1 Introduction

Pyrazinamide (C5H5N3O, pyrazine-2-carboxamide, Fig. 1) is one of the first-line drugs used in the treatment of tuberculosis [1,2,3]. It is noteworthy that this pathology is considered nowadays as the second most infectious disease worldwide after COVID 19, affecting 10 million people [4].

Fig. 1
figure 1

Molecular structure of pyrazinamide (PZA)

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In this senses, the excessive use of pyrazinamide has been contributing to the development of bacterial resistance [5], which consequently generated an increase in both the concentration and the number of doses required for adequate treatment. Thus, taking into account that the human body partially metabolizes antibiotics, drugs’ excretion is one of the main sources of discharge of active pharmaceutical ingredients (API) into the environment [6]. Besides, adding the waste management, the volume of drugs discharged into aquatic systems has become a serious environmental problem to such an extent that, in the case of pyrazinamide, this drug has commonly been found in wastewater and sludge [7]. For this reason, it has been categorized as a moderately toxic emerging pollutant by the NORMAN network [8].

On the other hand, solubility is one of the most important physicochemical properties useful in the pharmaceutical industry, because in addition to being a primary parameter when evaluating the most appropriate type of pharmaceutical form for drug delivery, it is also related to other important processes such as purification, crystallization, quantification, quality analysis, among others [9,10,11]. Additionally, in environmental sciences, solubility is also an important parameter because some other parameters of environmental interest could approximately be calculated from solubility data, such as: partition coefficient, bioaccumulation, and environmental toxicity levels [12,13,14]. In this context, the use of cosolvents is one of the most used strategies to increase the solubility of APIs. Thus, the (1,4-dioxane + water) cosolvent mixture is considered as one the most important cosolvent systems in pharmaceutical sciences as a physicochemical model owing its wide range of polarity (Ɛ1+2 from 2.25 to 80.1 at room temperature) [15]. From which, some important parameters such as the approximate dielectric requirement of APIs can be easily identified based on the mixtures polarity where maximum solubility peaks are obtained. However, it is important to note that 1,4-dioxane is highly toxic and therefore, it cannot included as excipient for liquid medicine formulations [16, 17]. Another relevant cosolvent mixed system is the (ethanol + water) binary mixture owing the fact that both solvents are the most used in the pharmaceutical industry, especially in the development of liquid pharmaceutical forms for every administration route [18,19,20].

Regarding pyrazinamide solubility, literature studies have been focused in the analysis of pyrazinamide equilibrium solubility in pure solvents [21,22,23,24], reflecting, in addition to the limited physicochemical information on the pyrazinamide dissolution processes, the lack of studies of this drug in aqueous cosolvent mixtures. Thus, the main objective of this study is to evaluate the solubility of pyrazinamide in two cosolvent mixtures of pharmaceutical interest (1,4-dioxane + water and ethanol + water) at 298.15 K, generating physicochemical information that could contribute to the possible understanding of molecular interactions of pyrazinamide in cosolvent systems that in turn, could be extrapolated to environmental conditions. In addition, the experimental data is correlated with the modified Nearly Ideal Binary Solvent/Redlich–Kister (NIBS/RK) model, which is one of the most widely used models in chemical and pharmaceutical industries [11, 25]. Furthermore, the inverse Kirkwood-Buff integrals method is also employed to evaluate the preferential solvation parameter of this drug by the components of the studied mixtures components [26].

2 Materials and Methods

2.1 Reagents

In this study, pyrazinamide (CAS number: 98-96-4, Sigma-Aldrich, Burlington, MA, United States, mass fraction purity: 0.99; compound 3), 1,4-dioxane (CAS number: 123-91-1, Merck, Burlington, MA, United States, mass fraction purity: 0.998; the solvent component 1), ethanol (CAS number: 64-17-5, Merck, Burlington, MA, United States, mass fraction purity: 0.995; the solvent component 1) and the double distilled water (CAS number: 7732-18-5, component 2) with conductivity lower than 2 µS·cm−1, were used.

2.2 Preparation of Cosolvent Mixtures

Nineteen {1,4-dioxane (1) + water (2)} and 19 {ethanol (1) + water (2)} mixtures were prepared, varying the mass fraction of the organic solvents in steps of 0.05 (from w1 = 0.05 to w1 = 0.95) from pure water to each of the organic solvents (1,4-dioxane or ethanol), using an analytical balance with sensitivity of ± 0.1 mg (RADWAG AS 220.R2, Poland). Three samples for each cosolvent composition were prepared in 10 mL capacity amber bottles.

2.3 Solubility Determination

The solubility of pyrazinamide in cosolvent mixtures (1,4-dioxane + water) and (ethanol + water) was determined according to the shake-bottle method proposed by Higuchi et al. [27, 28]. Initially, each one of the samples of the cosolvent mixtures were saturated by adding pyrazinamide in excess until two phases were obtained (namely, liquid phase: saturated solution, solid phase: undissolved solute). Later, the samples were deposited in a recirculating bath (thermostat) at T = (298.15 ± 0.1) K for 36 h, shaking periodically. It is noteworthy that previously was experimentally demonstrated that this time is enough to reach the drug saturation in neat water, where the lowest solubility is observed. Subsequently, using a syringe, an aliquot of each sample was taken, making the respective gravimetric dilution with absolute ethanol to later determine the concentration by UV/Vis spectrophotometry at 267 nm (λmax) (UV/Vis EMC-11- UV spectrophotometer, Germany). Prior to dilution, the sample was filtered through a membrane of 0.45 µm pore diameter (Millipore Corp. Swinnex-13, USA). The procedure is described in more detail in some open access published works of the research group [29,30,31].

2.4 Calorimetric Study

Differential Scanning Calorimetry (DSC) analyses were performed on six pyrazinamide samples (five bottom solid phases in equilibrium with the saturated liquid phases and one untreated commercial drug sample) (DSC 204 F1 Phoenix, Germany). The DSC equipment was calibrated by using Indium and Tin as standards and an empty sealed pan was used as reference. A mass of 10 mg was taken from each of the samples, depositing them in an aluminum crucible which was deposited in the calorimeter with a nitrogen flow of approximately 10 mL/min. A heating cycle was programmed from 50 °C to 250 °C with a heating ramp of 10 °C/min.

3 Results and Discussion

3.1 Pyrazinamide Solid Phases’ Analysis

Figure 2 depicts DSC thermograms of the original untreated pyrazinamide sample as well as those corresponding to the bottom solid phases in equilibrium with the saturated solutions in neat solvents and some aqueous mixed systems, namely water, 1,4-dioxane, ethanol, aqueous mixture of 0.50 mass fraction of 1,4-dioxane and aqueous mixture of 0.50 mass fraction of ethanol. As observed, all thermograms exhibit two endothermal peaks, being the first one at a temperature near to 422 K with an enthalpy of 1.7 kJ·mol–1 which corresponds to a polymorphic transition, whereas the second peak is observed at a temperature near to 464.0 K and enthalpy near to 25.4 kJ·mol–1 that correspond to pyrazinamide fusion. Particular thermodynamic values of fusion are shown in the Fig. 2 caption. Based on these values it is concluded that this drug does not suffer solid state transitions after saturation in the solvent systems considered. It is important to note that our calorimetric values were in good agreement with those reported in the literature as summarized in Table 1 [22, 32,33,34]. These values are characteristic of the stable α-form of pyrazinamide [33].

Fig. 2
figure 2

DSC thermograms of pyrazinamide solid phases. From top to bottom: a: original sample (Tf = 463.9 K, ΔHf = 25.3 kJ·mol–1); b: bottom solid phase in water (Tf = 463.9 K, ΔHf = 25.5 kJ·mol–1); c: bottom solid phase in 0.50 mass fraction of ethanol (Tf = 464.1 K, ΔHf = 25.4 kJ·mol–1); d: bottom solid phase in 0.50 mass fraction of 1,4-dioxane (Tf = 463.2 K, ΔHf = 25.3 kJ·mol–1); e: bottom solid phase in neat ethanol (Tf = 463.8 K, ΔHf = 25.5 kJ·mol–1); f: bottom solid phase in neat 1,4-dioxane (Tf = 463.6 K, ΔHf = 25.5 kJ·mol–1)

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Table 1 Comparison of thermodynamic quantities of polimorphic transition and fusion of pyrazinamide
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3.2 Equilibrium Solubility of Pyrazinamide

Table 2 reports the experimental solubility of pyrazinamide in both aqueous cosolvent systems expressed in mole fraction at 298.15 K. Footnotes c, d and e show that our solubility values in neat solvents are in good agreement with those reported earlier [21,22,23,24, 35]. However, our solubility values in neat 1,4-dioxane and neat ethanol are almost 30 and 25 % higher regarding literature, respectively. Up to our best knowledge no solubilities of this drug in these cosolvent mixtures have been reported and thus no more comparisons are possible. Moreover, Fig. 3 compares the mole fraction solubility of pyrazinamide in both aqueous cosolvent systems. As observed, mixtures of 1,4-dioxane exhibit better solubilizing properties than ethanol mixtures. Maximum solubility in aqueous mixtures of 1,4-dioxane and ethanol are observed in compositions of w1 = 0.85 and w1 = 0.70, respectively.

Table 2 Solubility, activity coefficient, and apparent thermodynamic quantities of dissolution, mixing and transfer, of pyrazinamide (3) in some {cosolvent (1) + water (2)} mixtures at T = 298.15 K and p = 96 kPa
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Fig. 3
figure 3

Mole fraction solubility of pyrazinamide (3) in some {cosolvent (1) + water (2)} mixtures at 298.15 K. ○: {1,4-dioxane (1) + water (2)}; ●: {ethanol (1) + water (2)}. Solid lines correspond to the best regular polynomials correlating the data

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Moreover, Fig. 4 depicts the pyrazinamide solubility as function of the Hildebrand solubility parameters of the aqueous mixtures of 1,4-dioxane and ethanol (δ1+2). δ1+2 is a polarity index widely used in pharmaceutical sciences whose values were calculated by using Eq. 1 from the corresponding δ values of the pure solvents, namely, δ1 = 20.5 MPa1/2 for 1,4-dioxane, δ1 = 26.5 MPa1/2 for ethanol, and δ2 = 47.8 MPa1/2 for water [36] and mixtures compositions expressed in volume fractions (fi) [18, 26]:

$$\delta_{{{1} + {2}}} = \sum\limits_{i = 1}^{2} {f_{i} \delta_{i} }$$
(1)
Fig. 4
figure 4

Mole fraction solubility of pyrazinamide (3) in some {cosolvent (1) + water (2)} mixtures as function of the Hildebrand solubility parameter of mixtures at 298.15 K. ○: {1,4-dioxane (1) + water (2)}; ●: {ethanol (1) + water (2)}. Lines correspond to the best regular polynomials correlating the data

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It is well-known that organic compounds reach maximum solubilities in solvent systems that exhibit the same or similar δ1+2 [9, 18]. Thus, the δ3 value of pyrazinamide would be variable between 24.7 MPa1/2 (of maximum peak in 1,4-dioxane mixtures) and 31.9 MPa1/2 (of maximum peak in ethanol mixtures) at T = 298.15 K. Nevertheless, the calculated Fedors δ3 value of this drug is 35.1 MPa1/2 as shown in Table 3 [37], which is higher than the δ3 values obtained based on maximum solubilities. This demonstrates that any other properties of solvents and solute apart of polarity must be involved in drug solubilities.

Table 3 Application of the Fedors’ method to estimate internal energy, molar volume, and Hildebrand solubility parameter of pyrazinamide (3)
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3.3 Correlation of Pyrazinamide Solubility

The simplest model to correlate and/or predict drug solubility in cosolvent mixtures is based on the algebraic rule of mixing [38]. For semipolar compounds in aqueous binary mixtures this model would take the following form:

$$\ln x_{3,1 + 2} \, = \,w_{1} \ln x_{3,1} \, + \,w_{2} \ln x_{3,2}$$
(2)

where x3,1+2 is the mole fraction drug solubility calculated in the respective cosolvent mixture, x3,1 is the mole fraction pyrazinamide solubility in neat cosolvent (component 1), x3,2 is the mole fraction pyrazinamide solubility in neat water (component 2), and w1 and w2 are the mass fractions of cosolvent (1) and water (2) in the solvent mixtures free of pyrazinamide (3). The main advantage of Eq. 2 is that only requires the solubility of pyrazinamide in both neat solvents and the mixtures composition to calculate the solubility of this drug in all the possible mixtures. Nevertheless, Eq. 2 is not adequate to correlate the pyrazinamide solubility values in aqueous mixtures of 1,4-dioxane and ethanol because the maximum logarithmic solubility is obtained in mixtures instead of neat cosolvent as shown in Fig. 5. As observed in this figure, positive deviations to additive logarithmic solubilities rule are observed in all the mixtures.

Fig. 5
figure 5

Logarithmic mole fraction solubility of pyrazinamide (3) in some {cosolvent (1) + water (2)} mixtures at 298.15 K. ○: {1,4-dioxane (1) + water (2)}; ●: {ethanol (1) + water (2)}. Solid lines correspond to the best regular polynomials correlating the data. Smoothed lines correspond to additive logarithmic behavior

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Figure 6 shows more clearly the positive differences between real and ideal logarithmic solubilities as function of mixtures’ composition for both aqueous cosolvent systems at 298.15 K. In this way, and additional term considering these deviations is required in Eq. 2 is required to make it more adequate for pyrazinamide solubility correlation and/or prediction.

Fig. 6
figure 6

Excess logarithmic mole fraction (x3) solubility of pyrazinamide (3) in some {cosolvent (1) + water (2)} mixtures at 298.15 K. ○: {1,4-dioxane (1) + water (2)}; ●: {ethanol (1) + water (2)}. Lines correspond to the best regular polynomials correlating the data

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Modified NIBS/R-K model represents the solubility of drugs in aqueous binary solvent mixtures at isothermal conditions as follows.

$$\ln x_{3,1 + 2} = w_{1} \ln x_{3,1} + w_{2} \ln x_{3,2} + w_{1}^{{}} w_{2}^{{}} \sum\limits_{i = 0}^{n} {A_{i} \left( {w_{1} - w_{2} } \right)^{i} }$$
(3)

where Ai terms are the model constants [11, 25]. Solubility values were adjusted to a fourth degree model for considering the observed positive deviations. Thus, the obtained constants for solubility behavior of pyrazinamide (3) in {1,4-dioxane (1) + water (2)} and {ethanol (1) + water (2)} mixtures and the computed mean percentage deviations (MPDs) are listed in Table 4. The MPD were calculated using:

$${\text{MPD}} = \frac{100}{N}\sum {\frac{{\left| {x_{3,1 + 2}^{{\exp {\text{tl}}}} - x_{3,1 + 2}^{{{\text{calc}}}} } \right|}}{{x_{3,1 + 2}^{{{\text{exptl}}}} }}}$$
(4)

where N is the number of experimental data points, namely 19 in every cosolvent system. Overall MPDs of 2.01 % and 0.80 % were obtained for {1,4-dioxane (1) + water (2)} and {ethanol (1) + water (2)}, respectively, when using Eq. 3. These deviations are in the same order of magnitude in experimental uncertainties which makes practical the use of this model for correlating pyrazinamide solubility in these mixtures.

Table 4 The modified NIBS-RK model constants, statistical parameters and mean percentage deviations (MPD)
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3.4 Gibbs Energies of Dissolution and Mixing of Pyrazinamide

The ideal solubility of pyrazinamide as a crystalline drug that suffers a polymorphic transition (ptr) is calculated by means of the following expression:

$$\ln x_{3}^{{{\text{id}}}} = - \frac{{\Delta_{{\text{f}}} H(T_{{\text{f}}} - T)}}{{RT_{{\text{f}}} T}} - \frac{{\Delta_{{{\text{ptr}}}} H(T_{{{\text{ptr}}}} - T)}}{{RT_{{{\text{ptr}}}} T}} + \left( {\frac{{\Delta C_{p} }}{R}} \right)\left[ {\frac{{(T_{{\text{f}}} - T)}}{T} + \ln \left( {\frac{T}{{T_{{\text{f}}} }}} \right)} \right]$$
(5)

where, ΔfH is the molar enthalpy of fusion of the solid at the melting point, Tf is the melting point in K, T is the absolute solution temperature in K, R is the gas constant (8.314 J·mol–1·K–1), ΔptrH is the molar enthalpy of polymorphic transition of the solid at the transition temperature, Tptr is the polymorphic transition temperature in K, and ΔCp is the difference between the molar heat capacity of the crystalline form and the molar heat capacity of the hypothetical super-cooled liquid form of the solute at 298.15 K [39]. Since ΔCp is not frequently reported in the literature for drugs, it may be approximated to the entropy of fusion, ΔSf, which is calculated as: ΔHf/Tf. The ideal solubility of pyrazinamide at 298.15 K calculated based on the calorimetric values reported in the caption of Fig. 2 for untreated sample was: \(x_{3}^{{{\text{id}}}}\) = 4.501 × 10–2 (as shown in Table 2). Another useful physicochemical dissolution property to analyze solute–solvent affinity is the asymmetric activity coefficient of pyrazinamide in the saturated solutions (\(\gamma_{3}^{{{\text{Asym}}}}\)) [39]. Here the \(\gamma_{3}^{{{\text{Asym}}}}\) values are defined on asymmetric basis (i.e. regarding the solute) and calculated as follows:

$$\gamma_{3}^{{{\text{Asym}}}} = \frac{{x_{3}^{{{\text{id}}}} }}{{x_{3} }}$$
(6)

Table 2 shows that the asymmetric activity coefficients of pyrazinamide are higher than unit in both aqueous cosolvent systems. They diminish from neat water to reach minimum values in the mixtures of w1 = 0.85 for {1,4-dioxane (1) + water (2)} and w1 = 0.70 for {ethanol (1) + water (2)}, respectively. Later, they increase with the cosolvent proportion.

Furthermore, the experimental solubilities expressed in mole fraction summarized in Table 2 allows the calculation of the standard Gibbs energy change relative to the dissolution processes of pyrazinamide according to the following expression [40]:

$$\Delta_{{{\text{soln}}}} G^{{\text{o}}} = - RT\ln a_{3} = - RT\ln \left( {\gamma_{3}^{{{\text{Sym}}}} \cdot x_{3} } \right)$$
(7)

Owing the low solubility values exhibited by pyrazinamide in all those solvent systems the symmetrical activity coefficients of the solute (\(\gamma_{3}^{{{\text{Sym}}}}\)) could be approximated to unit and thus, thermodynamic activities (a3) could be assumed as the drug concentrations at saturation (x3). When this calculation is performed upon this assumption the \(\Delta_{{{\text{soln}}}} G^{{\text{o}}}\) values are just considered as apparent. Table 2 shows that apparent \(\Delta_{{{\text{soln}}}} G^{{\text{o}}}\) values diminish continuously with the cosolvent (1) proportion in the composition intervals of 0.00 ≤ w1 ≤ 0.85 (for aqueous 1,4-dioxane) and 0.00 ≤ w1 ≤ 0.70 (for aqueous ethanol) indicating higher affinity of pyrazinamide by the cosolvent-rich media compared with neat water. After these compositions the \(\Delta_{{{\text{soln}}}} G^{{\text{o}}}\) values increases slightly with the cosolvent proportion in the mixtures.

On the other hand, the overall pyrazinamide dissolution process may be represented by the following hypothetic stages:

Solute(as solid) at T → Solute(as solid) at Tf → Solute(as liquid) at Tf → Solute(as liquid) at T → Solute(in solution) at T.

where the hypothetical dissolution stages are the heating and fusion of pyrazinamide, the cooling of the liquid pyrazinamide to T = 298.15 K, and the subsequent mixing of the hypothetical super-cooled liquid pyrazinamide with the cosolvent system at the same temperature [39]. Because the ideal solubility of pyrazinamide is directly related to the thermodynamic properties of fusion of the same solid forms, the apparent standard Gibbs energies of mixing (\(\Delta_{{{\text{mix}}}} G^{{\text{o}}}\)), were calculated by means of Eq. 8. Table 2 shows that all the \(\Delta_{{{\text{mix}}}} G^{{\text{o}}}\) values in cosolvent mixtures and neat water and neat 1,4-dioxane (or ethanol) are positive because real experimental solubilities are lower than ideal solubility.

$$\Delta_{{{\text{mix}}}} G^{{\text{o}}} = \Delta_{{{\text{soln}}}} G^{{\text{o}}} - \Delta_{{{\text{soln}}}} G^{{\text{o - id}}}$$
(8)

3.5 Preferential Solvation of Pyrazinamide

The preferential solvation parameter of pyrazinamide (compound 3) by cosolvent (compound 1) in {cosolvent (1) + water (2)} mixtures is defined as [22, 41]:

$$\delta x_{{1,3}} = x_{{1,3}}^{L} - x_{{1}} = - \delta x_{{2,3}}$$
(9)

where \(x_{{1,3}}^{L}\) is the local mole fraction of cosolvent (1) in the molecular vicinity of pyrazinamide (3) and x1 is the bulk mole fraction composition of cosolvent (1) in the initial binary solvent. If δx1,3 > 0 pyrazinamide is preferentially solvated by cosolvent (1). If δx1,3 < 0 it is preferentially solvated by water (2). Values of δx1,3 were obtained based on the IKBI method by using some thermodynamic quantities as defined in the following equations [26, 41]:

$$\delta x_{{1,3}} = \frac{{x_{{1}} x_{{2}} \left( {G_{{1,3}} - G_{{2,3}} } \right)}}{{x_{{1}} G_{{1,3}} + x_{{2}} G_{{2,3}} + V_{{{\text{cor}}}} }}$$
(10)

where,

$$G_{{1,3}} = RT\kappa_{T} - V_{{3}} + x_{{2}} V_{{2}} D/Q$$
(11)
$$G_{{2,3}} = RT\kappa_{T} - V_{{3}} + x_{{1}} V_{{1}} D/Q$$
(12)
$$V_{{{\text{cor}}}} = 2522.5\left( {r_{{3}} + 0.1363\left( {x_{{1,3}}^{L} V_{{1}} + x_{{2,3}}^{L} V_{{2}} } \right)^{1/3} - 0.085} \right)^{3}$$
(13)

In these equations κT is the isothermal compressibility of the aqueous cosolvent mixtures, which is calculated as an additive property by using the mixtures compositions and the reported κT values for neat solvents, V1 and V2 are the partial molar volumes of cosolvent and water in the mixtures, and V3 is the partial molar volume of pyrazinamide in these mixtures. The function D (Eq. 14) is the derivative of the standard molar Gibbs energies of transfer of pyrazinamide from neat water (2) to every {cosolvent (1) + water (2)} mixture with respect to the cosolvent proportion. The function Q (Eq. 15) involves the second derivative of the Gibbs energy of mixing of both solvents (\(G_{{{1} + 2}}^{Exc}\)) with respect to the water proportion. Vcor is the correlation volume and r3 is the molecular radius of pyrazinamide calculated by means of Eq. 16 where NAv as the Avogadro’s number (6.02 × 1023 molecules·mol–1).

$$D = \left( {\frac{{\partial \Delta_{{{\text{tr}}}} G_{{{3,2} \to 1 + 2}}^{{\text{o}}} }}{{\partial x_{{1}} }}} \right)_{T,p}$$
(14)
$$Q = RT + x_{{1}} x_{{2}} \left( {\frac{{\partial^{2} G_{{{1} + {2}}}^{Exc} }}{{\partial x_{{2}}^{2} }}} \right)_{T,p}$$
(15)
$$r_{{3}} = \left( {\frac{{3 \cdot 10^{21} V_{{3}} }}{{4\pi N_{{{\text{Av}}}} }}} \right)^{1/3}$$
(16)

Definitive correlation volumes require iteration because it depends on the local mole fractions of both solvents around pyrazinamide molecules. This iteration is performed by replacing δx1,3 and Vcor in Eqs. 9, 10 and 13 to recalculate \(x_{{1,3}}^{L}\) until a non-variant value of Vcor is obtained. Figure 7 shows the Gibbs energy of transfer behavior of pyrazinamide (3) from neat water (2) to all {cosolvent (1) + water (2)} systems at 298.15 K. These values were calculated from the x3 (by assuming x3 = a3) values summarized in Table 2 by using the following:

$$\Delta_{{{\text{tr}}}} G_{{{3,2} \to {1} + 2}}^{{\text{o}}} = RT\ln \left( {\frac{{x_{{3,2}} }}{{x_{{{3,1} + {2}}} }}} \right)$$
(17)
Fig. 7
figure 7

Gibbs energy of transfer of pyrazinamide from neat water (2) to some {cosolvent (1) + water (2)} mixtures at 298.15 K. ○: {1,4-dioxane (1) + water (2)}; ●: {ethanol (1) + water (2)}. Lines correspond to the best regular polynomials correlating the data

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The \(\Delta_{{{\text{tr}}}} G_{{{3,2} \to {1} + 2}}^{{\text{o}}}\) values were correlated according to sixth degree regular polynomials presented as Eq. 18. The obtained coefficients and statistical parameters are summarized in Table 5.

$$\Delta_{{{\text{tr}}}} G_{{{3,2} \to {1} + 2}}^{{\text{o}}} = a + bx_{{1}} + cx_{{1}}^{2} + dx_{1}^{3} + ex_{1}^{4} + fx_{1}^{5} + gx_{1}^{6}$$
(18)
Table 5 Gibbs energy of transfer correlation model constants and statistical parameters
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Thus, D values reported in Table 6 were calculated from the first derivative of the polynomial models solved according to the cosolvent proportion in the mixtures. Otherwise, Q and RT·κT values of both aqueous cosolvent systems as well as the partial molar volumes of cosolvents and water in the binary mixtures at 298.15 K, were taken from the literature [42]. Molar volume of pyrazinamide was considered in this research as independent of the cosolvent composition. This value was calculated as 71.9 cm3·mol–1 based on the Fedors’ method as shown in Table 3 [36, 37].

Table 6 Some solvation properties of pyrazinamide (3) in some {cosolvent (1) + water (2)} mixtures at 298.15 K
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Table 6 shows that D values for pyrazinamide are negative in compositions 0.00 ≤ x1 ≤ 0.55 and 0.00 ≤ x1 ≤ 0.40 for {1,4-dioxane (1) + water (2)} and {ethanol (1) + water (2)} mixtures, respectively, and positive in the other compositions. The G1,3 values are negative in both cosolvent systems in all compositions, whereas G2,3 values are negative in the composition interval of 0.00 ≤ x1 ≤ 0.55 in aqueous mixtures of 1,4-dioxane and also in all the compositions of the aqueous ethanol mixtures. Pyrazinamide radius (r3) value, required to calculate the correlation volumes, was calculated by means of Eq. 16 as 0.305 nm. Otherwise, Vcor values increase as the molar volume of the cosolvent mixtures also increase because of the higher molar volumes of 1,4-dioxane (1) and ethanol (1) regarding water (2).

Figure 8 and Table 6 show that the δx1,3 values for pyrazinamide vary non-linearly with the cosolvent (1) proportion for both aqueous cosolvent mixtures. Both systems exhibit three regions, two of them were preferential hydration is observed corresponding to water-rich and cosolvent-rich mixtures and one region where preferential solvation by cosolvent is apparently observed. Thus, in aqueous 1,4-dioxane mixtures pyrazinamide is preferentially hydrated in the interval from neat water to the mixture of x1 = 0.17 with a maximum negative δx1,3 value of − 1.54 × 10–2 in the mixture of x1 = 0.05. Because this value is higher than |0.01| it is considered as a real preferential solvation effect by water [43]. In the interval of 0.17 < x1 < 0.56 pyrazinamide is preferentially solvated by 1,4-dioxane owing the positive δx1,3 values that reach a maximum of 3.05 × 10–2 in the mixture of x1 = 0.45, which is also higher than |0.01|. Besides, in the interval of 0.56 < x1 < 1.00 pyrazinamide is again preferentially solvated by water with a maximum negative δx1,3 value of − 6.07 × 10–2 in the mixture of x1 = 0.70. Otherwise, in the case of aqueous ethanol mixtures, pyrazinamide is preferentially hydrated in the interval of 0.00 < x1 < 0.24 with a maximum negative δx1,3 value of − 1.40 × 10–2 in the mixture of x1 = 0.10. In the interval of 0.24 < x1 < 0.43 the δx1,3 values are positive but lower than |0.01| and thus they could be a consequence of uncertainties propagation in IKBI calculations instead of real preferential solvation effects by ethanol. Finally, in the interval of 0.43 < x1 < 1.00 pyrazinamide is preferentially hydrated again with a maximum negative δx1,3 value of − 4.25 × 10–2 in the mixture of x1 = 0.80. It is conjecturable that the structuring of water molecules around the aromatic ring of pyrazinamide (Fig. 1) could be contributing to make negative the net δx1,3 values in water-rich mixtures in both systems. However, hydrogen bonding between water and basic Lewis groups of solute could also play an important role.

Fig. 8
figure 8

δx1,3 values of pyrazinamide in some {cosolvent (1) + water (2)} mixtures at 298.15 K. ○: {1,4-dioxane (1) + water (2)}; ●: {ethanol (1) + water (2)}. Lines correspond to the δx1,3 values calculated with Eq. 14 according to the solvent mixtures composition

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Preferential solvation of pyrazinamide by 1,4-dioxane could be related to the breaking of the ordered structure of water around the non-polar moieties of this drug. Moreover, the preferential solvation by 1,4-dioxane molecules it is conjecturable that pyrazinamide would be acting as Lewis acid in front of the cosolvent molecules owing its non-shared higher electron density of its ether oxygen atoms or by polarization effects.

Finally, in cosolvent-rich mixtures, where pyrazinamide is preferentially solvated by water, it could be acting mainly as a Lewis base in front of the water owing the higher acidity behavior of water regarding 1,4-dioxane and ethanol as defined based on the Kamlet-Taft hydrogen bond donor parameters that are as follows, α = 1.17 for water, 0.86 for ethanol and 0.00 for 1,4-dioxane, respectively [44].

4 Conclusions

Equilibrium solubility of pyrazinamide (3) in some aqueous-cosolvent mixtures of 1,4-dioxane or ethanol at 298.15 K has been determined and reported. These solubilities were adequately correlated with the modified NIBS-RK model. Expressions for the local mole fraction of cosolvent (1) and water (2) around the molecules of this drug in solution were derived on the basis of the IKBI method. Thus, pyrazinamide is preferentially solvated by water in water-rich mixtures and cosolvent-rich mixtures, but preferentially solvated by cosolvent in mixtures with intermediate compositions. Briefly, one of the most relevant findings of the study was to demonstrate that the maximum solubility of the drug does not depend exclusively on the polarity of the cosolvent mixture, but also depends on the molecular interactions that may occur in relation to the molecular characteristics of the cosolvent (1,4-dioxane or ethanol). Moreover, this research expand the physicochemical information about antitubercular drugs that could be useful for understanding the dissolution mechanism in blended solvents as help for designing liquid dosage forms.