Numerical Analysis of Sulfamerazine Solubility in Acetonitrile + 1-Propanol Cosolvent Mixtures at Different Temperatures

Abstract

The current challenges of the pharmaceutical industry regarding the environmental impact caused by its waste have led to the design and development of more efficient industrial processes. In this context, solubility studies are at the core of different processes, such as formulation, preformulation, synthesis, purification, recrystallization, quantification, and quality control. This research evaluates the solubility of sulfamerazine (SMR) in acetonitrile + 1-propanol cosolvent mixtures at nine temperature levels with UV/vis spectrophotometry using the vial-shake method. According to the analysis of the solid phase in equilibrium using differential scanning calorimetry, there were no polymorphic changes. The minimal solubility of SMR was reached in 1-propanol at 278.15 K, and the maximal solubility in acetonitrile at 313.15 K. In all cases, the process was endothermic and dependent on the cosolvent composition, and the solution enthalpy drove the solution process. The solubility data were well correlated with the van’t Hoff, Yalkowsky–Roseman–van’t Hoff, Apelblat, Buchowski–Ksiazczak λh, Yaws, NRTL, Wilson, and modified Wilson models, with the YR model being one of the most attractive because it presented an excellent prediction percentage from four sets of experimental data. The solution process of SMR in acetonitrile + 1-propanol cosolvent mixtures depends on the affinity of SMR for acetonitrile and temperature increase.

1. Introduction

Sulfonamides were some of the first drugs used effectively in infection control. They significantly reduced morbidity and mortality from conditions caused by Gram-positive and -negative bacteria [1,2]. Due to their toxic nature and the introduction of safer antibiotics such as penicillin, their use decreased [3]. However, the uncontrolled use of antibiotics has led to bacterial resistance to most of these therapeutic agents [4]. Due to the above, the use of sulfonamides has increased again in human and veterinary therapy [5], which, in turn, has led to major environmental concerns [6,7,8].

Sulfonamides act by inhibiting the synthesis of dihydrofolic acid, as they are competitive antagonists of para-aminobenzoic acid (PABA), decreasing the synthesis of nucleotides and bacterial DNA [9,10].

Sulfamerazine (SMR; Figure 1) is used in the treatment of pathologies caused by bacteria, such as bronchitis, prostatitis, and urinary tract infections [11,12].

One of the major challenges when developing drug formulations whose active pharmaceutical ingredient (API) is SMR is the low solubility of the drug in aqueous systems; therefore, studies in cosolvent systems are highly relevant [13,14]. In addition, SMR is among the most widely used sulfonamides in the design of pharmaceutical forms (FPs) for veterinary use, and due to the increase in bacterial resistance to first-line antibiotics meant for humans, its use in the development of FP for human therapy has also increased. The solubility of SMR was studied in several cosolvent mixtures of pharmaceutical interest, such as water and methanol [15], water and ethanol [16], water and 1-propanol [17], acetonitrile and methanol [11], ethylene glycol and water [18], acetonitrile and water [12], and propylene glycol and water [19]. Nevertheless, it is important to conduct new studies in order to contribute to the understanding of the possible mechanisms of the SMR solution process.

Mathematical modeling is another important line of research in the field of solubility. In this area, the developed strategies focused on reducing experimental tests to a minimum, which generally optimizes processes and reduces the volume of polluting waste [20]. Some of the most widely used models are van’t Hoff [21], Apelblat [22], Buchowski–Ksiazaczak λh [23], and Yaws [24]. Each of these allows for the calculation of solubility as a function of temperature, which is very convenient when evaluating the solubility of an API at a different temperature from that determined experimentally. Other models, such as Yalkowsky–Roseman [25], modified Wilson [26], Jouyban–Acree [14], Wilson [27], and nonrandom two-liquid (NRTL) [28] correlate API solubility as a function of cosolvent mixtures, allowing for solubility to be calculated over ranges of polarity or cosolvent composition specific.

The aim of this research is to assess the thermodynamics of the SMR solution process in acetonitrile and 1-propanol cosolvent mixtures at nine temperature levels (278.15 up to 318.15 K). Acetonitrile (MeCN) and 1-propanol (1-PrOH) are both widely used solvents in industry, especially in crystallization and qualification processes. In addition, experimental data were correlated with different mathematical models used in the industry.

2. Materials and Methods

2.1. Reagents

Some characteristics of the reagents used in this research are shown in

Table 1. Trademark and related properties of the reagents used.

Chemical Name	CAS a	Chemical Formula	Purity b	Analy. Tech. c
Sulfamerazine d	127-79-7	C11H12N4O2S	>0.990	HPLC
Acetonitrile d	75-05-8	C2H3N	0.998	GC
1-PrOH d	71-23-8	C3H8O	0.998	GC
Sodium hydroxide d	1310-73-2	NaOH	≥97.0

^a Chemical Abstracts Service Registry Number; ^b in-mass fraction; ^c HPLC is high-performance liquid chromatography, and GC is gas chromatography. ^d Source: Sigma-Aldrich, Burlington, MA, USA.

.

2.2. Solubility Determination

To begin with, 19 cosolvent mixtures {MeCN (1) + 1-PrOH (2)} were prepared by varying 0.05 of their composition in an MeCN mass fraction. Once the cosolvent mixtures were prepared, enough SMR was added to obtain a saturated solution at each of the 9 study temperatures (278.15 K up to 318.15 K).

The measurement of the solubility in each mixture was performed according to the shake-flask method proposed by Higuchi and Connors [29], which consists of four phases: (1) saturation, (2) phase separation (filtration), (3) quantification (UV/Vis), and (4) solid phase analysis. This method is sufficiently described in some works conducted and published by the research group [20,30].

2.3. Mathematical Processing of Data

Each solubility value is presented as the result of the average of three measurements with its standard deviation, expressed by applying the 3–30 criterion [31]. These values were used to draw the van’t Hoff plots, which were fitted by the least squares method while applying uncertainty propagation methods using the software R version 4.2.3 (Shortstop Beagle).

3. Results

3.1. Solubility (x₃) of Sulfamerazine (3) in Some Cosolvent Mixtures {MeCN (1) + n-PrOH (2)}

The solubility data for SMR in cosolvent mixtures {MeCN (1) + 1-PrOH (2)} are reported in

Table 2. Experimental solubility of sulfamerazine (3) in {acetoitrile (1) + 1-PrOH (2)} cosolvent mixtures expressed in mole fraction (10⁴x₃) at different temperatures and p = 96 kPa ^ac.

w1 b	Temperatures
	278.15 K	283.15 K	288.15 K	293.15 K	298.15 K	303.15 K	308.15 K	313.15 K	318.15 K
0.00	0.911	1.148	1.435	1.779	2.148	2.647	3.074	3.755	4.511
0.05	1.036	1.311	1.629	1.943	2.425	2.963	3.493	4.261	5.084
0.10	1.185	1.489	1.844	2.198	2.727	3.332	3.936	4.809	5.785
0.15	1.354	1.694	2.090	2.485	3.070	3.742	4.446	5.436	6.564
0.20	1.546	1.928	2.368	2.810	3.456	4.203	5.020	6.143	7.45
0.25	1.764	2.196	2.685	3.178	3.894	4.715	5.681	6.952	8.434
0.30	2.022	2.496	3.043	3.596	4.381	5.313	6.400	7.846	9.625
0.35	2.311	2.825	3.430	4.060	4.911	5.968	7.176	8.821	10.95
0.40	2.646	3.221	3.898	4.597	5.536	6.719	8.117	9.985	12.46
0.45	3.007	3.672	4.417	5.194	6.241	7.513	9.206	11.31	14.03
0.50	3.464	4.191	5.037	5.894	7.046	8.503	10.42	12.82	16.10
0.55	3.962	4.715	5.660	6.620	7.907	9.453	11.71	14.43	18.23
0.60	4.521	5.371	6.412	7.495	8.892	10.64	13.21	16.29	20.70
0.65	5.168	6.107	7.264	8.472	10.01	11.95	14.91	18.41	23.50
0.70	5.906	6.945	8.229	9.578	11.27	13.42	16.84	20.80	26.69
0.75	6.759	7.887	9.323	10.82	12.69	15.03	19.04	23.52	30.30
0.80	7.714	9.002	10.57	12.27	14.29	16.97	21.47	26.59	34.43
0.85	8.770	10.24	11.96	13.87	16.00	19.10	24.12	29.89	39.03
0.90	10.04	11.68	13.56	15.72	18.07	21.51	27.27	33.87	44.37
0.95	11.50	13.21	15.35	17.69	20.34	23.97	30.87	38.28	50.33
1.00	13.09	15.10	17.40	20.09	22.86	27.14	34.73	43.19	57.18

^a is the atmospheric pressure in Neiva, Colombia, ^b is the mass fraction of MeCN (1) in the {acetoitrile (1) + 1-PrOH (2)} mixtures free of sulfamerazine (3), ^c is standard uncertainty in p is $u\left(p\right)=3.0$ kPa. Average relative standard uncertainty in $w_1$ is $u_r\left(w_1\right)=0.0008$. Standard uncertainty in T is $u\left(T\right)$ = 0.10 K. Average relative standard uncertainty in $x_3$ is $u_r\left(x_{3(1+2)}\right)=0.025$.

. In all cases, the solubility of SMR increases with increasing temperature and increasing MeCN mass fraction, reaching its minimum solubility in neat 1-PrOH at 278.15 K, and the maximum solubility in neat MeCN at 318.15 K (Figure 2).

Usually, one of the most influential factors in the solubility of a drug is the relationship between the solubility parameters ($\delta$) of the drug and the solvent. In this case, the $\delta$ from the MeCN and the 1-PrOH are similar (${\delta }_{\mathrm{MeCN}}$ = 24.8 MPa${}^{1/2}$ and ${\delta }_{1-\mathrm{PrOH}}$ = 24.9 MPa${}^{1/2}$ [20,32,33]) and, apparently, the solubility changes are not related to the ${\delta }_{\mathrm{mix}}$ from the cosolvent mixture. However, when considering the Kamlet–Taft acidity scale $\alpha$ [34], a relation can be deduced by increasing SMR solubility and increasing MeCN, since MeCN is less acidic than 1-PrOH (${\alpha }_{\mathrm{MeCN}}=0.29\pm 0.06$; ${\alpha }_{1-\mathrm{PrOH}}=0.776\pm 0.013$) and SMR is acidic [35].

A further factor that can affect the solubility of drugs is polymorphic changes, so it is important to analyze the solid phase in equilibrium with the cosolvent system.

Table 3. Thermophysical properties of SMR obtained by the DSC.

Sample	Enthalpy of Melting, ${\Delta }_{\mathit{m}}\mathit{H}$/kJ·mol−1	Melting Point ${\mathit{T}}_{\mathit{m}}$/K
Original sample a	41.3 ± 0.5	508.5 ± 0.5
	31.6 b	515.2 b
	24.75 c	509.3–510.3 c
	41.3 d	508.5 d
	41.3 ± 1.0 e	508.5 e
		508.9 f
		506.4 g
		508.95 h
		510.66 i
		508.5 j
		508.5 k
1-PrOH	41.2 ± 0.5	508.4 ± 0.5
$w_{0.50}$	41.31 ± 0.5	510.2 ± 0.5
Acetonitrile	40.9 ± 0.5	509.1 ± 0.5

^a High -purity commercial standard; ^b Sunwoo and Eisen [36]; ^c Lee et al. [37]; ^d Martínez and Gómez [38]; ^e Delgado and Martínez [16]; ^f Blanco et al. [12]; ^g Khattab [39]; ^h Delombaerde [40]; ⁱ Aloisio et al. [41]; ^j Cardenas et al. [11]; ^k Vargas et al. [18].

shows the enthalpy and melting temperatures from the original sample and the solid phases in equilibrium with neat 1-PrOH, $w_1=0.5$, and neat MeCN determined by DSC (Figure 3). It is observed that all four samples analyzed have similar enthalpy and melting temperature values, so it can be concluded that there are no polymorphic changes. Likewise, the values obtained in this research are consistent with those reported by other authors.

3.2. Thermodynamic Functions of Solution

From the experimental solubility data (Table 2), the thermodynamic solution functions were calculated according to Equations (1)–(5), proposed by the Gibbs–van’t Hoff–Krug model [42,43].

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

(1)

$${\Delta }_{\mathrm{soln}}G°=-R{T}_{\mathrm{hm}}.\mathrm{intercept}$$

(2)

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

(3)

$${\zeta }_H=|{\Delta }_{\mathrm{soln}}H°\left|\right(|T{\Delta }_{\mathrm{soln}}S°|+|{\Delta }_{\mathrm{soln}}S°{\left|\right)}^{-1}$$

(4)

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

(5)

where ${\Delta }_{\mathrm{soln}}H°$ (in kJ·mol⁻¹), ${\Delta }_{\mathrm{soln}}G°$ (en kJ·mol⁻¹), and ${\Delta }_{\mathrm{soln}}S°$ (in kJ·mol⁻¹·$T_{\mathrm{hm}}^{-1}$) are the thermodynamic functions: enthalpy, Gibbs energy, and entropy of solution. T is the study temperature (in K), $T_{\mathrm{hm}}$ is the harmonic temperature (in K), R is the gas constant (kJ·mol⁻¹·K⁻¹), and ${\zeta }_H$ and ${\zeta }_{TS}$ are the enthalpic and entropic contributions to the solution process.

$T_{\mathrm{hm}}$ is calculated through Equation (6).

$$T_{\mathrm{hm}}=\frac{n}{{\sum }_{i=1}^n\frac{1}{T_i}}$$

(6)

where n is the number of study temperatures ($n=9$) and T is the study temperatures (278.15, 283.15, 288.15, 293.15, 298.15, 303.15, 308.15, 313.15, and 318.15 K); therefore, $T_{\mathrm{hm}}$ is equal to 297.6 K.

Then, from each equation $(y=mx+b)$ in the van’t Hoff plot (Figure 4), ${\Delta }_{\mathrm{soln}}H°$ (from the slope values m) and ${\Delta }_{\mathrm{soln}}G°$ (from the intercept values b) were calculated.

The thermodynamic functions of the solution are presented in

Table 4. Thermodynamic functions of sulfamerazine solution process (3) in {acetonitrile (1) + water (2)} cosolvent mixtures at $T_{hm}=297.6$ K ^a.

${\mathit{w}}_1$b	${\Delta }_{\mathbf{soln}}\mathit{G}°$/ (kJ·mol−1)	${\Delta }_{\mathbf{soln}}\mathit{H}°$ (kJ·mol−1)	${\Delta }_{\mathbf{soln}}\mathit{S}°$ (J·mol−1·K−1)	${\mathit{T}}_{\mathbf{hm}}{\Delta }_{\mathbf{soln}}\mathit{S}°$ (kJ·mol−1)	${\mathit{\zeta }}_{\mathit{H}}$ c	${\mathit{\zeta }}_{\mathbf{TS}}$ c
0.00	20.96	29.25	27.85	8.29	0.779	0.221
0.05	20.66	29.09	28.33	8.43	0.775	0.225
0.10	20.35	28.94	28.86	8.59	0.771	0.229
0.15	20.04	28.78	29.37	8.74	0.767	0.233
0.20	19.74	28.63	29.88	8.89	0.763	0.237
0.25	19.43	28.47	30.38	9.04	0.759	0.241
0.30	19.12	28.32	30.92	9.20	0.755	0.245
0.35	18.82	28.18	31.43	9.35	0.751	0.249
0.40	18.51	28.02	31.96	9.51	0.747	0.253
0.45	18.21	27.86	32.44	9.65	0.743	0.257
0.50	17.89	27.71	33.01	9.82	0.738	0.262
0.55	17.59	27.58	33.55	9.99	0.734	0.266
0.60	17.29	27.42	34.05	10.13	0.730	0.270
0.65	16.98	27.27	34.57	10.29	0.726	0.274
0.70	16.67	27.12	35.09	10.44	0.722	0.278
0.75	16.37	26.97	35.62	10.6	0.718	0.282
0.80	16.06	26.81	36.13	10.75	0.714	0.286
0.85	15.76	26.64	36.56	10.88	0.710	0.290
0.90	15.45	26.49	37.11	11.04	0.706	0.294
0.95	15.14	26.35	37.65	11.2	0.702	0.298
1.0	14.83	26.18	38.13	11.35	0.698	0.302

^a Average relative standard uncertainty in $w_1$ is $u_r\left(w_1\right)=0.0008$. Standard uncertainty in T is $u\left(T\right)=0.10$ K. Average relative standard uncertainties in apparent thermodynamic quantities of real dissolution processes are $u_r({\Delta }_{\mathrm{soln}}G°)=0.015$, $u_r$(${\Delta }_{\mathrm{soln}}H°)=0.019$, $u_r$ (${\Delta }_{\mathrm{soln}}S°$) = 0.024, and $u_r$($T{\Delta }_{\mathrm{soln}}S°$) = 0.024. ^b$w_1$ is the mass fraction of acetonitrile (1) in the {acetonitrile (1) + 1-propanol (2)} mixtures free of sulfadiazine (3). ^c ${\zeta }_H$ and ${\zeta }_{TS}$ are the relative contributions by enthalpy and entropy toward the apparent Gibbs energy of dissolution.

, and ${\Delta }_{\mathrm{soln}}G°$ is positive in all cases and decreases with increasing MeCN concentration as a consequence of increasing solubility ($x_3$). Concerning the ${\Delta }_{\mathrm{soln}}H°$, it is also positive, indicating an endothermic solution process and, similar to Gibbs energy, the enthalpy of the solution decreases as the concentration of MeCN increases. This indicates an increase in molecular interactions, which agrees with the increase in SMR solubility. Regarding the ${\Delta }_{\mathrm{soln}}S°$, it is positive as well, favoring the solution process, and it increases with increasing MeCN concentration.

When analyzing the contribution of enthalpy and entropy to the solution process using Equations (4) and (5) (Table 4), the ${\zeta }_H$ values are greater than ${\zeta }_{TS}$ in all cases, indicating that the solution process is driven by the enthalpy of the solution. This analysis was verified using the Perlovich graphical method (Figure 5), where all values were found in sector I (${\Delta }_{\mathrm{soln}}H°$ > $T{\Delta }_{\mathrm{soln}}S°$ [44,45]), validating that the solution process is driven by enthalpy.

3.3. Thermodynamic Functions of Mixing

The solution process involves the molecular rearrangement of the solute and solvent (Figure 6); hence, there must be a diffusion process of the solute and solvent molecules, which involves a hypothetical melting process for the solute.

Therefore, the solution process thermodynamics can be described as follows:

$${\Delta }_{\mathrm{soln}}{f°}^{-297.6}={\Delta }_{\mathrm{m}}{f°}^{-297.6}+{\Delta }_{\mathrm{mix}}{f°}^{-297.6}$$

(7)

where f represents the thermodynamic functions (Gibbs energy, enthalpy, and entropy), and the subscripts soln, m, and mix indicate the solution, melting, and mixing, respectively.

In this research, the thermodynamic fusion (${\Delta }_{\mathrm{m}}{f°}^{-297.6}$) functions were replaced by the thermodynamic functions of the ideal (${\Delta }_{\mathrm{id}}{f°}^{-297.6}$) process, as described by Mora and Martínez [46]

$${\Delta }_{\mathrm{mix}}{f°}^{-297.6}={\Delta }_{\mathrm{sol}}{f°}^{-297.6}-{\Delta }_{\mathrm{id}}{f°}^{-297.6}$$

(8)

In

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

${\mathit{w}}_1$b	${\Delta }_{\mathbf{mix}}\mathit{G}°$ (kJ·mol−1)	${\Delta }_{\mathbf{mix}}\mathit{H}°$ (kJ·mol−1)	${\Delta }_{\mathbf{mix}}\mathit{S}°$ (J·mol−1·K−1)	$\mathit{T}{\Delta }_{\mathbf{mix}}\mathit{S}°$ (kJ·mol−1)
0.00	−3.730	5.107	29.696	8.837
0.05	−4.038	4.944	30.180	8.981
0.10	−4.344	4.793	30.703	9.137
0.15	−4.650	4.639	31.214	9.289
0.20	−4.957	4.485	31.726	9.441
0.25	−5.264	4.327	32.227	9.590
0.30	−5.572	4.178	32.765	9.750
0.35	−5.872	4.032	33.280	9.904
0.40	−6.184	3.878	33.809	10.061
0.45	−6.488	3.715	34.284	10.203
0.50	−6.807	3.565	34.856	10.373
0.55	−7.100	3.434	35.401	10.535
0.60	−7.407	3.276	35.898	10.683
0.65	−7.714	3.124	36.417	10.837
0.70	−8.020	2.971	36.933	10.991
0.75	−8.326	2.824	37.468	11.150
0.80	−8.636	2.664	37.972	11.300
0.85	−8.937	2.494	38.410	11.430
0.90	−9.248	2.345	38.957	11.593
0.95	−9.551	2.203	39.498	11.754
1.00	−9.860	2.036	39.973	11.896

^a Average relative standard uncertainty in $w_1$ is $u_r(w_1$) = 0.0008. Standard uncertainty in T is $u\left(T\right)$ = 0.10 K. Average relative standard uncertainties in apparent thermodynamic quantities of real dissolution processes are $u_r$ (${\Delta }_{\mathrm{mix}}G°$) = 0.015, $u_r$(${\Delta }_{\mathrm{mix}}H°$) = 0.019, $u_r$ (${\Delta }_{\mathrm{mix}}S°$) = 0.024, and $u_r$($T{\Delta }_{\mathrm{mix}}S°$)= 0.024. ^b $w_1$ is the mass fraction of acetonitrile (1) in the {acetonitrile (1) + water (2)} mixtures free of sulfamerazine (3).

, the thermodynamic functions of SMR mixing in {MeCN (1) + 1-PrOH (2)} are presented. The Gibb energy of mixing is positive in all cases, indicating a favoring of the mixing process over the solution process. The enthalpy of mixing is positive and decreases with increasing MeCN concentration in the solution; hence, it can be assumed that cavity formation (Figure 6) involves less energy in MeCN-rich mixtures. The enthalpy of mixing is positive in all cases, indicating an entropic favoring of mixing to the solution process, and it increases with increasing MeCN mass fraction.

By analyzing the enthalpic and entropic contributions to the mixing process using the Perlovich plot (Figure 7), the mixing process is driven by the entropy of mixing. When plotting the mixing entropy values as a function of mixing enthalpy, all points were in sector II: (${\Delta }_{\mathrm{soln}}H°$ < $T{\Delta }_{\mathrm{soln}}S°$; $|T{\Delta }_{\mathrm{soln}}S°|$ > $|{\Delta }_{\mathrm{soln}}H°|$ [44,45]).

3.4. Enthalpy–Entropy Compensation Analysis

In the solution process, enthalpy changes occur that are compensated by entropy changes as a consequence of non-covalent interactions between the solute and the solvent, promoting the solution process to take place [47]. Plotting this compensation yields linear trends that indicate which thermodynamic function is driving the process [48,49].

Sharp suggests that changes in $T{\Delta }_{\mathrm{soln}}G°$ with linear relations between $T{\Delta }_{\mathrm{soln}}S°$ and ${\Delta }_{\mathrm{soln}}H°$ indicate strongly compensated processes [48].

Usually, the enthalpy–entropy compensation can be evaluated by plotting ${\Delta }_{\mathrm{soln}}H°$ vs. ${\Delta }_{\mathrm{soln}}G°$, with negative slopes indicating entropic compensation and positive slopes indicating enthalpic compensation to the solution process [50,51].

Thus, according to Figure 8, the solution process is driven by the solution enthalpy since the slope in the plot is positive.

3.5. Mathematical Assessment of Solubility

The experimental solubility of SMR in the system {MeCN (1) + 1-PrOH (2)} was correlated with some models used in the pharmaceutical industry.

Accordingly, the models evaluated were van’t Hoff (Equation (9)), Yalkowsky–Roseman–van’t Hoff (Equation (10)), Apelblat (Equation (11)), Buchowski–Ksiazczak λh (Equation (12)), Yaws (Equation (13)), NRTL (Equation (14)), Wilson (Equation (15)), and modified Wilson (Equation (15)). The models van’t Hoff, Apelblat, Buchowski–Ksiazczak $\lambda$ h, and Yaws correlated the solubility as a function of the temperature, and the NRTL, Wilson, and modified Wilson models correlated the solubility as a function of solvent or cosolvent mixtures.

A particularly versatile model was the combined Yalkowsky–Roseman–van’t Hoff (Equation (10)), since it allowed calculating, simultaneously, the solubility of SMR at different temperatures and MeCN mass fractions. In addition, the model was developed from four sets of experimental data (solubility of SMR in MeCN and 1-PrOH at 278.18 and 318.15 K, respectively).

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

(9)

$$ln{x}_{3,1+2}=w_1\left[A_1+\frac{B_1}{T}\right]+(1-w_1)\left[A_2+\frac{B_2}{T}\right]$$

(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)

$$ln{x}_3=A+\frac{B}{T}+\frac{C}{T^2}$$

(13)

$$\begin{array}{c}ln{\gamma }_3=\frac{x_2{\tau }_{23}{G}_{23}+x_1{\tau }_{13}{G}_{13}}{x_3+x_2{G}_{23}+x_3{G}_{13}}-x_3\frac{(x_2{\tau }_{23}{G}_{23}+x_1{\tau }_{13}{G}_{13})}{{(x_3+x_2{G}_{23}+x_3{G}_{13})}^2}+\frac{x_2{G}_{32}}{x_3{G}_{32}+x_2+x_1{G}_{12}}\hfill \\ \hfill \left({\tau }_{32}-\frac{x_3{\tau }_{32}{G}_{32}+x_1{\tau }_{12}{G}_{12}}{x_3{G}_{32}+x_2+x_1{G}_{12}}\right)+\frac{x_1{G}_{31}}{x_3{G}_{31}+x_2{G}_{21}+x_1}\left({\tau }_{31}-\frac{x_3{\tau }_{31}{G}_{31}+x_2{\tau }_{21}{G}_{21}}{x_3{G}_{31}+x_2{G}_{21}+x_1}\right)\end{array}$$

(14)

$$\begin{array}{cc}ln{\gamma }_3=1-ln(x_3+x_2{\Lambda }_{32}+x_1{\Lambda }_{31})-\hfill & \\ \hfill \frac{x_3}{x_3+x_2{\Lambda }_{32}+x_1{\Lambda }_{31}}-& \frac{x_2{\Lambda }_{23}}{x_3{\Lambda }_{23}+x_2+x_1{\Lambda }_{21}}-\frac{x_1{\Lambda }_{13}}{x_3{\Lambda }_{13}+x_2{\Lambda }_{12}+x_1}\hfill \end{array}$$

(15)

$$-ln{x}_{3,1+2}=1-w_1\frac{1-ln{x}_{3,1}}{w_1+w_2{\lambda }_{12}}-w_2\frac{1-ln{x}_{3,2}}{w_2+w_1{\lambda }_{21}}$$

(16)

In order to evaluate the accuracy of each model, the mean percentage deviation (MPD) was used (Equation (17)).

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

(17)

where N is the number of solubility data points, and $x_3$ and $x_3^c$ represent the experimentally measured and calculated solubility, respectively [14].

The models were developed with Python, and the functions of each model are fitted by nonlinear least squares using the library scipy.optimize.curve_fit. In order to validate the model fit, the experimental data were partitioned into training data and test data using the library sklearn.model_selection. The fitting was conducted with 60% of the experimental data taken randomly, and the testing or validation was conducted with the remaining experimental data.

The correlation statistics of the models are presented in

Table 6. Statistical analysis of the models correlation.

Model	Correlation Coefficient	F	F Critic	p
van’t Hoff	0.987	4727.9	1.33$\times {10}^{-100}$	≪0.001
Yalkowsky–Roseman–van’t Hoff	0.994	14,801.7	5.63$\times {10}^{-180}$	≪0.001
Apelblat	0.995	9753.9	5.08$\times {10}^{-87}$	≪0.001
Buchowski–Ksiazczak	0.998	23,781.1	8.39$\times {10}^{-103}$	≪0.001
Yaws	0.999	100,234.7	2.25$\times {10}^{-128}$	≪0.001
NRTL	1.000	438,496.5	8.73$\times {10}^{-208}$	≪0.001
Wilson	1.000	6,076,060.4	9.01$\times {10}^{-195}$	≪0.001
modified Wilson	1.000	16,746,788.2	9.65$\times {10}^{-299}$	≪0.001

. In all cases, the correlation coefficients are approximately 1.0, the F values are high, the critical F values are low, and the p values are <$0.001$, indicating that the correlation and model constants are significant.

Moreover, Figure 9 shows the correlation between the experimental data and those calculated with each model. Then, the lowest accuracy is reached with the van’t Hoff and Yalkowsky–Roseman–van’t Hoff models (MPD: 5.53 and 6.65, respectively). However, it is noteworthy that for the Yalkowsky–Roseman–van’t Hoff model, only 2.1% of the experimental data (four data sets) were used for fitting, which makes it an excellent predictive model. In relation to the other models, all showed an excellent correlation with MPD <3.0. Overall, the MPD for all models is within a reasonable range, where an error of 30% is acceptable for correlating solubility data in the pharmaceutical field [52,53].

In general, models correlating solubility as a function of temperature, such as van’t Hoff, Apelblat, Buchowski–Ksiazczak, and Yaws, present better correlation coefficients [54,55] compared to those reported in this work. However, the deviation from the logarithm linearity of the SMR solubility in mixtures rich in MeCN and neat MeCN leads to small dispersions in the models (Figure 9).

Regarding the NRTL, Wilson, and modified Wilson models, they usually present high correlations between experimental and calculated data [56]. In this case, the correlation coefficients were equal to 1.000 (Table 6), demonstrating the versatility of these models.

The parameters of the equations of each of the evaluated models and the MPD are presented in Tables S1–S8 (Supplementary Materials).

4. Conclusions

The solution process of SMR in cosolvent mixtures {MeCN (1) + 1-PrOH (2)} is endothermic and depends on the cosolvent composition. Due to the acidic character of SMR, it reaches its maximum solubility in neat MeCN possibly because it is less acidic than 1-PrOH.

Concerning the thermodynamic characterization, the Gibbs energy of the solution decreases with increasing MeCN in the cosolvent mixture as a consequence of increasing solubility. The enthalpy is positive, which disfavors the solution process; however, an increase in entropy is observed, which is positive in all cases, favoring the solution process. On the other hand, the thermodynamic mixing functions indicate that the mixing process favors the overall solution process.

Finally, regarding the mathematical modeling, the proposed models presented a good accuracy, highlighting the combined Yalkowsky–Roseman–van’t Hoff model, which presented an MPD lower than 6.65 using only 2.1% of the experimental data when developing the model.