Solubility determination, mathematical modeling, and thermodynamic analysis of naproxen in binary solvent mixtures of (1-propanol/2-propanol) and ethylene glycol at different temperatures

Abstract

This study investigates the solubility behavior of Naproxen (NAP) in binary solvent mixtures of 1-propanol (1-PrOH) and 2-propanol (2-PrOH) with ethylene glycol (EG) across a range of temperatures. The solubility of NAP was experimentally determined at five different temperatures (293.15 to 313.15 K), and the data were correlated using various thermodynamic models, including the van’t Hoff, Jouyban-Acree, modified Wilson, mixture response surface, Jouyban-Acree-van’t Hoff. The results demonstrated that NAP’s solubility increases with temperature in both solvent systems. Notably, NAP exhibited higher solubility in mixtures with 1-PrOH compared to 2-PrOH, despite the lower polarity of 2-PrOH. This unexpected trend is attributed to the distinct molecular interactions, including hydrogen bonding, influenced by the structural differences between 1-PrOH and 2-PrOH. The X-ray diffraction analysis confirmed that no polymorphic transformation occurred in NAP during dissolution, maintaining its crystalline structure. The solubility data were well-correlated by the applied models, with overall MRDs% (mean relative deviation percentage) below 6.1.

Introduction

The solubility behavior of pharmaceutical compounds in different solvents plays a vital role in drug formulation and delivery [1]. Naproxen (NAP, Fig. 1) is a nonsteroidal anti-inflammatory medication renowned for its potent anti-inflammatory and analgesic effects. It is extensively prescribed in medical practice to address various pain and inflammation-related conditions [2]. Despite the widespread therapeutic use of NAP in recent years, comprehensive physicochemical information regarding important properties like solubility and molar volume remains limited. This knowledge plays a fundamental role in facilitating the procedure to produce drug dosage forms [3]. Understanding the solubility characteristics of NAP in various solvents is also crucial for optimizing its extraction and purification procedures.

Fig. 1

2D (left) and 3D (right) images of the molecular structure of naproxen (NAP)

The concept of cosolvency is crucial for enhancing the aqueous solubility of pharmaceuticals. Cosolvency refers to the phenomenon where the combined presence of multiple solvents in a mixture leads to a higher solubility of a solute compared to individual solvents alone [4]. This effect may be attributed to the alteration of interactions involved in solute-solvent, solvent-solvent, and solvent polarity [5]. By carefully selecting binary solvent mixtures, one can harness the cosolvency effect to improve the solubility of substances that exhibit poor aqueous solubility like NAP, ultimately increasing bioavailability and therapeutic effectiveness. Several studies have investigated the solubility of NAP in aqueous media and organic solvents [6,7,8,9,10,11,12,13,14]. Recent studies have explored the solubility behavior of NAP in binary solvent mixtures including, ethanol (EtOH) + water [6], ethylene glycol (EG) + water [7], 2-propanol (2-PrOH) + water [8], acetone + water [10], ethanol + propylene glycol [11], ethyl acetate + ethanol [12], and polyethylene glycol 200 (PEG 200) + water [13] recognizing their importance in enhancing drug solubility. For instance, Moradi et al. [7] demonstrated the efficacy of binary aqueous solvent mixtures in improving the solubility of poorly water-soluble NAP, emphasizing the role of solvent composition in modulating solubility. Similarly, Rodríguez et al. [12] investigated the solubility of NAP in mixed organic solvent systems, concluding that specific solvent combinations significantly affect drug solubility and bioavailability. However, despite these advances, there remains a notable gap in the literature regarding the solubility of NAP in binary solvent mixtures, particularly those involving 1-PrOH/2-PrOH and EG. To our knowledge, no comprehensive studies have been conducted on how these solvent systems affect the solubility and thermodynamic behavior of NAP across a range of temperatures. Addressing this gap is crucial, as understanding the solubility profile of NAP in these mixtures could lead to better formulation strategies for this widely used non-steroidal anti-inflammatory drug. In addition, extending the solubility database of NAP in various solvent mixtures provides the possibility of presenting a general cosolvency model for NAP at various temperatures [15].

Mathematical models could provide a comprehensive understanding of the dissolution process. Several models have been commonly used, including van’t Hoff, Jouyban-Acree, mixture response surface (MRS), Jouyban-Acree-van’t Hoff, and modified Wilson models [16]. Each of these models offers unique insights into the solute-solvent interactions and thermodynamic parameters. Van’t Hoff model, based on the ideal gas law, relates the solubility of a solute to temperature through the calculation of the equilibrium constant [17]. Jouyban-Acree model considers the influence of solvent parameters on solute solubility [18]. MRS combines statistical regression analysis with mathematical optimization techniques to analyze the effect of solvent composition on solubility [19]. Jouyban-Acree-van’t Hoff model integrates both van’t Hoff and Jouyban-Acree models, providing a comprehensive analysis of the thermodynamics and solvent effects on solubility [20]. Lastly, modified Wilson model considers the deviation from ideal behavior and accounts for the non-ideal mixing effects in binary solvent mixtures [21, 22].

The solubility of a substance in solvent blends is influenced by factors such as intermolecular interactions, solvent polarity, and temperature [23]. Thermodynamic functions encompassing entropy, enthalpy, and Gibbs free energy, provide valuable insights into the solute-solvent interactions and the dissolution process. The range of temperatures investigated in this study, specifically 293.15 K (20 °C) to 313.15 K (40 °C), is highly relevant to practical pharmaceutical applications. These temperatures are commonly encountered in various stages of drug development and storage. For example, 298.15 K represents room temperature conditions under which many drugs are formulated, tested, and stored in temperate climates. The higher end of the range, 313.15 K, simulates conditions that might be encountered during manufacturing processes or in tropical climates, where ambient temperatures can reach similar levels. Understanding the solubility behavior of NAP within this temperature range is crucial for optimizing its formulation, ensuring stability, and predicting its performance under different storage conditions. This knowledge can guide the selection of appropriate solvent systems and conditions to maintain the drug’s efficacy and safety throughout its shelf life.

This study explored the solubility of NAP in binary solvent mixtures of 1-PrOH, 2-PrOH, and EG combinations that have not been extensively studied before, offering new insights into solubility behavior driven by solvent polarity and hydrogen bonding. Secondly, our investigation spans a comprehensive temperature range (293.15 K to 313.15 K), providing valuable data relevant to pharmaceutical formulation under varying environmental conditions. Additionally, the application of multiple thermodynamic models, including the van’t Hoff, Jouyban-Acree, Jouyban-Acree- van’t Hoff, modified Wilson, and MRS models, allows for a robust analysis of the solubility data. Furthermore, the use of X-ray diffraction (XRD) analysis to confirm the crystalline structure of NAP after dissolution adds an extra layer of depth to our study, ensuring that no polymorphic transformation occurs. These innovative elements collectively enhance the understanding of NAP solubility and contribute valuable knowledge to the field of pharmaceutical formulation. Finally, by combining mathematical modeling with sustainable pharmacy practices, this research aimed to contribute to more environmentally responsible and efficient drug development processes.

Experimental section

Materials

NAP (CAS No.: 22204-53-1), with a claimed purity of 98.5%, was acquired from Daana Pharmaceutical Company (Tabriz, Iran) and utilized as received without any additional purification step. EG (CAS No.: 107-21-1) with a mass fraction purity of 0.995 was obtained from Scharlau Chemie (Spain). 2-PrOH (CAS No.: 67-63-0) with a mass fraction purity of 0.997 and 1-PrOH (CAS No.: 71-23-8) with a mass fraction purity of 0.995 were purchased from Merck (Germany).

Solubility determination

Shake-flask method was employed to determine the solubility of NAP at the point of solid-liquid equilibrium. The shake-flask method was selected for this study due to its widespread acceptance and reliability in determining the equilibrium solubility of pharmaceutical compounds. It allows for precise control of experimental conditions, such as temperature and solvent composition, ensuring reproducibility of results. Additionally, this method is well-suited for handling the binary solvent mixtures used in our study, providing accurate solubility data necessary for subsequent thermodynamic analysis. Further details of the shake-flask method was provided in a recent video article [24]. Blends of 2-PrOH or 1-PrOH and EG were meticulously formulated by mixing suitable masses of solvents at 0.10 intervals, ranging from 0.10 to 0.90 in mass fractions. A surplus quantity of NAP was introduced into each flask and shaken under incubation in an incubator-shaker (Heidolph Unimax 1010, Germany). Saturation was reached after solid-liquid equilibrium within 48 h. For quantitative analysis, the undissolved fraction of NAP in saturated solutions was settled using centrifugation, and an aliquot of the supernatant was analyzed by a UV/Vis spectrophotometer (Cecil BioAquarius 7250 CE, UK) at 262 nm (if necessary dilution with ethanol 50% (v/v) was used). To avoid precipitation of solute due to temperature change, both centrifugation and dilution steps were accomplished at the desired temperature in an incubator (Kimia Idea Pardaz Azarbayjan (KIPA) Co., Tabriz, Iran). The density of the saturated solution was also determined. All experiments were carried out under the atmospheric pressure of the lab (861 mbar).

XRD analysis

Raw and equilibrated NAP solutions in neat 1-PrOH, 2-PrOH, and EG were analyzed using X-ray analysis. This was done to confirm that there were no polymorphic transformations or solvate formations during the solubility experiments conducted in the study. The XRD data were collected in the range of 10° to 60° (2θ) at 30 mA and 40 kV under atmospheric pressure using a Philips PW 1730.

Computational and thermodynamic analysis

In this study, solubility models applied encompass van’t Hoff, Jouyban-Acree, modified Wilson, Jouyban-Acree-van’t Hoff, and MRS models. van’t Hoff equation (Eq. (1)) was employed to articulate the dissolution process of a solute in a particular solvent mixture, establishing a connection with temperature. Equation (2) (MRS model) was utilized to correlate the drug’s solubility at different solvent ratios under a given temperature. Jouyban-Acree and Jouyban-Acree-van’t Hoff models (Eqs. (3) and (4)) were used to establish links between solubility data, temperatures of mixtures, and solvent composition. Lastly, Eq. (5) (modified Wilson model) represented a non-linear equation applied to fit the acquired data, considering the blend composition at a specific temperature. Details regarding the equations and relations can be found in our previous works [21, 25]. For the assessment of models’ predictive accuracy, mean relative deviation (MRD%) (as defined in Eq. (6)) is utilized.

$$\:\text{l}\text{n}x=A+\frac{B}{T}$$

(1)

$$\text{l}\text{n}{x}_{m}={\beta}_{1}{w}_{1}^{{\prime}}+{\beta}_{2}{w}_{2}^{{\prime}}+{\beta}_{3}\left(\frac{1}{{w}_{1}^{{\prime}}}\right)+{\beta}_{4}\left(\frac{1}{{w}_{2}^{{\prime}}}\right)+{\beta}_{5}{w}_{1}^{{\prime}}.{w}_{2}^{{\prime}}$$

(2)

$$\text{l}\text{n}{x}_{m,T}={w}_{1}\text{l}\text{n}{x}_{1,T}+{w}_{2}\text{l}\text{n}{x}_{2,T}+\frac{{w}_{1}.{w}_{2}}{T}\sum\:_{i=0}^{2}{J}_{i}.{({w}_{1}-{w}_{2})}^{i}$$

(3)

$$\:\text{l}\text{n}{x}_{m,T}={w}_{1}({A}_{1}+\frac{{B}_{1}}{T})+{w}_{2}({A}_{2}+\frac{{B}_{2}}{T})+\frac{{w}_{1}.{w}_{2}}{T}\sum\:_{i=0}^{2}{J}_{i}.{({w}_{1}-{w}_{2})}^{i}$$

(4)

$$\:-\text{l}\text{n}{x}_{m}=1-\frac{{w}_{1}[1+\text{l}\text{n}{x}_{1}]}{{w}_{1}+{w}_{2}{\lambda}_{12}}-\frac{{w}_{2}[1+\text{l}\text{n}{x}_{2}]}{{w}_{1}{\lambda}_{21}+{w}_{2}}$$

(5)

$$\:MRD\%=\frac{100}{N}\sum\:\left(\frac{\left|Calculated\:Value-Observed\:Value\right|}{Observed\:Value}\right)$$

(6)

In this context, solubilities of NAP in mono-solvents 1 and 2, as well as solvent blend, are denoted as x₁, x₂, and x_m, respectively. Mass fractions of the mono-solvents 1 and 2 are represented by w₁ and w₂, respectively. Variable N signifies the number of data points in each set.

Results and discussions

XRD analysis

As depicted in Fig. 2, the XRD results for both the raw NAP and the processed NAP derived from three saturated solutions were examined. Upon comparison with the raw NAP, the plot of NAP in neat solvents did not reveal any new peaks. Consequently, it can be inferred that the crystalline structure of NAP remained unchanged following dissolution in the investigated solvents. This suggests that no polymorphic transformation occurred during the dissolution process. This analysis aimed to ascertain whether the solid forms of NAP in the saturated solutions gave rise to solvated compounds or polymorphs. Polymorphism refers to the capacity of a compound to exist in multiple crystalline forms, where the molecules are arranged differently within the crystal lattice. Since different polymorphs can exhibit markedly distinct pharmaceutically relevant properties, the study of these forms are crucial step in the pre-formulation phase of pharmaceutical research and development. Typically, the most stable polymorphic form is preferred for use in marketed formulations because other polymorphs are metastable and may transform into the more stable form during storage. Such a phase transition can lead to formulation issues, such as precipitation from solution, physical instability of solid dosage forms, and changes in bioavailability. During drug dissolution in different solvents, the drug may recrystallize into various forms, a process known as solvent-mediated polymorphic transformation. Given that different polymorphs have different solubilities, it is essential to assess the form of drug crystals in the solvents being studied. When it comes to the onset of crystallization, XRD is a valuable tool for detecting crystallinity within an amorphous matrix, as the crystalline and amorphous patterns are distinctly different [26].

Fig. 2

The XRD patterns of raw NAP and equilibrated NAP in 1-PrOH, 2-PrOH, and EG

Solubility of NAP in binary mixtures

Obtained experimental data for the solubility of NAP in solvent mixtures containing (1-PrOH/2-PrOH) and EG at five investigated temperatures are presented in Table 1. All solubility data are presented in mole fraction units. Solubility values are reported as mean values calculated from three experiments, along with standard deviation (SD) indicated in parentheses. In both solvent systems, generally, more NAP got solubilized by elevated temperature and 1-PrOH (or 2-PrOH) concentration. This trend is evident as we observe higher solubility values at higher temperatures and for higher mass fractions of 1-PrOH or 2-PrOH.

Table 1 Experimental mole fraction solubility ($\:{\chi\:}_{m,T}$) values as the mean of three experiments (± SD) measured for NAP in two binary mixtures of (1-PrOH + EG) and (2-PrOH + EG) at different temperatures

In the (1-PrOH + EG) mixture, the minimum solubility of NAP was observed at the lowest 1-PrOH concentration (w₁ = 0.0; neat EG). At this composition, the mole fraction solubility of NAP was determined to be 3.54 × 10^− 3 at 293.2 K. As 1-PrOH concentration increased, the solubility of NAP also increased steadily, reaching a maximum value of 2.55 × 10^− 2 at 313.2 K and w₁ = 1.0 (neat 1-PrOH). This maximum solubility represents the highest amount of NAP that can be dissolved in any specified quantity of the (1-PrOH + EG) mixture under given experimental conditions. In the (2-PrOH + EG) mixture, a similar trend was observed. Minimum solubility of NAP was observed at the lowest 2-PrOH concentration (neat EG). As 2-PrOH concentration increased, the solubility of NAP also increased gradually, reaching a maximum value of 2.15 × 10^− 2 at 313.2 K and w₁ = 0.8. Afterward, with subsequent increases in 2-PrOH concentration, a decrease in solubility was observed.

Given that NAP is almost a non-polar drug (log P = 3.18 [27]) and EG is a relatively polar solvent (dielectric constant of 41.2 [28]), it is anticipated that the solubility of NAP would increase with the addition of 1-PrOH (or 2-PrOH), a less polar component than EG. This expectation aligns well with experimental evidence. A more non-polar cosolvent (1-PrOH/2-PrOH) better dissolves non-polar NAP via hydrophobic interactions of hydrophobic moieties. On the other hand, as evident from Table 1, the solubility of NAP at all corresponding temperatures is higher in (1-PrOH + EG) compared to (2-PrOH + EG) at almost all studied fractions except in 0.1 ≤ w1 ≤ 0.3 which is higher in 2-PrOH + EG (Fig. 3). However, when comparing polarities of 1-PrOH (dielectric constant of 21) and 2-PrOH (dielectric constant of 19) [29], it might be assumed that more NAP would dissolve in the less polar 2-PrOH. Surprisingly, results demonstrate the opposite trend. Although 1-PrOH and 2-PrOH are isomers, structural differences between them and associated variances in molecular interactions with NAP and EG can be used to explain the differences in NAP’s solubility in each compound. 1-PrOH is a linear structure with a single hydroxyl (-OH) group connected to the end of a carbon chain. As opposed to this, 2-PrOH has a branching structure with an attached hydroxyl group to the middle carbon atom. Besides hydrophobic interactions, polar functional groups in NAP (namely, carboxyl and ether groups, Fig. 1) and the hydroxyl group in 1-PrOH (or 2-PrOH) may form hydrogen bonds, further affecting solubility. In addition, the hydrogen-bonding network between EG molecules can be affected by the presence of cosolvents in EG. This could thus affect NAP’s solvation and change how it behaves when it comes to solubility in mixtures. Mendez-Bermudez et al. [30] utilized molecular dynamics simulations to reveal that aqueous mixtures of 1-PrOH and 2-PrOH exhibit unique interactions with water molecules and these interactions are concentration-dependent. The dynamics of alcohol-alcohol, alcohol-water, and water-water interactions were notably different between the aqueous mixtures of 1-PrOH and 2-PrOH. Considering these findings, similar scenarios may be involved in binary mixtures of EG with 1-PrOH or 2-PrOH. Therefore, the main reason for the difference in the solubility power of 1-PrOH and 2-PrOH is likely to depend on their ability to form hydrogen bonds with NAP and EG molecules. Even though 2-PrOH may still establish hydrogen bonds with NAP and EG, the branching structure may hinder some interactions, resulting in a somewhat altered solubility behavior from 1-PrOH [31]. Similar observations have been documented with various drugs when 1-PrOH and 2-PrOH cosolvents are present [31, 32]. Additionally, a report indicates that the branched structure of 2-PrOH may influence the interaction between silanol groups on glass surfaces in a manner distinct from 1-PrOH, especially when hydrogen bonding is the primary mode of interaction [33]. In general, explaining solubility patterns outlined in Table 1 is challenging because they are influenced by multiple factors. These factors include interactions between solutes and solvents, interactions among solvents, as well as molecular shapes and sizes, among others [32].

Fig. 3

Mole fraction solubility of NAP in both cosolvent mixtures at 298.2 K. ○: 1-PrOH + EG, ●: 2-PrOH + EG

As previously noted, all solubility data are reported in terms of mole fraction; however, to facilitate comparison with certain previously published studies, Fig. 4 illustrates the NAP solubility represented in molarity in both cosolvent mixtures at various temperatures.

Fig. 4

NAP solubility expressed in molarity in both cosolvent mixtures at different temperatures. ○: 293.2 K, ●: 298.2 K, Δ: 303.2 K, ▲: 308.2 K, ◊: 313.2 K

We compiled the solubility data for NAP reported in the literature, presenting it in Table 2 based on mole fraction solubility. The solubility power in individual solvents is as follows order: ethyl acetate > EtOH > PEG 200 > 1-PrOH > propylene glycol (PG) > 2-PrOH > EG > water. This ranking aligns with the general understanding of solvent-solute interactions and the polarity of solvents. Ethyl acetate, being a relatively non-polar solvent with a moderate dielectric constant, provides a favorable environment for the dissolution of NAP, a lipophilic drug. Ethanol and PEG 200 also demonstrate high solubility power, likely due to their intermediate polarity and ability to form hydrogen bonds with the NAP molecules.

Table 2 Experimental mole fraction solubility ($\:{\chi}_{m,T}$± SD) for NAP in different binary mixtures at 298.15 K

Table 3 presents parameters obtained from van’t Hoff model and corresponding mean relative deviation percentages (MRDs%) for NAP in investigated blends of (1-PrOH + EG) and (2-PrOH + EG). Parameters A and B represent the intercept and slope of the van’t Hoff equation, respectively. MRD% values indicate the accuracy of the model predictions compared to the experimental data. In both mixtures, MRD% values range from 0.6 to 5.0, suggesting good agreement between van’t Hoff model and experimental solubility data. Table 4 presents the calculated parameters for Jouyban-Acree and Jouyban-Acree-van’t Hoff models, depicting NAP solubility in (2-PrOH + EG) and (1-PrOH + EG) mixtures. Parameters J₀, J₁, and J₂ are specific to the Jouyban-Acree model, while A₁, B₁, A₂, and B₂ correspond to Jouyban-Acree-van’t Hoff model. The MRD% values for both models in both mixtures range from 1.9 to 3.6, indicating a reasonably good fit between models and experimental data. Table 5 presents constants (β₁-β₅) obtained from the MRS model for NAP solubility. MRD% values range from 0.7 to 3.8, suggesting a satisfactory agreement between the MRS model predictions and experimental solubility data. Table 6 provides modified Wilson model parameters (λ₁₂ and λ₂₁) for NAP solubility in (1-PrOH + EG) and (2-PrOH + EG) mixtures at various temperatures. MRD% values range from 0.7 to 6.4, indicating a good fit between modified Wilson model and experimental solubility data.

Table 3 Van’t Hoff model parameters and corresponding MRD% for NAP in two investigated binary mixtures of (1-PrOH + EG) and (2-PrOH + EG)

Alongside correlation and back-calculation computations, Jouyban-Acree-van’t Hoff model’s predictive ability as a semi-predictive model for solubility data was assessed. The model was trained using limited data points, specifically the solubility data for solvent mixtures at mass fractions of 0.3, 0.5, and 0.7 at 298.2 K and mono-solvents at low and high temperatures. Subsequently, the model was utilized to predict the remaining data for other mass fractions and temperatures. Prediction MRDs% for various temperatures of the (1-PrOH + EG) system were 5.6, 7.8, 7.8, 7.8 and 5.7 and (2-PrOH + EG) system were 6.3, 2.4, 5.6, 3.0 and 3.0 for 293.2, 298.2, 303.2, 308.2 and 313.2 K, respectively. Overall, the application of mathematical models shows promising results in predicting and describing the solubility of NAP in investigated blends. These models provide valuable insights into the dissolution process and offer reasonable accuracy in representing the experimental solubility data.

Table 4 Parameters calculated for Jouyban-Acree, and Jouyban-Acree-Van’t Hoff model for NAP solubility in two binary mixtures of (1-PrOH + EG) and (2-PrOH + EG)

Mathematical models used for solute solubility in cosolvent mixtures are categorized as theoretical, semi-theoretical, or empirical based on chemical theory. Theoretical models contribute to understanding solubility behavior, while semi-theoretical or empirical models correlate experimental solubilities with independent variables like cosolvent concentration. Two practical categories are predictive and correlative models. Predictive cosolvency models can predict solute solubility without using experimental data or with minimal input points, but their low prediction capability is noted. Correlative cosolvency models use curve-fitting parameters to correlate experimental solubility data with cosolvent concentration. The practical preference is for models with minimal curve-fitting parameters. The overall goal of developing cosolvency equations is to enable solute solubility prediction with a minimum number of experiments or even without experimental data, emphasizing the potential failure of correlative equations with insufficient data points [34]. The van’t Hoff equation is commonly employed to establish a correlation between the logarithm of a solute’s mole fraction solubility and the reciprocal of absolute temperature within a specific solvent composition at various temperatures [35]. However, models correlating a drug’s solubility in a given mono-solvent (or mixed solvent) as a function of temperature have a notable limitation. The trained versions of these models are applicable solely to the original solvent, lacking the capability to extend predictions to other mono-solvents or solvent compositions [34]. Alternatively, the modified Wilson and MRS models can correlate the logarithm of the mole fraction solubility of a drug in solvent mixtures at a constant temperature and various solvent mass fractions. Nonetheless, a drawback of these models is their restricted use under isothermal conditions, requiring training for each temperature of interest. The Jouyban-Acree model addresses non-ideal mixing behavior by introducing additional solute-solvent and solvent-solvent interaction terms, correlating solubility data with temperature and solvent composition. The Jouyban-Acree-van’t Hoff model is a more practical iteration, enabling the calculation of drug solubility in solvent mixtures across different temperatures and offering a generally trained model for a given drug in various solvent compositions [36, 37].

Table 5 MRS model constants at the investigated temperatures and MRD% for back-calculated NAP solubility in two investigated binary mixtures of (1-PrOH + EG) and (2-PrOH + EG)

Table 6 Modified Wilson model parameters at the investigated temperatures and MRD% for back-calculated NAP solubility in two investigated binary mixtures of (1-PrOH + EG) and (2-PrOH + EG)

Table 7 provides measured densities of NAP-saturated solutions in two investigated cosolvent blends at all studied temperatures. Density values are reported with their corresponding standard deviations (± SD). These density measurements provide important information about the physical properties of the NAP solutions in binary mixtures.

Apart from solubility data, Jouyban-Acree model was also utilized to correlate with the density values [38], resulting in the following trained equations for the models:

$$\:\text{l}\text{n}{\rho}_{m,T}={w}_{1}\text{l}\text{n}{\rho}_{1,T}+{w}_{2}\text{l}\text{n}{\rho}_{2,T}-14.932\frac{{w}_{1}.{w}_{2}}{T}$$

(7)

$$\begin{aligned}\text{l}\text{n}{\rho}_{m,T}&={w}_{1}\text{l}\text{n}{\rho}_{1,T}+{w}_{2}\text{l}\text{n}{\rho}_{2,T}-5.851\frac{{w}_{1}.{w}_{2}}{T}\\ &\quad+10.144\frac{{w}_{1}.{w}_{2}{\:(w}_{1}-{w}_{2})}{T}\end{aligned}$$

(8)

Equations (7) and (8) were trained models for density data of NAP saturated solutions in both blends of (1-PrOH + EG) and (2-PrOH + EG), respectively. ρ_{m, t} stands for NAP saturated solvents mixture density at the studied temperature. MRD% values for back-calculated data were found to be 0.2 and 0.4 for Eqs. (7) and (8) respectively, indicating the high reliability of Jouyban-Acree model for density prediction.

Table 7 Measured density (g·cm^–3) of NAP saturated solutions in two investigated binary mixtures of (1-PrOH + EG) and (2-PrOH + EG) at different temperatures

Apparent thermodynamic properties of NAP dissolution

Table 8 presents the apparent thermodynamic properties of NAP dissolution in the two investigated binary mixtures at a mean harmonic temperature (T_hm) of 303.0 K. The following thermodynamic functions are provided: ΔG° (standard Gibbs free energy change), ΔH° (standard enthalpy change), ΔS° (standard entropy change), ζ_H (enthalpy contribution to the dissolution process), and ζ_TS (entropy contribution to the dissolution process). In the (1-PrOH + EG) mixture, as the concentration of 1-PrOH (w₁) increases, values of ΔG° decrease. This indicates that the dissolution of NAP becomes more favorable with the addition of 1-PrOH. The ΔH° values show a slight decrease with increasing 1-PrOH concentration, suggesting that the dissolution process needs slightly less energy. However, ΔS° values increase, indicating an increase in disorderliness of the system during dissolution. Positive ΔS° values confirm that the dissolution process is driven by an increase in entropy at the given temperature. ζ_H and ζ_TS values provide insight into the contributions of enthalpy and entropy to the dissolution process. In the (1-PrOH + EG) mixture, both ζ_H and ζ_TS values are positive, suggesting that both enthalpy and entropy contribute to the dissolution process. However, ζ_H values are larger than ζ_TS values, indicating that enthalpy contribution is dominant. Overall, these thermodynamic properties indicate that the dissolution of NAP in the (1-PrOH + EG) mixture is an endothermic process driven by entropy.

In the (2-PrOH + EG) mixture, similar trends are observed. ΔG° values decrease with increasing 2-PrOH concentration (w₁), indicating a more favorable dissolution process (except w₁ = 0.9 and 1.0). ΔH° values show an increasing trend, suggesting that dissolution of NAP at higher concentrations of 2-PrOH becomes more endothermic. ΔS° values also increase, indicating a higher level of disorderliness during dissolution. Positive ΔS° values further confirm the importance of entropy in driving the dissolution process. ζ_H and ζ_TS values in the (2-PrOH + EG) mixture are again positive, indicating contributions from both enthalpy and entropy. However, similar to the (1-PrOH + EG) mixture, ζ_H values are larger than ζ_TS values, emphasizing the dominance of enthalpy in the dissolution process. Overall, apparent thermodynamic properties reveal that the dissolution of NAP in both binary mixtures is influenced by both enthalpy and entropy. The addition of either 1-PrOH or 2-PrOH leads to more favorable dissolution conditions, with enthalpy contribution playing a significant role. An increase in entropy indicates a greater degree of disorderliness, supporting the solubilization of NAP in binary solvent mixtures at the given temperature.

Table 8 Apparent thermodynamic functions of dissolution of NAP in two investigated binary mixtures of (1-PrOH + EG) and (2-PrOH + EG) at T_hm = 303.0 K

Enthalpy–entropy compensation analysis

Enthalpy-entropy compensation analysis is a concept employed to understand the relationship between enthalpy and entropy changes in a given process [39]. In the context of NAP solubility in the investigated blends, this analysis can supply an understanding of the interplay between entropy and enthalpy contributions to the dissolution process [40].

In the case of NAP dissolution, enthalpy-entropy compensation analysis can help determine whether changes in enthalpy and entropy are correlated. By plotting the values of ΔH° and ΔG° for different compositions of the binary mixtures, it is possible to analyze the relationship between enthalpy and entropy changes. Based on the enthalpy-entropy compensation plot for (1-PrOH + EG) shown in Fig. 5a, it can be observed that two linear correlation lines with different slopes fit the data. This suggests that both enthalpy-driven and entropy-driven processes contribute to the solubility of NAP. In mixtures with 0 ≤ w₁ ≤ 0.8, the plot shows a positive slope. This indicates that the transfer of NAP in these mixtures is primarily driven by enthalpy effects. A decrease in enthalpy is accompanied by a corresponding decrease in free energy, suggesting that the transfer process in these mixtures is mainly influenced by enthalpic interactions between NAP and solvent components. On the other hand, for mixtures with 0.8 ≤ w₁ ≤ 1.0, the plot shows a negative slope. This suggests that the solubility of NAP in these mixtures is predominantly influenced by entropy effects. A decrease in ΔG° is accompanied by an increase in ΔH°, implying that in these mixtures, the dissolution process is forced by entropic factors such as increased disorder or solvation effects. For (2-PrOH + EG) mixture, exactly opposite trends are seen, as illustrated in Fig. 5b. In this fashion, the curve exhibits a negative slope, indicating that the solubility of NAP is pushed by entropy for mixtures with 0 ≤ w₁ ≤ 0.8. A positive slope for mixtures with w₁ values between 0.8 and 1.0 is observed. This indicates that enthalpy effects are the main factor affecting the solubility of NAP in these mixes. Overall, the enthalpy-entropy compensation plot demonstrates that the solubility of NAP is influenced by both enthalpic and entropic contributions, with different driving forces depending on the composition of the solvent mixtures.

Fig. 5

Enthalpy-entropy compensation plot for the solubility of NAP in (a) 1-PrOH + EG and (b) 2-PrOH + EG mixtures at 303.0 K. The solid points correspond to the mass fraction of 1-PrOH/2-PrOH in the binary solvent mixtures (before the addition of NAP)

Conclusions

In this study, solubility and thermodynamics of NAP in binary solvent mixtures of (1-PrOH + EG) and (2-PrOH + EG) were investigated across a range of temperatures. Results indicated that both (1-PrOH + EG) and (2-PrOH + EG) mixtures demonstrated improved solubility of NAP compared to neat EG. Furthermore, the solubility of NAP increased with higher cosolvent concentration and temperature in both solvent systems. Mathematical models employed in this study, including van’t Hoff, Jouyban-Acree, MRS, Jouyban-Acree-van’t Hoff, and modified Wilson models, accurately predicted the solubility of NAP in binary solvent mixtures. These models serve as valuable tools for estimating NAP solubility in various solvent systems and calculating relevant thermodynamic parameters. Thermodynamic analysis revealed that the dissolution of NAP in investigated solvent mixtures was an endothermic process, indicating an elevated energy requirement for NAP to dissolve in these systems. Notably, results highlighted the superior solubilizing power of (1-PrOH + EG) mixtures for NAP compared to (2-PrOH + EG) mixtures at all temperatures examined. This finding suggests that (1-PrOH + EG) mixtures hold particular promise for enhancing NAP solubility and potentially improving its bioavailability in oral drug formulations.

The findings of this study have significant implications for pharmaceutical formulation. The enhanced solubility of NAP in studied binary mixtures, suggests that these solvent systems could be leveraged to improve the bioavailability of naproxen in oral and topical formulations. This knowledge can be applied in the formulation of more stable and effective drug products, particularly for poorly soluble drugs. Additionally, the use of multiple thermodynamic models to accurately predict solubility behaviors sets a precedent for future research aimed at understanding solubility in more complex solvent environments. Future studies could build on this work by exploring other drug compounds or solvent mixtures, potentially leading to the discovery of even more effective formulations. The insights gained from this study thus have the potential to inform both current pharmaceutical practices and ongoing research in drug solubility and formulation science.