Dissolution Thermodynamics and Preferential Solvation of Phenothiazine in Some Aqueous Cosolvent Systems

Abstract

Published equilibrium mole fraction solubilities of phenothiazine in ethanol, propylene glycol and water as mono-solvents at several temperatures were investigated to find standard apparent thermodynamic quantities of dissolution mixing and solvation based on the van’t Hoff and Gibbs equations. Further, by processing the reported mole fraction solubility values of phenothiazine in some aqueous cosolvent mixtures at T/K = 298.2, the inverse Kirkwood–Buff integrals treatment demonstrated preferential hydration of phenothiazine in water-rich mixtures and preferential solvation of this agent by cosolvents in mixtures of 0.24 < x₁ < 1.00 in the {ethanol (1) + water (2)} mixed system and mixtures of 0.18 < x₁ < 1.00 in the {propylene glycol (1) + water (2)} mixed system.

1. Introduction

Phenothiazine (molecular structure shown in Figure 1; IUPAC name: 10H-phenothiazine, molar mass: 199.27 g·mol⁻¹, CAS number: 92-84-2, PubChem CID: 7108) is a heterocyclic compound related to thiazine [1]. Some phenothiazine derivatives exhibit highly bioactive behavior. In particular, its derivatives chlorpromazine and promethazine are commonly employed in psychiatry and allergy treatment, respectively [2]. It is noteworthy that the widely used agent methylene blue also has the basic heterocyclic structure of phenothiazine [3]. From a non-medical point of view, it is interesting to note that phenothiazine is commonly used during acrylic acid polymerization as an anaerobic inhibitor. Moreover, it is also used as an in-process inhibitor in some purification processes of acrylic acid [4]. Although this agent was prepared a long time ago, its physicochemical properties have not been completely studied, including the equilibrium solubility in mono- or mixed solvents. Thus, the solubility of phenothiazine in 38 organic solvents at T/K = 298.2 was reported by Hoover et al. [5], whereas its solubility in neat water, ethanol and propylene glycol at several temperatures as well as in some aqueous mixtures of ethanol and propylene glycol at T/K = 298.2 was reported by Ahmadian et al. [6]. These studies were performed in order to understand the effect of different kinds of solvents on its equilibrium solubility. However, a complete thermodynamic study including the effect of mixture composition on preferential solvation has not been reported yet. As has been described in the specialized literature, systematic data about the equilibrium solubility of drugs or drug-alike compounds in different aqueous cosolvent mixtures, as well as a deep understanding of the respective solute dissolution mechanisms, are very important from theoretical and practical points of view [7,8,9,10].

Owing to the lack of this information for phenothiazine, based on published solubility and phase-equilibrium values [6,11], the main aims of this research were as follows: (i) analyze the effect of temperature on the solubility of phenothiazine in ethanol, propylene glycol and water as mono-solvents; (ii) calculate the apparent standard dissolution, mixing and solvation thermodynamic parameters of these neat solvents; (iii) correlate equilibrium solubility data in aqueous cosolvent mixtures with several well-known thermodynamic models; and (iv) evaluate the preferential solvation parameters of phenothiazine in aqueous binary mixtures of ethanol or propylene glycol at T/K = 298.2.

2. Results and Discussion

2.1. Ideal Mole Fraction Solubility and Asymmetrical Activity Coefficients of Phenothiazine in Neat Solvents

The ideal solubility ($x_{\mathrm{s}}^{\mathrm{id}}$) of crystalline drugs is calculated as follows:

$$\ln x_{\mathrm{s}}^{\mathrm{id}}=-\frac{{\mathrm{\Delta }}_{\mathrm{f}}H(T_{\mathrm{f}}-T)}{R{T}_{\mathrm{f}}T}+\left(\frac{\mathrm{\Delta }{C}_p}{R}\right)\left[\frac{(T_{\mathrm{f}}-T)}{T}+\ln \left(\frac{T}{T_{\mathrm{f}}}\right)\right]$$

(1)

where Δ_fH is the molar enthalpy of fusion of the solid drug at the melting point, T_f is the absolute melting point, T is the absolute solution temperature, R is the universal gas constant (8.314 J·mol⁻¹·K⁻¹), and ΔC_p denotes the difference between the molar heat capacity of the hypothetical super-cooled liquid form of the solute and the molar heat capacity of its crystalline form at the solution temperature [11]. Since ΔC_p is not frequently reported for drugs in the literature, it may be approximated to the entropy of fusion, Δ_fS, which, in turn, is calculated as Δ_fH/T_f. Ideal solubilities of phenothiazine at temperatures from T/K = (298.2 to 338.2) summarized in

Table 1. Ideal solubility and activity coefficients of phenothiazine in some pure solvent mixtures at several temperatures and p/kPa = 101.3. ^a,b.

	T/K b
	298.2	308.2	318.2	328.2	338.2
	$x_{\mathrm{s}}^{\mathrm{id}}$
	5.52 × 10−2	6.90 × 10−2	8.55 × 10−2	0.1053	0.1289
Solvent a,b			γs
EtOH	6.31	6.90	7.01	7.27
PG	13.28	12.78	13.04	13.81	15.13
Water	3.54 × 105	3.48 × 105	3.39 × 105	2.94 × 105	2.63 × 105

^a p is the atmospheric pressure in Tabriz, Iran. ^b Standard uncertainty in p is u(p)/kPa = 3.0. Average relative uncertainty in γ_s is u_r(γ₃) = 0.027. Standard uncertainty in T is u(T)/K = 0.2.

were calculated based on the calorimetric values reported by Sabbah and El Watik [12], namely, Δ_fH/kJ·mol⁻¹ = 22.66 and T_f/K = 458.4.

Table 1 also summarizes the asymmetrical activity coefficients (γ_s) of phenothiazine in ethanol, propylene glycol and water as mono-solvents at several temperatures. These γ_s values were calculated as the quotient $x_{\mathrm{s}}^{\mathrm{id}}/x_{\mathrm{s}}$ from the ideal solubilities ($x_{\mathrm{s}}^{\mathrm{id}}$) summarized in Table 1 and the experimental mole fraction solubilities reported by Ahmadian et al. for these neat solvents [6]. Reported solubility values were determined by means of the shake-flask method [6,13]. As observed, phenothiazine γ_s values vary from a maximum of 3.54 × 10⁵ in neat water at T/K = 298.2 to a minimum of 6.31 in neat ethanol at T/K = 298.2. In all solvents at all temperatures, phenothiazine exhibits γ_s values higher than the unity because the ideal solubilities are higher than the experimental ones. Moreover, γ₃ values decrease with the temperature increase in water but increase in ethanol and propylene glycol. This implies some distancing of real behavior from the ideal dissolution hypothetical process when the temperature increase for both cosolvents.

On the other hand, Equation (2) allows a rough estimation of the magnitudes of solute–solvent intermolecular interactions from γ_s values [14].

$$\ln {\gamma }_{\mathrm{s}}=(e_{11}+e_{\mathrm{ss}}-2e_{1\mathrm{s}})\frac{V_{\mathrm{s}}{\phi }_1^2}{RT}$$

(2)

Subscript 1 stands for the solvent system and e₁₁, e_ss and e_1s represent the magnitudes of solvent–solvent, solute–solute and solvent–solute interaction energies, respectively. V_s denotes the molar volume of the hypothetical super-cooled liquid phenothiazine, whereas φ₁ denotes the volume fraction of every solvent. For low x_s solubility values, $V_{\mathrm{s}}{\phi }_1^2/RT$ may be considered as constant despite the solvent. Hence, γ_s values should depend mainly on e_11, e_ss and e_1s [14]. As is well known, e₁₁ and e_ss are unfavorable for drug solubility and dissolution, whereas e_1s favors the respective drug solubility and dissolution rate increasing. The contribution of e₃₃ was planned as constant in the different solvents studied. Thus, from a qualitative point of view, based on the energetic quantities described in Equation (2), the next analysis could be established by considering that e₁₁ was highest in neat water (because of its high Hildebrand solubility parameter, δ₁/MPa^1/2 = 47.8), followed by propylene glycol (δ₁/MPa^1/2 = 30.2) and, finally, ethanol (δ₁/MPa^1/2 = 26.5) [15,16]. It is noteworthy that the Hildebrand solubility parameter is one of the most used polarity indices in pharmaceutical sciences [7,8,9]. Thus, because pure water reveals γ_s values higher than 2.5 × 10⁵, it should have high e₁₁ and low e_1s values, whereas, in pure propylene glycol exhibiting γ_s values near 12.0 and ethanol exhibiting γ_s values near 7.0, the e₁₁ values should be relatively low, while the e_1s values should be high. In this way, a higher solvation of phenothiazine in ethanol and propylene glycol was expected.

2.2. Apparent Thermodynamic Properties of the Dissolution, Mixing and Solvation of Phenothiazine in Mono-Solvents

2.2.1. Apparent Thermodynamic Functions of Phenothiazine Dissolution in Mono-Solvents

All the apparent standard thermodynamic quantities relative to phenothiazine dissolution processes were calculated at T/K = 298.2. Hence, all the apparent standard enthalpy changes (∆_solnH°) were obtained by means of the modified van’t Hoff equation, as shown in Equation (3) [17]:

$${\left(\frac{\partial \ln x_s}{\partial \left(1/T-1/298.2\right)}\right)}_P=-\frac{{\mathrm{\Delta }}_{\mathrm{soln}}H°}{R}$$

(3)

The apparent standard Gibbs energy changes relative to all the phenothiazine dissolution processes (∆_solnG°) were calculated by means of the following equation [17]:

$${\mathrm{\Delta }}_{\mathrm{soln}}G°=-R\cdot 298.2\cdot \mathrm{intercept}$$

(4)

The intercepts used in Equation (4) were those obtained as a result of the linear regressions of ln x_s as a function of (1/T − 1/298.2). Therefore, Figure 2 depicts the phenothiazine solubility linear van’t Hoff behavior of all three neat solvents. It is noteworthy that linear regressions with r² > 0.98 were observed in all cases, as shown in

Table 2. Coefficients and statistical parameters of the van’t Hoff linear regressions of mole fraction solubility of phenothiazine (3) in some pure solvents at different temperatures and p/kPa = 101.3. ^a,b.

Solvent a,b	Intercept	Slope	Adjusted r2	Typical Error	F
EtOH	−4.758 ± 0.023	−1671 ± 116	0.986	0.026	207.6
PG	−5.444 ± 0.033	−1804 ± 132	0.979	0.041	187.7
Water	−15.729 ± 0.050	−2899 ± 202	0.981	0.063	205.9

^a p is the atmospheric pressure in Tabriz, Iran. ^b Standard uncertainty in T is u(T) = 0.2 K. Standard uncertainty in p is u(p)/kPa = 3.0. Average relative uncertainty in phenothiazine solubility is u_r(x_s) = 0.027.

[18,19,20].

Finally, the apparent standard changes in entropy for all the studied phenothiazine dissolution processes (∆_solnS°) were calculated based on the respective ∆_solnH° and ∆_solnG° values by using Equation (5) [17]:

$${\mathrm{\Delta }}_{\mathrm{soln}}{S}^{\mathrm{o}}=\frac{\left({\mathrm{\Delta }}_{\mathrm{soln}}H°-{\mathrm{\Delta }}_{\mathrm{soln}}G°\right)}{298.2}$$

(5)

Table 3. Apparent thermodynamic functions relative to dissolution processes of phenothiazine (3) in some pure solvents at T/K = 298.2 and p/kPa = 101.3. ^a,b.

Solvent a,b	∆solnG°/ kJ·mol−1 b	∆solnH°/ kJ·mol−1 b	∆solnS°/ J·mol−1·K−1 b	T∆solnS°/ kJ·mol−1 b	ζH c	ζTS c
EtOH	11.80	13.89	7.03	2.10	0.869	0.131
PG	13.50	15.00	5.05	1.50	0.909	0.091
Water	38.99	24.11	−49.93	−14.89	0.618	0.382
Ideal	7.20	17.76	35.42	10.56	0.627	0.373

^a p is the atmospheric pressure in Tabriz, Iran. ^b Standard uncertainty in T is u(T) = 0.2 K. Standard uncertainty in p is u(p)/kPa = 3.0. Average relative standard uncertainty in apparent thermodynamic quantities of real dissolution processes are u_r(∆_solnG°) = 0.032, u_r(∆_solnH°) = 0.070, u_r(∆_solnS°) = 0.072, and u_r(T∆_solnS°) = 0.072. ^c ζ_H and ζ_TS are the relative contributions by enthalpy and entropy to the apparent Gibbs energy of dissolution.

summarizes all the apparent standard thermodynamic quantities for the dissolution processes of phenothiazine in all the neat solvents at T/K = 298.2, including those corresponding to the ideal dissolution process. As expected, all the apparent standard Gibbs energies and apparent enthalpies of dissolution of phenothiazine were positive in every solvent. The apparent standard entropy of dissolution was negative in neat water but positive in ethanol and propylene glycol. Thus, the global dissolution processes of phenothiazine are always endothermic in nature and entropy-driven for those occurring in ethanol and propylene glycol (with ∆_solnH° > 0 and ∆_solnS° > 0), whereas, in neat water, neither enthalpy nor entropy-driving are observed because ∆_solnH° > 0 and ∆_solnS° < 0. As observed, the highest Δ_solnH° and lowest ∆_solnS° values were observed in neat water. The negative apparent dissolution entropy obtained in neat water could be a consequence of the possible hydrophobic hydration around the phenylene groups of phenothiazine (Figure 1).

Moreover, to calculate the magnitude contributions by enthalpy (ζ_H) and entropy (ζ_TS) toward the dissolution processes, the subsequent equations were used [21]:

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

(6)

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

(7)

As observed in Table 3, the higher contributions to the positive apparent standard molar Gibbs energies of phenothiazine dissolution processes were given by the positive enthalpies.

2.2.2. Apparent Thermodynamic Quantities of Phenothiazine Mixing in Mono-Solvents

The overall dissolution processes of phenothiazine in all mono-solvent systems may be represented by the following hypothetical stages:

Solute_{(Solid state)} at T → Solute_{(Solid state)} at T_fus → Solute_{(Liquid state)} at T_fus → Solute_{(Liquid state)} at T → Solute_{(Solution state)} at T

Here, the hypothetical stages were considered as follows: (i) the heating and fusion of phenothiazine at T_f/K = 458.4, (ii) the cooling of the liquid fused phenothiazine to the considered temperature (i.e., T/K = 298.2), and (iii) the subsequent mixing of both the hypothetical super-cooled liquid phenothiazine and the pure solvents at T/K = 298.2 [22]. This treatment allowed us to calculate every individual thermodynamic contribution toward the overall phenothiazine dissolution processes using the following equations:

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

(8)

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

(9)

where ${\mathrm{\Delta }}_{\mathrm{fus}}{H}^T$ and ${\mathrm{\Delta }}_{\mathrm{fus}}{S}^T$ represent the thermodynamic quantities relative to phenothiazine melting and its cooling at T/K = 298.2, which, in turn, were calculated as follows [23]:

$${\mathrm{\Delta }}_{\mathrm{f}}{H}^T={\mathrm{\Delta }}_{\mathrm{f}}{H}^T-\mathrm{\Delta }{C}_p\left(T_{\mathrm{f}}-T\right)$$

(10)

$${\mathrm{\Delta }}_{\mathrm{f}}{S}^T={\mathrm{\Delta }}_{\mathrm{f}}{S}^T-\mathrm{\Delta }{C}_p\ln \left(\frac{T_{\mathrm{f}}}{T}\right)$$

(11)

where ΔC_p denotes the difference in the heat capacities of hypothetical liquid and solid states at the temperature of melting. Owing to the difficulties of ΔC_p experimental determinations, the entropy of fusion (ΔS_f) was used instead [23].

Table 4. Apparent thermodynamic functions relative to mixing processes of phenothiazine (3) in some pure solvents at T/K = 298.2 and p/kPa = 101.3. ^a,b.

Solvent a,b	∆mixG°/ kJ·mol−1 b	∆mixH°/ kJ·mol−1 b	∆mixS°/ J·mol−1·K−1 b	T∆mixS°/ kJ·mol−1 b	ζH c	ζTS c
EtOH	4.60	−3.87	−28.39	−8.47	0.313	0.687
PG	6.30	−2.76	−30.38	−9.06	0.233	0.767
Water	31.80	6.35	−85.35	−25.45	0.200	0.800

^a p is the atmospheric pressure in Tabriz, Iran. ^b Standard uncertainty in T is u(T) = 0.2 K. Standard uncertainty in p is u(p)/kPa = 3.0. Average relative standard uncertainty in apparent thermodynamic quantities of mixing processes are u_r(∆_mixG°) = 0.035, u_r(∆_mixH°) = 0.080, u_r(∆_mixS°) = 0.082, and u_r(T∆_mixS°) = 0.082. ^c ζ_H and ζ_TS are the relative contributions by enthalpy and entropy to the apparent Gibbs energy of mixing.

summarizes all the apparent standard thermodynamic quantities of mixing of the hypothetical super-cooled liquid phenothiazine with all the pure solvents, as calculated at T/K = 298.2. As observed, the Gibbs energies of mixing were positive in all three solvents. The contributions by the thermodynamic quantities of mixing subprocesses to the overall dissolution processes of phenothiazine were variable depending on the solvent. Thus, Δ_mixH° values were negative for ethanol and propylene glycol but positive for pure water. Moreover, Δ_mixS° values were negative in all three solvents. In this way, the mixing processes of phenothiazine in ethanol and propylene glycol were enthalpy-driven because of the exothermic character exhibited. However, in pure water, neither enthalpy nor entropy-driving was observed for mixing. Furthermore, in order to evaluate the relative contributions of enthalpy (ζ_H) and entropy (ζ_TS) to the mixing processes in these mono-solvent systems, two equations analogous to Equations (6) and (7) were also employed. As observed in Table 4, in all cases, the main contributor to the Gibbs energies of mixing was entropy.

As described earlier in the literature, the net value of Δ_mixH° values depends on the contribution of various kinds of intermolecular interactions. Hence, the cavity formation in the solvent system, required for the solute accommodation, is endothermic because some quantity of energy must be supplied to overcome the respective cohesive forces of the solvent. This contribution diminishes the phenothiazine equilibrium solubility. Conversely, the solvent–solute interactions, which result mainly from van der Waals and Lewis acid–base interactions, like hydrogen bonding, are clearly exothermic in nature. This contribution increases the phenothiazine solubility and dissolution rate. Furthermore, the structuring of water molecules around the phenylene groups of phenothiazine (Figure 1) diminishes the net Δ_mixH° quantity to small or even negative values in water-rich mixtures, as indicated above [24]. However, this event is not observed with phenothiazine in pure water.

2.2.3. Apparent Thermodynamic Quantities of Phenothiazine Solvation in Mono-Solvents

In addition to the previously exposed fusion-mixing process, the phenothiazine dissolution process may also be represented by the following hypothetic stages: Solute_(Solid) → Solute_(Vapor) → Solute_(Solution), where the respective partial processes of the global dissolution processes are phenothiazine sublimation and solvation [25]. This treatment permits the calculation of the partial thermodynamic contributions to dissolution processes by means of Equations (12) and (13), respectively, while the standard Gibbs energy of solvation is calculated by means of Equation (14):

∆_solnH° = ∆_sublH° + ∆_solvH°

(12)

∆_solnS° = ∆_sublS° + ∆_solvS°

(13)

∆_solnG° = ∆_sublG° + ∆_solvG°

(14)

where ∆_sublH°/kJ mol⁻¹ = 111.45 at T/K = 298.2, which was reported by Sabbah and El Watik [12]; moreover, from phenothiazine vapor pressure (P_v/mmHg = 5.0 × 10⁻³ at T/K = 580 [12]), the other thermodynamic quantities of sublimation at T/K = 298.2 were calculated as summarized in

Table 5. Apparent thermodynamic functions relative to sublimation and solvation processes of phenothiazine (3) in some pure solvents at T/K = 298.2 and p/kPa = 101.3. ^a,b.

	Sublimation
	∆sublG°/ kJ·mol−1	∆sublH°/ kJ·mol−1	∆sublS°/ J·mol−1·K−1	T∆sublS°/ kJ·mol−1	ζH	ζTS
	53.57	111.45	194.15	57.88	0.658	0.342
	Solvation
Solvent a,b	∆solvG°/ kJ·mol−1 b	∆solvH°/ kJ·mol−1 b	∆solvS°/ J·mol−1·K−1 b	T∆solvS°/ kJ·mol−1 b	ζH c	ζTS c
EtOH	−41.77	−97.56	−187.12	−55.79	0.636	0.364
PG	−40.07	−96.45	−189.10	−56.38	0.631	0.369
Water	−14.57	−87.34	−244.07	−72.77	0.546	0.454

^a p is the atmospheric pressure in Tabriz, Iran. ^b Standard uncertainty in T is u(T) = 0.2 K. Standard uncertainty in p is u(p)/kPa = 3.0. Average relative standard uncertainty in apparent thermodynamic quantities of solvation processes are u_r(∆_mixG°) = 0.040, u_r(∆_mixH°) = 0.090, u_r(∆_mixS°) = 0.093, and u_r(T∆_mixS°) = 0.093. ^c ζ_H and ζ_TS are the relative contributions by enthalpy and entropy to the apparent Gibbs energy of solvation.

. Thermodynamic functions of phenothiazine solvation in all three pure solvents are also summarized in Table 5. Owing to the negative values observed in all thermodynamic quantities, it follows that drug solvation in all three solvents is exothermic and, thus, enthalpy-driven. On the other hand, with the aim of comparing the relative contributions by enthalpy (ζ_H) and entropy (ζ_TS) to the solvation processes, two equations analogous to Equations (6) and (7) were employed again. The ζ_H and ζ_TS values shown in Table 5 show that enthalpy was the main contributing force to the standard Gibbs energy of solvation in all mono-solvents.

2.3. Effect of Mixed Solvents Polarity on Phenothiazine Solubility in Aqueous Cosolvent Mixtures

Figure 3 depicts the phenothiazine solubility profiles as a function of the Hildebrand solubility parameters (δ₁₊₂) of both {ethanol (1) + water (2)} and {propylene glycol (1) + water (2)} mixtures at T/K = 298.2. As widely described, δ₁₊₂ is a very important polarity descriptor of aqueous cosolvent mixtures [7,8,9,10]. Numerical values of this descriptor were calculated considering the Hildebrand solubility parameter of all pure solvents mentioned above and the volume fraction (f_i) of each solvent [7,26]:

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

(15)

As observed, both solubility curves exhibited maximum phenothiazine solubilities in neat cosolvents. Because organic solutes normally reach their maximum solubilities in solvent systems exhibiting similar polarity [7,8], it was expected that the phenothiazine δ₃ value would be lower than 26.5 MPa^1/2 at T/K = 298.2.

Table 6. Application of the Fedors’ method to estimate the molar volume and Hildebrand solubility parameter of phenothiazine.

Group	Group Number	ΔU°/kJ·mol−1	V/cm3·mol−1
Phenylene	2	2 × 31.9 = 63.8	2 × 52.4 = 104.8
Ring closure of 6 atoms	1	1.05	16
(-NH-)	1	8.4	4.5
(-S-)	1	14.15	12
		∑ ΔU° = 87.4	∑ V = 137.3
		δ3 = (87,400/137.3)1/2 = 25.2 MPa1/2

shows a phenothiazine δ₃ value of 25.2 MPa^1/2, as calculated by means of the Fedors’ method [27], which was lower than the ethanol δ₁ value of 26.5 MPa^1/2 [15,16].

2.4. Preferential Solvation Analysis of Phenothiazine in Aqueous Cosolvent Mixtures

The preferential solvation parameter of phenothiazine (component 3) by ethanol or propylene glycol (component 1) in the {cosolvent (1) + water (2)} mixtures at saturation was defined as follows:

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

(16)

where $x_{1,3}^L$ is the local mole fraction of cosolvent (1) in the molecular environment around phenothiazine (3) and x₁ is the bulk mole fraction of cosolvent in the initial aqueous cosolvent mixture in the absence of phenothiazine. If δx_1,3 values were positive, phenothiazine (3) was preferentially solvated by cosolvent (1), but if they were negative, phenothiazine was preferentially solvated by water (2). Thus, the respective δx_1,3 values were obtained by means of the inverse Kirkwood–Buff integrals (IKBI) for both solvent components, based on the following [28,29,30]:

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

(17)

with

$$G_{1,3}=RT{\kappa }_T-{\overline{V}}_3+x_2{\overline{V}}_2\left(\frac{D}{Q}\right)$$

(18)

$$G_{2,3}=RT{\kappa }_T-{\overline{V}}_3+x_1{\overline{V}}_1\left(\frac{D}{Q}\right)$$

(19)

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

(20)

Here, κ_T represents the isothermal compressibility of every {cosolvent (1) + water (2)} mixture. ${\overline{V}}_1$ and ${\overline{V}}_2$ denote the partial molar volumes of cosolvent and water in the aqueous cosolvent mixtures. ${\overline{V}}_3$ denotes the partial molar volume of phenothiazine. The function D corresponds to the first derivative of the variation of standard molar Gibbs energies of transfer of phenothiazine from neat water to {cosolvent (1) + water (2)} mixtures regarding the cosolvent proportion in the mixtures free of solute, as shown in Equation (21). The function Q involves the second derivative of the variation of excess molar Gibbs energy of mixing of cosolvent and water ($G_{1+2}^{Exc}$/kJ·mol⁻¹) regarding the water proportion in the aqueous cosolvent mixtures, as shown in Equation (22). V_cor is the correlation volume and r₃ is the hydrodynamic molecular radius of phenothiazine, which is commonly calculated by means of Equation (23), where N_Av is the number of Avogadro.

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

(21)

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

(22)

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

(23)

As reported in the literature, the definitive V_cor values require iteration because they depend on the local mole fractions of cosolvent and water around the phenothiazine molecules in the equilibrated solutions. Hence, these iterations were performed by substituting δx_1,3 and V_cor values in Equations (16), (17) and (20) in order to recalculate the $x_{1,3}^L$ value until almost invariant values of V_cor were obtained.

Figure 4 depicts the apparent Gibbs energies of transfer of phenothiazine from neat water to both {cosolvent (1) + water (2)} systems (${\mathrm{\Delta }}_{\mathrm{tr}}{G}_{3,2\to 1+2}^{\mathrm{o}}$) at T/K = 298.2. These ${\mathrm{\Delta }}_{\mathrm{tr}}{G}_{3,2\to 1+2}^{\mathrm{o}}$ values were calculated from the mole fraction solubilities reported in [6], as follows:

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

(24)

${\mathrm{\Delta }}_{\mathrm{tr}}{G}_{3,2\to 1+2}^{\mathrm{o}}$ values were correlated according to the regular third-degree polynomial presented as Equation (25). Coefficients and statistical parameters of Equation (25) for both aqueous cosolvent systems are summarized in

Table 7. Coefficients and statistical parameters of Equation (23) applied to Gibbs energy of transfer from neat water (2) to {cosolvent (1) + water (2)} mixtures at T/K = 298.2.

Coefficient or Parameter	Ethanol (1) + Water (2)	Propylene Glycol (1) + Water (2)
a	0.24 ± 0.91	−0.13 ± 0.54
b	−63.63 ± 7.89	−47.22 ± 4.97
c	73.10 ± 18.65	57.85 ± 11.95
d	−37.09 ± 12.07	−35.85 ± 7.64
Adjusted r2	0.985	0.994
Statistical error	0.938	0.545
F-statistical	254.6	613.3

.

$${\mathrm{\Delta }}_{\mathrm{tr}}{G}_{3,2\to 1+2}^{\mathrm{o}}=a+b{x}_1+c{x}_1^2+d{x}_1^3$$

(25)

In this way, the D values shown in

Table 8. Some properties associated with the preferential solvation of phenothiazine (3) in {ethanol (1) + water (2)} mixtures at T/K = 298.2.

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	−63.63	−600.1	−136.2	698	0.00
0.05	−56.60	−573.1	−204.4	708	−3.61
0.10	−50.12	−534.1	−269.7	731	−5.47
0.15	−44.20	−488.0	−327.1	772	−4.87
0.20	−38.84	−440.3	−374.3	827	−2.41
0.25	−34.03	−395.1	−411.6	885	0.64
0.30	−29.78	−354.9	−441.0	942	3.43
0.35	−26.09	−320.9	−465.7	994	5.70
0.40	−22.95	−293.2	−489.7	1042	7.48
0.45	−20.37	−271.5	−517.2	1088	8.93
0.50	−18.34	−255.0	−552.9	1132	10.23
0.55	−16.88	−243.0	−602.6	1176	11.53
0.60	−15.96	−234.4	−672.0	1221	12.94
0.65	−15.61	−227.7	−764.5	1266	14.37
0.70	−15.81	−220.5	−875.2	1307	15.44
0.75	−16.57	−210.1	−981.9	1342	15.42
0.80	−17.88	−195.0	−1048.3	1365	13.66
0.85	−19.75	−176.9	−1048.5	1377	10.39
0.90	−22.17	−159.3	−990.6	1386	6.54
0.95	−25.16	−145.0	−903.7	1395	2.97
1.00	−28.69	−134.4	−813.4	1409	0.00

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

and

Table 9. Some properties associated with the preferential solvation of phenothiazine (3) in {propylene glycol (1) + water (2)} mixtures at T/K = 298.2.

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	−47.22	−480.7	−136.2	698	0.00
0.05	−41.71	−426.2	−195.4	730	−2.10
0.10	−36.73	−377.9	−241.7	780	−2.34
0.15	−32.29	−335.9	−276.3	842	−1.36
0.20	−28.39	−300.1	−301.1	907	0.03
0.25	−25.02	−270.1	−318.1	970	1.36
0.30	−22.20	−245.5	−329.5	1029	2.43
0.35	−19.91	−225.8	−337.7	1084	3.24
0.40	−18.16	−210.4	−345.0	1137	3.82
0.45	−16.94	−198.5	−353.9	1188	4.25
0.50	−16.27	−189.7	−366.8	1238	4.62
0.55	−16.13	−183.1	−385.8	1286	4.96
0.60	−16.53	−178.3	−413.3	1335	5.31
0.65	−17.47	−174.7	−451.7	1383	5.67
0.70	−18.94	−171.7	−503.3	1430	6.01
0.75	−20.96	−168.8	−570.9	1476	6.25
0.80	−23.51	−165.5	−657.8	1519	6.28
0.85	−26.60	−161.3	−767.8	1558	5.92
0.90	−30.22	−155.6	−906.1	1593	4.96
0.95	−34.39	−147.5	−1079.6	1619	3.11
1.00	−39.09	−136.1	−1297.9	1635	0.00

^a x₁ is the mole fraction of propylene glycol (1) in the {propylene glycol (1) + water (2)} mixtures free of phenothiazine (3).

were calculated from the first derivative of Equation (23) by considering the variation of aqueous cosolvent mixture composition in incremental x₁ = 0.05 steps for all the mixture composition intervals. The required Q, RT·κ_T, ${\overline{V}}_1$ and ${\overline{V}}_2$ values corresponding to {ethanol (1) + water (2)} and {propylene glycol (1) + water (2)} mixtures were taken from the literature [31]. Because ${\overline{V}}_3$ values are not available for phenothiazine in these {cosolvent (1) + water (2)} mixtures, they were considered as those calculated based on the Fedors’ method, i.e., 137.3 cm³·mol⁻¹, as shown in Table 6 [27].

G_1,3 and G_2,3, shown in Table 8 and Table 9, were negative in all cases, indicating the affinity of phenothiazine to both cosolvents and water. An approximated hydrodynamic radius of phenothiazine (r₃) was calculated as 0.379 nm by means of Equation (23). In turn, the preferential solvation parameters of phenothiazine by ethanol or propylene glycol molecules are also summarized in Table 8 and Table 9. According to Figure 5, the initial addition of a cosolvent to water gave negative δx_1,3 values of phenothiazine in the interval from neat water to the mixture of x₁ = 0.24 for aqueous ethanol mixtures and the mixture of x₁ = 0.18 for aqueous propylene glycol mixtures. The maximum negative values of this parameter were obtained for the mixture of x₁ = 0.10, with δx_1,3 = −5.47 × 10⁻² for ethanolic mixtures and δx_1,3 = −2.34 × 10⁻² for propylene glycol mixtures. These values were higher than the considered absolute values of 1.00 × 10⁻². Therefore, they were a consequence of real preferential solvation effects of water on phenothiazine, rather than a consequence of the uncertainties propagation in the respective IKBI calculations [32,33]. The cosolvent action of both cosolvents for increasing the phenothiazine solubility in these water-rich mixtures was associated with the breaking of the “iceberg”-like ordered structure exhibited by water molecules around the phenylene groups of phenothiazine.

In mixtures of 0.24 < x₁ < 1.00 for {ethanol (1) + water (2)} and 0.18 < x₁ < 1.00 for {propylene glycol (1) + water (2)} mixtures, the δx_1,3 values were positive, indicating the preferential solvation of phenothiazine by ethanol or propylene glycol. Maximum δx_1,3 values were obtained for the mixture of x₁ = 0.70 (δx_1,3 = 0.1544) for aqueous ethanol mixtures and the mixture of x₁ = 0.80 (δx_1,3 = 6.28 × 10⁻²) for aqueous propylene glycol mixtures. The maximum positive δx_1,3 values were also higher than |1.00 × 10⁻²|, as a consequence of real preferential solvation effects of both cosolvents [32,33]. From a mechanistic viewpoint, the preferential solvation by both cosolvents could have been due to phenothiazine acting as a Lewis acid in front of the ethanol or propylene glycol molecules owing to the unshared electrons of the hydroxyl atoms of these cosolvents. Notably, these cosolvents are more basic than water, noted in the magnitude of their Kamlet–Taft hydrogen bond acceptor parameters, namely, β = 0.75 for ethanol, 0.78 for propylene glycol and 0.47 for water [16]. By comparing the phenothiazine behavior in aqueous mixtures of ethanol and propylene glycol, it follows that the region of preferential hydration is wider with ethanol and preferential hydration magnitude is also higher with this cosolvent. Nonetheless, the region of preferential solvation with a cosolvent is wider with propylene glycol, although the preferential solvation magnitude is higher with ethanol.

3. Conclusions

Reported equilibrium mole fraction solubilities of phenothiazine in ethanol, propylene glycol and water were processed to find apparent thermodynamic quantities of dissolution mixing and solvation based on van’t Hoff and Gibbs equations. IKBI treatment demonstrated the preferential hydration of phenothiazine in water-rich mixtures and preferential solvation by cosolvents in mixtures of 0.24 < x₁ < 1.00 for the {ethanol (1) + water (2)} system and mixtures of 0.18 < x₁ < 1.00 for the {propylene glycol (1) + water (2)} system.