3]; and Peña et al. and Paruta et al. have also reported this in solubility studies of other drugs, such as benzocaine, salicylic acid and xanthines in mixtures {dioxane (1) + water (2)} [27,28].

Within $w_1=0.80$ to $w_1=1.0$ cosolvent concentration range, the PP solubility increases from $w_1=0.80$ up to $w_1=0.90$, where the maximum solubility is reached; and from $w_1=0.90$ to pure MeCN, the PP solubility decreases indicating a negative cosolvent effect of MeCN within this range of concentration. According to Hildebrand, the maximum solubility is reached in cosolvent mixtures where the solubility parameter of the drug is similar to that of the cosolvent system [29].

In this case, the solubility parameter of PP is 25.4 MPa${}^{1/2}$ [4], and that of the cosolvent mixture ($w_1=0.9$) is 28.99 MPa${}^{1/2}$ [20], thus, showing the utility of Hildebrand’s theory when choosing solvents or cosolvent mixtures to improve the solubility of a substance.

Table 1. Experimental solubility of propylparaben (3) in {acetonitrile (1) + water (2)} cosolvent mixtures expressed as a mole fraction (${10}^4{x}_3$) at different temperatures and $p=96$ kPa ^ac.

${\mathit{w}}_1$ b	Temperatures
	278.15	283.15	288.15	293.15	298.15	303.15	308.15	313.15	318.15
0.00	0.1468	0.1873	0.2176	0.2804	0.3422	0.4265	0.5617	0.7132	0.842
0.05	0.1498	0.1995	0.2502	0.3121	0.3849	0.4743	0.5856	0.7717	0.8939
0.10	0.1603	0.2013	0.2543	0.343	0.4307	0.533	0.6699	0.8298	1.0109
0.15	0.1771	0.2149	0.2625	0.365	0.4679	0.5712	0.698	0.8956	1.1508
0.20	0.2053	0.2819	0.3545	0.4612	0.6085	0.7737	0.9582	1.2116	1.5053
0.80	387.3	502.2	631.8	813.6	959.2	1224.4	1527.4	1934.9	2238.3
0.85	419.9	576.8	741.8	1053.1	1342.9	1723.9	2322.8	3019.9	3642.3
0.90	438.0	613.2	795.4	1118.4	1477.2	1925.1	2535.5	3365.1	4148.8
0.95	429.5	572.0	786.6	1019.1	1364.4	1765.4	2363.6	2966.9	3804.4
1.00	403.2	516.1	693.4	817.8	1058.8	1373.0	1877.7	2115.1	2834.2

^a The atmospheric pressure in Neiva, Colombia. ^b The mass fraction of acetonitrile (1) in the acetonitrile (1) + water (2) mixtures free of propylparaben (3). ^c The standard uncertainty in p is u(p) = 3.0 kPa. The average relative standard uncertainty in w₁ is u_r(w₁) = 0.0008. The standard uncertainty in T is u(T) = 0.10 K. The average relative standard uncertainties in x₃ is u_r(x₃₍₁₊₂₎) = 0.025.

Table 1. Experimental solubility of propylparaben (3) in {acetonitrile (1) + water (2)} cosolvent mixtures expressed as a mole fraction (${10}^4{x}_3$) at different temperatures and $p=96$ kPa ^ac.

${\mathit{w}}_1$ b	Temperatures
	278.15	283.15	288.15	293.15	298.15	303.15	308.15	313.15	318.15
0.00	0.1468	0.1873	0.2176	0.2804	0.3422	0.4265	0.5617	0.7132	0.842
0.05	0.1498	0.1995	0.2502	0.3121	0.3849	0.4743	0.5856	0.7717	0.8939
0.10	0.1603	0.2013	0.2543	0.343	0.4307	0.533	0.6699	0.8298	1.0109
0.15	0.1771	0.2149	0.2625	0.365	0.4679	0.5712	0.698	0.8956	1.1508
0.20	0.2053	0.2819	0.3545	0.4612	0.6085	0.7737	0.9582	1.2116	1.5053
0.80	387.3	502.2	631.8	813.6	959.2	1224.4	1527.4	1934.9	2238.3
0.85	419.9	576.8	741.8	1053.1	1342.9	1723.9	2322.8	3019.9	3642.3
0.90	438.0	613.2	795.4	1118.4	1477.2	1925.1	2535.5	3365.1	4148.8
0.95	429.5	572.0	786.6	1019.1	1364.4	1765.4	2363.6	2966.9	3804.4
1.00	403.2	516.1	693.4	817.8	1058.8	1373.0	1877.7	2115.1	2834.2

^a The atmospheric pressure in Neiva, Colombia. ^b The mass fraction of acetonitrile (1) in the acetonitrile (1) + water (2) mixtures free of propylparaben (3). ^c The standard uncertainty in p is u(p) = 3.0 kPa. The average relative standard uncertainty in w₁ is u_r(w₁) = 0.0008. The standard uncertainty in T is u(T) = 0.10 K. The average relative standard uncertainties in x₃ is u_r(x₃₍₁₊₂₎) = 0.025.

When comparing the solubility data in water with those reported by other authors, a good match is observed with the data given by Alexander et al. [30], which present relative deviations between 2% and 8% ($\mathrm{RD}=|100(x_3-x_3^R)/x_3|$, where $x_3$ is the data reported in this paper, and $x_3^R$ is that reported in the literature). Furthermore, when comparing with the data reported by Prankerd [31], the most significant differences were found at 313.15 and 318.15 K, with RDs of 19% and 12%, respectively. At other temperatures, the RDs were between 2% and 8%. Regarding the data reported by Yang and Rasmuson [32], at 313.15 K, there is a significant difference with an RD of 42.9%.

Finally, the MeCN solubility data were compared with those reported by Ouyang et al. [33], and RDs between 23% and 31% were observed. These differences may be due to different equilibrium times, PP water content when preparing the sample, among other experimental factors. An important factor in drug solubility studies is the analysis of the solid phase in equilibrium with different cosolvent compositions in order to identify possible polymorphic changes.

Differential scanning calorimetry (DSC) and X-ray powder diffraction (XRPD) analyses of the commercial sample and three samples of solid phases in equilibrium with water ($w_1=0.80$) and MeCN are presented in Figure 3 and Figure 4. According to the DSC analysis (Table 2), PP does not show polymorphic transitions, since in each case, both the temperature and enthalpy of melting are similar. When comparing the DSC analysis results with the data reported in the literature, good similarity between the data is observed.

In order to verify the calorimetric analysis results, an XRPD analysis was performed (Figure 4), where for each of the four samples, six well-defined peaks at 10, 11, 16, 24, 25 and 33 $2\theta$ degrees were found, indicating that there were no polymorphic changes. This is consistent with the results reported by Rudyanto et al. [5] and Yang et al. [34], and it can be observed that the patterns match in the number of peaks. However, the relative intensity varies, possibly due to the way the sample was prepared—in particular, the homogenization process.

Table 2. The thermophysical properties of propylparaben obtained by DSC.

Sample	Enthalpy of Melting, ${\mathsf{\Delta }}_{\mathbf{m}}\mathit{H}$/kJ·mol${}^{-1}$	Melting Point ${\mathit{T}}_{\mathbf{m}}$/K
Original sample	27.0 ± 0.5	369.6 ± 0.5
	26.150 a	369.7 a
	27.9 ± b	369.5 ± 0.5 b
	28.4 ± 0.6 c	369.4 ± 0.5 c
	27.2 ± 0.8 d	369.4 ± 0.5 d
	28.0 e	369.65 e
	26.51 f	369.6 f
	24.75 g	370.92 g
Water	26.8 ± 0.5	369.4 ± 0.5
$w_{0.80}$	27.1 ± 0.5	369.9 ± 0.5
Acetonitrile	26.9 ± 0.5	370.1 ± 0.5

^a Martin and Carstensen [35]. ^b Ouyang et al. [33]. ^c Cárdenas et al. [4]. ^d Perlovich et al. [36]. ^e Prankerd [31]. ^f Giordano et al. [37]. ^g Rudyanto et al. [5].

Table 2. The thermophysical properties of propylparaben obtained by DSC.

Sample	Enthalpy of Melting, ${\mathsf{\Delta }}_{\mathbf{m}}\mathit{H}$/kJ·mol${}^{-1}$	Melting Point ${\mathit{T}}_{\mathbf{m}}$/K
Original sample	27.0 ± 0.5	369.6 ± 0.5
	26.150 a	369.7 a
	27.9 ± b	369.5 ± 0.5 b
	28.4 ± 0.6 c	369.4 ± 0.5 c
	27.2 ± 0.8 d	369.4 ± 0.5 d
	28.0 e	369.65 e
	26.51 f	369.6 f
	24.75 g	370.92 g
Water	26.8 ± 0.5	369.4 ± 0.5
$w_{0.80}$	27.1 ± 0.5	369.9 ± 0.5
Acetonitrile	26.9 ± 0.5	370.1 ± 0.5

^a Martin and Carstensen [35]. ^b Ouyang et al. [33]. ^c Cárdenas et al. [4]. ^d Perlovich et al. [36]. ^e Prankerd [31]. ^f Giordano et al. [37]. ^g Rudyanto et al. [5].

3.2. Thermodynamic Functions of Solution

The solution thermodynamic functions were calculated from the experimental solubility data (Table 1) by the Gibbs–va not Hoff–Krug Equations [38,39] (Equations (1)–(3)).

$${\Delta }_{\mathrm{soln}}{H}^{\circ }=-R{\left(\frac{\partial ln{x}_3}{\partial \left(T^{-1}-T_{\mathrm{hm}}^{-1}\right)}\right)}_p$$

(1)

$${\Delta }_{\mathrm{soln}}{G}^{\circ }=-R{T}_{\mathrm{hm}}·\mathrm{intercept}$$

(2)

$${\Delta }_{\mathrm{soln}}{S}^{\circ }=\left({\Delta }_{\mathrm{soln}}{H}^{\circ }-{\Delta }_{\mathrm{soln}}{G}^{\circ }\right)T_{\mathrm{hm}}^{-1}$$

(3)

$$T_{\mathrm{hm}}=\frac{n}{{\displaystyle \sum _{i=1}^n\left(\frac{1}{T}\right)}}$$

(4)

$${\zeta }_H=\frac{|{\Delta }_{\mathrm{soln}}{H}^{\circ }|}{|{\Delta }_{\mathrm{soln}}{H}^{\circ }|+|T{\Delta }_{\mathrm{soln}}{S}^{\circ }|}$$

(5)

$${\zeta }_{TS}=1-{\zeta }_H$$

(6)

where ${\Delta }_{\mathrm{soln}}{H}^{\circ }$, ${\Delta }_{\mathrm{soln}}{G}^{\circ }$ and ${\Delta }_{\mathrm{soln}}{S}^{\circ }$ are the thermodynamic functions of enthalpy, the Gibbs energy and entropy of solution (in $\mathrm{kJ}{\mathrm{mol}}^{-1}$), $x_3$ is the solubility as a mole fraction, T is the temperature (in K), $T_{\mathrm{hm}}$ is the harmonic mean for n study temperatures (in K), R is the molar gas constant (in $\mathrm{J}{\mathrm{K}}^{-1}{\mathrm{mol}}^{-1}$), “intercept” is the intercept of each equation in the va not Hoff plot, and ${\zeta }_H$ and ${\zeta }_{TS}$ are the relative contributions of the enthalpy and entropy of solution to the Gibbs energy.

Following the va not Hoff–Krug model, the enthalpy (Equation (1)) and Gibbs energy (Equation (2)) of solution were calculated from the slope and the intercept of the equation that describes the behavior of the logarithm of solubility as a function of the inverse of temperature ($T^{-1}-T_{\mathrm{hm}}^{-1}$).

As shown in Table 3, the Gibbs energy is positive for each case and decreases from pure water to the $w_1=0.2$ mixture. Within the $w_1=0.2$ up to the $w_1=0.8$ cosolvent concentration range, no values were reported due to the liquid phase separation. Between $w_1=0.8$ and $w_1=0.9$, the ${\Delta }_{\mathrm{soln}}{G}^{\circ }$ continuously decreases as a consequence of the increasing solubility; and from $w_1=0.9$ up to pure MeCN, there is an increase due to a solubility decrease as a consequence of adding MeCN to the mixture.

Table 3. Thermodynamic functions of propylparaben solution process (3) in {acetonitrile (1) + water (2)} cosolvent mixtures at $T_{\mathrm{hm}}=297.6$ K ^a.

${\mathit{w}}_1$b	${\mathsf{\Delta }}_{\mathbf{soln}}{\mathit{G}}^{°}$/ (kJ·mol${}^{-1}$)	${\mathsf{\Delta }}_{\mathbf{soln}}{\mathit{H}}^{°}$ (kJ·mol${}^{-1}$)	${\mathsf{\Delta }}_{\mathbf{soln}}{\mathit{S}}^{°}$ (J·mol${}^{-1}$·K${}^{-1})$	${\mathit{T}}_{\mathbf{hm}}{\mathsf{\Delta }}_{\mathbf{soln}}{\mathit{S}}^{°}$ (kJ·mol${}^{-1}$)	${\mathit{\zeta }}_{\mathit{H}}$ c	${\mathit{\zeta }}_{\mathbf{TS}}$ c
0.00	25.37	32.62	24.35	7.25	0.818	0.182
0.05	25.17	32.70	25.30	7.53	0.813	0.187
0.10	24.97	34.36	31.57	9.40	0.785	0.215
0.15	24.79	34.76	33.51	9.97	0.777	0.223
0.20	24.12	36.47	41.48	12.34	0.747	0.253
0.80	5.77	32.50	89.83	26.73	0.549	0.451
0.85	5.03	40.23	118.28	35.2	0.533	0.467
0.90	4.82	41.65	123.76	36.83	0.531	0.469
0.95	5.00	40.29	118.60	35.29	0.533	0.467
1.00	5.52	35.67	101.32	30.15	0.542	0.458

^a The average relative standard uncertainty in w₁ is u_r(w₁) = 0.0008. The standard uncertainty in T is u(T) = 0.10 K. The average relative standard uncertainty in apparent thermodynamic quantities of real dissolution processes are u_r(Δ_solnG°) = 0.017, u_r(Δ_solnH°) = 0.020, u_r(Δ_solnS°) = 0.022, and u_r(TΔ_solnS°) = 0.022. ^b w₁ is the mass fraction of acetonitrile (1) in the {acetonitrile (1) + 1-propanol (2)} mixtures free of propylparaben (3). ^c ζ_H and ζ_TS are the relative contributions by enthalpy and entropy toward the apparent Gibbs energy of dissolution.

Table 3. Thermodynamic functions of propylparaben solution process (3) in {acetonitrile (1) + water (2)} cosolvent mixtures at $T_{\mathrm{hm}}=297.6$ K ^a.

${\mathit{w}}_1$b	${\mathsf{\Delta }}_{\mathbf{soln}}{\mathit{G}}^{°}$/ (kJ·mol${}^{-1}$)	${\mathsf{\Delta }}_{\mathbf{soln}}{\mathit{H}}^{°}$ (kJ·mol${}^{-1}$)	${\mathsf{\Delta }}_{\mathbf{soln}}{\mathit{S}}^{°}$ (J·mol${}^{-1}$·K${}^{-1})$	${\mathit{T}}_{\mathbf{hm}}{\mathsf{\Delta }}_{\mathbf{soln}}{\mathit{S}}^{°}$ (kJ·mol${}^{-1}$)	${\mathit{\zeta }}_{\mathit{H}}$ c	${\mathit{\zeta }}_{\mathbf{TS}}$ c
0.00	25.37	32.62	24.35	7.25	0.818	0.182
0.05	25.17	32.70	25.30	7.53	0.813	0.187
0.10	24.97	34.36	31.57	9.40	0.785	0.215
0.15	24.79	34.76	33.51	9.97	0.777	0.223
0.20	24.12	36.47	41.48	12.34	0.747	0.253
0.80	5.77	32.50	89.83	26.73	0.549	0.451
0.85	5.03	40.23	118.28	35.2	0.533	0.467
0.90	4.82	41.65	123.76	36.83	0.531	0.469
0.95	5.00	40.29	118.60	35.29	0.533	0.467
1.00	5.52	35.67	101.32	30.15	0.542	0.458

^a The average relative standard uncertainty in w₁ is u_r(w₁) = 0.0008. The standard uncertainty in T is u(T) = 0.10 K. The average relative standard uncertainty in apparent thermodynamic quantities of real dissolution processes are u_r(Δ_solnG°) = 0.017, u_r(Δ_solnH°) = 0.020, u_r(Δ_solnS°) = 0.022, and u_r(TΔ_solnS°) = 0.022. ^b w₁ is the mass fraction of acetonitrile (1) in the {acetonitrile (1) + 1-propanol (2)} mixtures free of propylparaben (3). ^c ζ_H and ζ_TS are the relative contributions by enthalpy and entropy toward the apparent Gibbs energy of dissolution.

The ${\Delta }_{\mathrm{soln}}{H}^{\circ }$ increases from pure water up to $w_1=0.2$. Generally, in hydroalcoholic mixtures, this increase in water-rich mixtures is due to the water molecules destructuring around the non-polar groups of the drug [40]. However, in {MeCN (1) + W (2)} mixtures, MeCN works as a structuring agent for water molecules; thus, in this case, the enthalpy increase is possibly due to a breakdown in PP molecules.

From $w_1=0.8$ up to $w_1=0.9$, the solution enthalpy increases and then decreases from $w_1=0.9$ up to pure MeCN. When analyzing the behavior of ${\Delta }_{\mathrm{soln}}{S}^{\circ }$, it is positive in all cases, thus, favoring the PP solution process. The solution entropy increases from pure water up to $w_1=0.2$ and from $w_1=0.8$ up to $w_1=0.9$, indicating a positive cosolvent effect of MeCN. From $w_1=0.9$ up to pure MeCN, the ${\Delta }_{\mathrm{soln}}{S}^{\circ }$ decreases, thereby, reducing the entropic benefit.

Finally, by using the results obtained from Equations (5) and (6), the relative contributions of enthalpy (energy factor) and entropy (organizational factor) were identified. According to the values shown in Table 3, ${\zeta }_H$ presents values higher than 0.5 in each case, indicating that the solution enthalpy contributes in a high proportion to the PP solution process.

3.3. Thermodynamic Functions of Mixing

The solution process involves breaking and forming bonds (Figure 5). Initially, the solute and solvent molecules must disperse, then the solute must change state, and finally the solvent molecules must restructure to form the cavity that will host the solute molecule. In both cases, energy must be supplied, and thus these sub-processes are highly endothermic and, therefore, energetically unfavorable for the solution process. Once both solute and solvent molecules are dispersed, the mixing process takes place, and the solution is formed [41].

Mathematically, the solution process can be expressed as follows [42]:

$${\Delta }_{\mathrm{soln}}{f}^{\circ }={\Delta }_{\mathrm{m}}{f}^{\circ }+{\Delta }_{\mathrm{mix}}{f}^{\circ }$$

(7)

where f stands for thermodynamic functions (Gibbs energy, enthalpy or entropy), and the subscripts m and mix refer to the melting and mixing processes, respectively.

From Equation (7), ${\Delta }_{\mathrm{mix}}{f}^{\circ }$ is solved, obtaining Equation (8).

$${\Delta }_{\mathrm{mix}}{f}^{\circ }={\Delta }_{\mathrm{soln}}{f}^{\circ }-{\Delta }_{\mathrm{m}}{f}^{\circ }$$

(8)

In this work, the thermodynamic melting functions were replaced by the ideal process thermodynamic functions, calculated from the ideal solubility (Table 4) by Equation (9).

$$ln{x}_3^{id}=-\frac{{\Delta }_{\mathrm{m}}H}{R}\left(\frac{T_{\mathrm{m}}-T}{T_{\mathrm{m}}T}\right)+\frac{\Delta C_p}{R}\left(\frac{T_{\mathrm{m}}-T}{T}\right)-\frac{\Delta Cp}{R}ln\left(\frac{T_{\mathrm{m}}}{T}\right)$$

(9)

where T and $T_{\mathrm{m}}$ are the study temperature and the melting temperature, respectively (in K), ${\Delta }_{\mathrm{m}}H$ is the melting enthalpy (in kJ mol${}^{-1}$) of the solute, R is the gas constant (in kJ mol${}^{-1}$K${}^{-1}$), and $\Delta C_p$ is the differential heat capacity of melting (in kJ K${}^{-1}$mol${}^{-1}$) [43]. Some researchers, such as Hildebrand [29], Neau and Flynn [44], Neau et al. [45] and Opperhuizen et al. [46], assume $\Delta C_p$ as the entropy of melting (${\Delta }_{\mathrm{m}}S$), which is calculated as ${\Delta }_{\mathrm{m}}H/T_{\mathrm{m}}$.

According to the thermodynamic functions shown in Table 5, in water-rich mixtures (from pure water up to $w_1=0.2$), the Gibbs energy is positive and decreases as the MeCN concentration increases, thereby, indicating a lower energy requirement for cavity formation [47]. Within this cosolvent concentration range, enthalpic (+) and entropic (−) disfavor is observed.

Table 4. Solubility and thermodynamic functions of the PP ideal solution process.

Variable	Temperatures
	278.15	283.15	288.15	293.15	298.15	303.15	308.15	313.15	318.15
IS a	0.1468	0.1873	0.2176	0.2804	0.3422	0.4265	0.5617	0.7132	0.842
${\Delta }_{\mathbf{Soln}}{G}^{\mathbf{Id}\mathbf{b}}$	4.84 kJ·mol${}^{-1}$
${\Delta }_{\mathbf{Soln}}{H}^{\mathbf{Id}}$	22.45 kJ·mol${}^{-1}$
${\Delta }_{\mathbf{Soln}}{S}^{\mathbf{Id}}$	59.16 J·mol${}^{-1}$·K${}^{-1}$
$T{\Delta }_{\mathbf{Soln}}{S}^{\mathbf{Id}}$	17.60 kJ·mol${}^{-1}$

^a IS: Ideal solubility. ^b Id: Ideal.

Table 4. Solubility and thermodynamic functions of the PP ideal solution process.

Variable	Temperatures
	278.15	283.15	288.15	293.15	298.15	303.15	308.15	313.15	318.15
IS a	0.1468	0.1873	0.2176	0.2804	0.3422	0.4265	0.5617	0.7132	0.842
${\Delta }_{\mathbf{Soln}}{G}^{\mathbf{Id}\mathbf{b}}$	4.84 kJ·mol${}^{-1}$
${\Delta }_{\mathbf{Soln}}{H}^{\mathbf{Id}}$	22.45 kJ·mol${}^{-1}$
${\Delta }_{\mathbf{Soln}}{S}^{\mathbf{Id}}$	59.16 J·mol${}^{-1}$·K${}^{-1}$
$T{\Delta }_{\mathbf{Soln}}{S}^{\mathbf{Id}}$	17.60 kJ·mol${}^{-1}$

^a IS: Ideal solubility. ^b Id: Ideal.

Table 5. Thermodynamic functions relative to the mixing processes of propylparaben (3) in {acetonitrile (1) + 1-propanol (2)} co-solvent mixtures at $T_{\mathrm{hm}}=297.6$ K ^a.

${\mathit{w}}_1$b	${\mathsf{\Delta }}_{\mathbf{mix}}{\mathit{G}}^{\circ }$	${\mathsf{\Delta }}_{\mathbf{mix}}{\mathit{H}}^{\circ }$	${\mathsf{\Delta }}_{\mathbf{mix}}{\mathit{S}}^{\circ }$	$\mathit{T}{\mathsf{\Delta }}_{\mathbf{mix}}{\mathit{S}}^{\circ }$
	(kJ·mol${}^{-1}$)	(kJ·mol${}^{-1}$)	(J·mol${}^{-1}·{\mathbf{K}}^{-1}$)	(kJ·mol${}^{-1}$)
0.00	20.53	10.17	−34.81	−10.36
0.05	20.33	10.25	−33.86	−10.08
0.10	20.12	11.92	−27.58	−8.21
0.15	19.94	12.31	−25.64	−7.63
0.20	19.28	14.02	−17.68	−5.26
0.80	0.92	10.05	30.68	9.13
0.85	0.19	17.79	59.13	17.60
0.90	−0.02	19.21	64.60	19.22
0.95	0.16	17.85	59.45	17.69
1.00	0.67	13.22	42.16	12.55

^a The average relative standard uncertainty in w₁ is u_r(w₁) = 0.0008. The standard uncertainty in T is u(T) = 0.10 K. The average relative standard uncertainties in apparent thermodynamic quantities of real dissolution processes are u_r(Δ_mixG°) = 0.238, u_r(Δ_mixH°) = 0.028, u_r(Δ_mixS°) = 0.034, and u_r(TΔ_mixS°) = 0.034. ^b w₁ is the mass fraction of acetonitrile (1) in the {acetonitrile (1) + water (2)} mixtures free of propylparaben (3).

Table 5. Thermodynamic functions relative to the mixing processes of propylparaben (3) in {acetonitrile (1) + 1-propanol (2)} co-solvent mixtures at $T_{\mathrm{hm}}=297.6$ K ^a.

${\mathit{w}}_1$b	${\mathsf{\Delta }}_{\mathbf{mix}}{\mathit{G}}^{\circ }$	${\mathsf{\Delta }}_{\mathbf{mix}}{\mathit{H}}^{\circ }$	${\mathsf{\Delta }}_{\mathbf{mix}}{\mathit{S}}^{\circ }$	$\mathit{T}{\mathsf{\Delta }}_{\mathbf{mix}}{\mathit{S}}^{\circ }$
	(kJ·mol${}^{-1}$)	(kJ·mol${}^{-1}$)	(J·mol${}^{-1}·{\mathbf{K}}^{-1}$)	(kJ·mol${}^{-1}$)
0.00	20.53	10.17	−34.81	−10.36
0.05	20.33	10.25	−33.86	−10.08
0.10	20.12	11.92	−27.58	−8.21
0.15	19.94	12.31	−25.64	−7.63
0.20	19.28	14.02	−17.68	−5.26
0.80	0.92	10.05	30.68	9.13
0.85	0.19	17.79	59.13	17.60
0.90	−0.02	19.21	64.60	19.22
0.95	0.16	17.85	59.45	17.69
1.00	0.67	13.22	42.16	12.55

^a The average relative standard uncertainty in w₁ is u_r(w₁) = 0.0008. The standard uncertainty in T is u(T) = 0.10 K. The average relative standard uncertainties in apparent thermodynamic quantities of real dissolution processes are u_r(Δ_mixG°) = 0.238, u_r(Δ_mixH°) = 0.028, u_r(Δ_mixS°) = 0.034, and u_r(TΔ_mixS°) = 0.034. ^b w₁ is the mass fraction of acetonitrile (1) in the {acetonitrile (1) + water (2)} mixtures free of propylparaben (3).

In MeCN-rich mixtures ($0.8\ge w_1\le 1.0$), the mixing Gibbs energy is close to zero and even reaches negative values ($w_1=0.9$), thus, cavity formation requires less energy than in water-rich mixtures. This is possibly due to the fact that W–W interactions are more energy-rich than MeCN–MeCN, and although enthalpic disfavor is observed, the process is enhanced by the entropy of mixing.

In order to evaluate the energy and organizational contribution to the mixing process, the data in Table 5 are evaluated using Perlovich’s method [48,49]. Thus, in pure water, the mixing process is driven by the entropy of mixing ($T_{hm}{\Delta }_{\mathrm{mix}}{S}^{\circ }<0;{\Delta }_{\mathrm{mix}}{H}^{\circ }>0;|T_{hm}{\Delta }_{\mathrm{mix}}{S}^{\circ }|>|{\Delta }_{\mathrm{mix}}{H}^{\circ }|$). From $w_1=0.1$ up to $w_1=0.2$, the process is driven by the enthalpy of mixing ($T_{hm}{\Delta }_{\mathrm{mix}}{S}^{\circ }<0;{\Delta }_{\mathrm{mix}}{H}^{\circ }>0$). From $w_1=0.8$ up to $w_1=0.85$ and from $w_1=0.95$ up to pure MeCN, the mixing process is also driven by the enthalpy (${\Delta }_{\mathrm{mix}}{H}^{\circ }>T_{hm}{\Delta }_{\mathrm{mix}}{S}^{\circ }$). Finally, in the $w_1=0.9$ mixture, where the maximum PP solubility is reached, the process is driven by the entropy of mixing (${\Delta }_{\mathrm{mix}}{H}^{\circ }<T_{hm}{\Delta }_{\mathrm{mix}}{S}^{\circ };|T_{hm}{\Delta }_{\mathrm{mix}}{S}^{\circ }|>|{\Delta }_{\mathrm{mix}}{H}^{\circ }|$).

3.4. Enthalpy–Entropy Compensation Analysis

Ryde stated that the increase in entropy is one of the consequences of non-covalent interactions between molecules, which thermodynamically disfavors the process performance. However, in order to compensate for the entropy effect, there is a simultaneous decrease in the energy factor (enthalpy) [50]. This compensation phenomenon between enthalpy and entropy produces a linear relation that allows for the evaluation of the influence of the energy (enthalpy) and organizational (entropy) factors in the process [51,52].

This relation can be evaluated by plotting ${\Delta }_{\mathrm{soln}}{G}^{\circ }$ Vs ${\Delta }_{\mathrm{soln}}{H}^{\circ }$, where positive slopes indicate enthalpy-driven processes and negative slopes indicate entropy-driven processes.

In this context, according to Figure 6, the process is entropy driven, since the three trends (from water up to $w_1=0.2$, from $w_1=0.8$ up to $w_1=0.9$ and from $w_1=0.9$ up to pure MeCN) have negative slopes in the case ${\Delta }_{\mathrm{soln}}{G}^{\circ }$ Vs ${\Delta }_{\mathrm{soln}}{H}^{\circ }$ (Figure 6).

3.5. Mathematical Assessment of Solubility

One of the most versatile tools for process optimization is mathematical modeling, which allows making approximations under different conditions than experimental ones [53].

In general, the mathematical models used to correlate the drug solubility in cosolvent systems can be classified into two groups. In the first group, there are models that allow correlating the solubility as a function of the cosolvent composition, and the most commonly used models are Yalkowsky–Roseman [7,54], Wilson [55], Modifyed Wilson [56], NRTL [57] and Extended Hildebrand [29,58]. In the second group are the models that correlate solubility as a function of temperature; these models, in addition to allowing correlation of solubility data, have an important predictive potential as long as the data to be predicted are interpolated. In the last, the most commonly used models are va not Hoff [19], Apelblat [59] and Buchowski–Ksiazaczak $\lambda h$ [60].

Since PP exhibits liquid phase separation in intermediate mixtures, models that correlate solubility as a function of cosolvent composition cannot be used; therefore, models whose independent variable is temperature were used.

Accordingly, the experimental solubility data (Table 1) were correlated with the models proposed by va not Hoff (Equation (10)), Apelblat (Equation (11)) and Buchowski–Ksiazaczak $\lambda h$ (Equation (12)),

$$ln{x}_3=A+\frac{B}{T}$$

(10)

$$ln{x}_3=A+\frac{B}{T}+ClnT$$

(11)

$$ln\left[1+\frac{\lambda (1-x_3)}{x_3}\right]=\lambda h\left[\frac{1}{T}-\frac{1}{T_{\mathrm{m}}}\right]$$

(12)

where $x_3$ is the calculated solubility; A, B, C, $\lambda$ and h are model-specific parameters; T is the study temperature; and $T_{\mathrm{m}}$ is the PP melting temperature.

The model equation parameters assessed and the average relative deviation (ARD) (Equation (13)) are presented in Table 6, Table 7 and Table 8.

$${\displaystyle ARD=\frac{100}{N}\sum _{i=1}^N\left|\frac{x_3-x_3^{cal}}{x_3}\right|}$$

(13)

where N is the number of solubility data points, and $x_3$ and $x_3^{cal}$ represent the experimentally measured and calculated solubility, respectively.

Table 6. The va not Hoff model parameters.

${\mathit{w}}_1$ a	A	B	ARD b
0.00	3.56 ± 0.36	−4116.30 ± 111.63	3.79
0.05	3.22 ± 0.32	−3987.13 ± 99.51	2.07
0.10	3.73 ± 0.11	−4112.51 ± 33.57	1.30
0.15	4.52 ± 0.33	−4330.91 ± 101.26	3.43
0.20	4.91 ± 0.10	−4362.04 ± 30.01	1.21
0.80	10.82 ± 0.27	−3913.52 ± 83.91	1.49
0.85	14.00 ± 0.33	−4770.38 ± 101.79	2.30
0.90	14.72 ± 0.24	−4958.96 ± 73.08	1.33
0.95	14.31 ± 0.13	−4858.76 ± 40.13	0.99
1.00	12.63 ± 0.51	−4424.53 ± 158.78	4.17
Overall ARD			2.08

^a w₁ is the mass fraction of MeCN (1) in the {MeCN (1) + W (2)} mixtures free of propylparaben (3). ^b The average relative deviation (ARD).

Table 6. The va not Hoff model parameters.

${\mathit{w}}_1$ a	A	B	ARD b
0.00	3.56 ± 0.36	−4116.30 ± 111.63	3.79
0.05	3.22 ± 0.32	−3987.13 ± 99.51	2.07
0.10	3.73 ± 0.11	−4112.51 ± 33.57	1.30
0.15	4.52 ± 0.33	−4330.91 ± 101.26	3.43
0.20	4.91 ± 0.10	−4362.04 ± 30.01	1.21
0.80	10.82 ± 0.27	−3913.52 ± 83.91	1.49
0.85	14.00 ± 0.33	−4770.38 ± 101.79	2.30
0.90	14.72 ± 0.24	−4958.96 ± 73.08	1.33
0.95	14.31 ± 0.13	−4858.76 ± 40.13	0.99
1.00	12.63 ± 0.51	−4424.53 ± 158.78	4.17
Overall ARD			2.08

^a w₁ is the mass fraction of MeCN (1) in the {MeCN (1) + W (2)} mixtures free of propylparaben (3). ^b The average relative deviation (ARD).

Table 7. The Apelblat model parameters.

${\mathit{w}}_1$ a	A	B	C	ARD b
0.00	−0.052 ± 0.004	2.174 ± 0.177	0.008 ± 0.001	3.70
0.05	−0.049 ± 0.005	2.035 ± 0.232	0.007 ± 0.001	4.01
0.10	−0.054 ± 0.003	2.255 ± 0.121	0.008 ± 0.0	1.85
0.15	−0.07 ± 0.007	2.916 ± 0.329	0.01 ± 0.001	3.81
0.20	−0.087 ± 0.006	3.608 ± 0.246	0.013 ± 0.001	3.18
0.80	−116.6 ± 10.0	4835.8 ± 444.1	17.636 ± 1.494	3.21
0.85	−235.6 ± 16.1	9871.4 ± 713.2	35.57 ± 2.399	4.14
0.90	−280.3 ± 18.1	11771.6 ± 802.1	42.288 ± 2.698	4.73
0.95	−251.8 ±  17.1	10576.2 ± 757.7	37.992 ± 2.548	4.41
1.00	−174.6 ± 22.5	7322.6 ± 998.7	26.346 ± 3.359	5.01
Overall ARD				3.81

^a w₁ is the mass fraction of MeCN (1) in the {MeCN (1) + W (2)} mixtures free of propylparaben (3). ^b The average relative deviation (ARD).

Table 7. The Apelblat model parameters.

${\mathit{w}}_1$ a	A	B	C	ARD b
0.00	−0.052 ± 0.004	2.174 ± 0.177	0.008 ± 0.001	3.70
0.05	−0.049 ± 0.005	2.035 ± 0.232	0.007 ± 0.001	4.01
0.10	−0.054 ± 0.003	2.255 ± 0.121	0.008 ± 0.0	1.85
0.15	−0.07 ± 0.007	2.916 ± 0.329	0.01 ± 0.001	3.81
0.20	−0.087 ± 0.006	3.608 ± 0.246	0.013 ± 0.001	3.18
0.80	−116.6 ± 10.0	4835.8 ± 444.1	17.636 ± 1.494	3.21
0.85	−235.6 ± 16.1	9871.4 ± 713.2	35.57 ± 2.399	4.14
0.90	−280.3 ± 18.1	11771.6 ± 802.1	42.288 ± 2.698	4.73
0.95	−251.8 ±  17.1	10576.2 ± 757.7	37.992 ± 2.548	4.41
1.00	−174.6 ± 22.5	7322.6 ± 998.7	26.346 ± 3.359	5.01
Overall ARD				3.81

^a w₁ is the mass fraction of MeCN (1) in the {MeCN (1) + W (2)} mixtures free of propylparaben (3). ^b The average relative deviation (ARD).

Table 8. The Buchowski–Ksiazaczak $\lambda h$ model parameters.

${\mathit{w}}_1$ a	$\mathit{\lambda }$	h	ARD b
0.00	0.0003 ± 2$·{10}^{-5}$	11,301,162 ± 525,076	2.97
0.05	0.0003 ± 2$·{10}^{-5}$	10,420,554 ± 353,844	2.01
0.10	0.0004 ± 2$·{10}^{-5}$	8,606,763 ± 278,840	1.88
0.15	0.0005 ± 3$·{10}^{-5}$	7,765,353 ± 348,425	2.62
0.20	0.0008 ± 3$·{10}^{-5}$	5,352,830 ± 161,122	1.72
0.80	1.42 ± 0.08	2846.3 ± 101.3	1.60
0.85	5.83 ± 0.46	954.6 ± 55.6	3.06
0.90	8.31 ± 0.76	714.0 ± 49.5	3.81
0.95	6.10 ± 0.51	919.4 ± 57.4	2.90
1.00	2.49 ± 0.27	1877.5 ± 150.7	4.55
Overall ARD			2.71

^a w₁ is the mass fraction of MeCN (1) in the {MeCN (1) + W (2)} mixtures free of propylparaben (3). ^b The average relative deviation (ARD).

Table 8. The Buchowski–Ksiazaczak $\lambda h$ model parameters.

${\mathit{w}}_1$ a	$\mathit{\lambda }$	h	ARD b
0.00	0.0003 ± 2$·{10}^{-5}$	11,301,162 ± 525,076	2.97
0.05	0.0003 ± 2$·{10}^{-5}$	10,420,554 ± 353,844	2.01
0.10	0.0004 ± 2$·{10}^{-5}$	8,606,763 ± 278,840	1.88
0.15	0.0005 ± 3$·{10}^{-5}$	7,765,353 ± 348,425	2.62
0.20	0.0008 ± 3$·{10}^{-5}$	5,352,830 ± 161,122	1.72
0.80	1.42 ± 0.08	2846.3 ± 101.3	1.60
0.85	5.83 ± 0.46	954.6 ± 55.6	3.06
0.90	8.31 ± 0.76	714.0 ± 49.5	3.81
0.95	6.10 ± 0.51	919.4 ± 57.4	2.90
1.00	2.49 ± 0.27	1877.5 ± 150.7	4.55
Overall ARD			2.71

^a w₁ is the mass fraction of MeCN (1) in the {MeCN (1) + W (2)} mixtures free of propylparaben (3). ^b The average relative deviation (ARD).

The data calculated using the three mathematical models are well-correlated with the experimental data. Thus, the va not Hoff model is the best correlated with an ARD of 2.2, and the Apelblat model has the highest dispersion with an ARD of 3.8. Figure 7 shows the correlation between the experimental and calculated data, obtaining r${}^2$ values greater than 0.99, validating the results obtained through the ARD analysis and confirming that the three models are well-correlated, considering that a model is useful if it reports ARD values below 30.0 [61,62].

4. Conclusions

The PP solubility in cosolvent mixtures {MeCN (1) + W (2)} is an endothermic process that is dependent on the cosolvent composition. In intermediate mixtures, liquid phase separation occurs, possibly due to a higher affinity of PP for one of the solvents that form the cosolvent mixture. In addition, the mixture {MeCN (1) + W (2)} presents microheterogeneity in intermediate mixtures, which would also favor the separation of liquid phases.

The Gibbs energy of the solution is positive in all cases and lower in MeCN-rich mixtures due to a higher solubility of PP with these mixtures. Regarding the enthalpy and entropy of the solution, they are positive in all cases. This indicates an enthalpic depletion compensated by an entropic enhancement that allows the development of the solution process.

Finally, the PP experimental solubility is well-correlated with the va not Hoff, Apelblat and Buchowski–Ksiazaczak $\lambda h$ models with ARD values below 4.0.