Solubility, Solution Thermodynamics, and Preferential Solvation of Amygdalin in Ethanol + Water Solvent Mixtures

Abstract

The equilibrium solubility of amygdalin in [ethanol (1) + water (2)] mixtures at 293.15 K to 328.15 K was reported. The thermodynamic properties (standard enthalpy Δ_solnH°, standard entropy Δ_solnS°, and standard Gibbs energy of solution Δ_solnG°) were computed using the generated solubility data via van’t Hoff and Gibbs equations. The dissolution process of amygdalin is endothermic and the driving mechanism in all mixtures is entropy. Maximal solubility was achieved in 0.4 mole fraction of ethanol at 328.15 K and the minimal one in neat ethanol at 293.15 K. Van’t Hoff, Jouyban–Acree–van’t Hoff, and Buchowski–Ksiazczak models were used to simulate the obtained solubility data. The calculated solubilities deviate reasonably from experimental data. Preferential solvation parameters of amygdalin in mixture solvents were analyzed using the inverse Kirkwood–Buff integrals (IKBI) method. Amygdalin is preferentially solvated by water in ethanol-rich mixtures, whereas in water-rich mixtures, there is no clear evidence that determines which of water or ethanol solvents would be most likely to solvate the molecule.

1. Introduction

Amygdalin (Figure 1) is a naturally occurring cyanogenic diglycoside with a molecular formula of C₂₀H₂₇NO₁₁ and a molecular mass of 457.4 g mol⁻¹. It is a major bioactive component present mostly in kernels and seeds of “Rosaceae” plants such as peaches, apples, cherries, and more [1,2]. The use of amygdalin can lead to the release of toxic hydrogen cyanide (HCN) through the action of emulsin enzyme from the human intestinal microflora [3]. The HCN selectively decomposes cancer cells in the tumor site inside the body [4,5]. Several studies have demonstrated the antitumor activities of amygdalin on prostate cancer, bladder cancer, lung cancer, rectal cancer, and colon cancer [6]. Furthermore, highly purified amygdalin used in therapeutic dosage levels has antioxidant, anti-fibrosis [7], anti-inflammatory, analgesic [8,9], anti-atherosclerosis [10,11,12], anti-cardiac hypertrophy [13], anti-ulcer [14], anti-tussive, and anti-asthmatic effects [15].

Besides the degradation of amygdalin caused by enzymes from the gut microflora, plant enzymes (β-glucosidases and α-hydroxynitrile lyases) can lead to the production of cyanide when plant tissue is damaged or seeds are crushed or macerated. Enzymatic degradation of amygdalin to gentibiose, benzaldehyde, and HCN usually takes place in an alkaline solution [16].

In addition to enzymatic hydrolysis mentioned above, amygdalin degradation can also occur in boiling water through the process of epimerization, particularly under mild basic conditions as well as in a long extraction time [17,18,19]. In Bolarinwa et al.’s research [17], it was demonstrated that at 100 °C of boiling water, an extended extraction period can result in reduced extraction yield due to the conversion of amygdalin into neoamygdalin (amygdalin epimer).

Extracting a high rate of amygdalin from food plants without causing degradation of the molecule is challenging to achieve. Therefore, the selectivity of a solvent for this compound is a crucial parameter, as this will have a paramount influence on the extraction process. Extraction rate and time can be considerably affected by amygdalin solubility in solvents [17]. It is therefore of some interest to know the solubility of amygdalin in different mixtures of solvents. On the other hand, the knowledge of the solubility behavior in different solvent systems is of high importance in the pharmaceutical industry as it influences the drug efficacy and its pharmacokinetics [20]. Solubility data are also useful for drug purification, refining procedures, and method development [21,22,23].

For the above-mentioned reasons, the solubility and solution thermodynamics of amygdalin in pure and solvent mixtures are quite essential and must be determined. The binary water and ethanol mixtures are the most versatile and most used solvent systems for these previous purposes [24,25,26,27].

Thus, the goals of this study were (1) to extend the database on the solubility of amygdalin in several ethanol (1) + water (2) mixtures over a temperature range of 298.105 to 328.15 K, (2) to study the effect of solvent composition on the solubility and solution thermodynamics of amygdalin in aqueous ethanol mixtures, (3) to calculate the apparent thermodynamic functions of solution in the investigated solvents using the van’t Hoff and Gibbs equations, and (4) to estimate the preferential solvation of amygdalin in these solvents through the method of inverse Kirkwood–Buff integrals (IKBI), which describes the local solvent proportions around the dissolved substance concerning to the composition of the cosolvent mixtures [28,29]. Some models were used to predict the solubility of amygdalin in ethanol–water mixtures at different temperatures.

2. Results and Discussion

2.1. Solubility of Amygdalin in [Ethanol (1) + Water (2)] Cosolvent Mixtures

Table 1. Experimental solubility of amygdalin (3) expressed in molar fraction (10³x₃ ^a) in [ethanol (1) + water (2)] mixtures at different temperatures. Experimental pressure p: 0.1 MPa ^b.

x1c,d	Temperature/K e
	293.15	298.15	303.15	308.15	310.15	313.15	318.15	323.15	328.15
0.00	2.54	3.25	4.19	5.49	6.07	7.04	7.91	9.73	11.0
0.10	2.80	3.55	4.61	6.16	6.69	7.83	8.86	10.8	12.9
0.20	3.07	3.99	5.21	7.12	7.42	8.84	10.4	12.2	14.6
0.30	3.34	4.48	5.84	8.08	8.32	9.90	11.8	13.6	16.9
0.40	3.56	5.06	6.60	8.39	9.32	11.0	13.3	15.3	18.3
0.50	3.12	4.06	4.84	5.90	5.99	7.26	8.67	9.52	12.1
0.60	2.20	2.58	3.17	3.94	4.14	4.81	5.55	6.30	7.89
0.70	1.53	1.88	2.14	2.52	2.83	3.25	3.77	4.12	5.45
0.80	1.00	1.21	1.36	1.70	1.76	2.02	2.44	2.62	2.93
0.90	0.71	0.83	0.91	1.12	1.15	1.27	1.35	1.57	1.70
1.00	0.51	0.56	0.61	0.65	0.69	0.77	0.91	0.94	1.03

^a Average relative uncertainty in mole fraction solubility is u(x₃) = 0.025. ^b Standard uncertainty in pressure u(p) = 0.001 MPa. ^c x₁ is the mole fraction of ethanol (1) in the {ethanol (1) + water (2)} mixtures free of amygdalin (3). ^d Average relative standard uncertainty in x₁ is ur(x₁) = 0.01. ^e T is the absolute temperature. Standard uncertainty in temperature is u(T) = 0.05 K.

shows the experimental solubility of amygdalin (3) in {ethanol (1) + water (2)} cosolvent mixtures including EtOH and water neat solvents at nine temperatures (293.15–328.15 K).

Experimental results demonstrate that the solubility increases with temperature indicating endothermic dissolution (Figure 2A). The maximum solubility of amygdalin was observed in the cosolvent mixture x₁ = 0.4 at 328.15 K (Figure 2B). The addition of ethanol (in water-rich mixtures) has a positive cosolvent effect, enhancing amygdalin solubility. Indeed, the presence of the non-polar phenyl group in the amygdalin chemical structure may cause the formation of a structured water layer around it. As the proportion of ethanol in the solvent mixture increases, the solvation water shell will be ruptured [30,31,32], therefore, increasing amygdalin’s solubility in the system.

The solubility profiles of amygdalin in {ethanol (1) + water (2)} binary mixtures at different temperatures were plotted as a function of the Hildebrand solubility parameter δ_{1 + 2} of the mixtures (Figure 2C).

For binary mixtures, δ_{1 + 2} is calculated as [33]:

$${\delta }_{1 + 2}=f_1{\delta }_1+\left(1-f_1\right){\delta }_2$$

(1)

where δ₁ and δ₂ are the Hildebrand solubility parameters of the pure solvents (δ₁ = 26.5 MPa^1/2 for ethanol (1) [34] and δ₂ = 47.8 MPa^1/2 for water (2) [34]; f is the solute-free volume fraction which is calculated assuming additive volumes as:

f = V₁/(V₁ + V₂)

(2)

where V₁ and V₂ are the volumes of cosolvent and water, respectively.

Considering the entire polarity region, the solubility increases from pure water (δ = 47.8 MPa^1/2) up to the mixture with x₁ = 0.40 (δ_mix = 33.2 MPa^1/2), where the curve shows a maximum solubility peak; from this mixture up to pure ethanol, the solubility decreases in all cases (Figure 2C).

This behavior is commonly observed in compounds whose polarity coincides with the polarity of a mixture of solvents (δ₁ > δ₃ > δ₂ or δ₂ > δ₃ > δ₁) [35,36].

According to the literature, solutes reach their maximum solubility in solvents with the same solubility parameter [36] and thus, the δ₃ value of amygdalin (3) would be 33.2 MPa^1/2. However, the solubility parameter of amygdalin (3), estimated in accordance with the group contribution methods proposed by Fedors and van Krevelen, is δ₃ = 29.9 MPa^1/2 (

Table 2. Estimation of the solubility parameter of amygdalin by Fedor’s method [34].

Group or Atom	Quantity	∆V (cm3 mol−1)	∆U (kJ mol−1)
–CH2	2	16.1	4.94
–CH<	11	13.5	4.31
–OH	7	10	29.8
–O–	4	3.8	3.35
–C≡N	1	24	25.5
Phenyl	1	71.4	31.9
Ring closure	1	16	1.05
Total		377.3	337.74
			δ3 = (337,740/377.3)1/2 = 29.9 MPa1/2

), which is lower than the experimental value obtained in this work at the solubility maximum (δ₃ = 33.2 MPa^1/2).

It is important to note that the group contribution methods only provide a rough estimation of δ₃; however, this calculation is relevant to identify the most suitable solvent or solvent mixture to dissolve the drug, which is useful information in experimental and industrial designs.

2.2. Computational Validation

The use of calculation models to predict the solubility of chemicals in mixed solvents is one of the lines of research that have evolved the most in recent years. Some of the most widely implemented models are those of, van’t Hoff, Jouyban–Acree–van’t Hoff and Buchowski–Ksiazczak λh.

Thus, the van’t Hoff equation (Equation (3)) presents a relationship between solubility (expressed in mole fraction) and temperature.

$$x_3=e^{\left(A + \frac{B}{T}\right)}$$

(3)

A and B are parameters, which can be related to thermodynamic parameters such as dissolution enthalpy and dissolution entropy [37].

Jouyban and Acree developed a specific model for the prediction of the solubility of drugs in {ethanol (1) + water (2)} cosolvent mixtures at a specific temperature (Equation (4)) [38,39,40]:

$$\begin{gathered} \ln x_{3,1+2}=x_1\ln x_{3,1}+ x_2\ln x_{3,2}+ x_1{x}_2\left[724.21T^{-1}+485.17\left(x_1 \\ -x_2\right)T^{-1}+194.41{\left(x_1-x_2\right)}^2{T}^{-1}\right] \end{gathered}$$

(4)

Introducing the van’t Hoff model, the Jouyban–Acree would be left, with the advantage of being able to calculate solubility at various temperatures [41].

$$\begin{gathered} \ln x_{3,1+2}=x_1\left({\mathrm{A}}_1+{\mathrm{B}}_1{T}^{-1}\right)+x_2\left({\mathrm{A}}_2+{\mathrm{B}}_2{T}^{-1}\right)+x_1{x}_2\left[724.21T^{-1}+485.17\left(x_1 \\ -x_2\right)T^{-1}+194.41{\left(x_1-x_2\right)}^2{T}^{-1}\right] \end{gathered}$$

(5)

For this investigation, when calculating A and B coefficients by linear regression, the following equation is obtained:

$$\begin{gathered} \ln {\mathrm{x}}_{3,1+2}=x_1\left(-0.69\pm 0.22-2048T^{-1}\right)+x_2\left(8.06\pm 0.28-4101\pm 87T^{-1}\right) \\ +x_1{x}_2\left[724.21T^{-1}+485.17\left(x_1-x_2\right)T^{-1} \\ +194.41{\left(x_1-x_2\right)}^2{T}^{-1}\right] \end{gathered}$$

(6)

The Buchowski–Ksiazczak λh equation, (Equation (8)), is another way to describe the solubility behavior:

$$x_3=\frac{\lambda e^{\lambda h{T}_f^{-1}}}{\lambda e^{\lambda h{T}_f^{-1}}-e^{\lambda h{T}_f^{-1}}+e^{\lambda h{T}^{-1}}}$$

(7)

where λ and h are the two parameters of the Buchowski–Ksiazczak model, and T_f represents the melting point of drug [42,43,44].

The mean percentage deviation (MPD) was calculated from Equation (8) [45,46]:

$$\mathrm{MPD}=\frac{100}{N}{\displaystyle \sum }\frac{\left|{\mathrm{x}}_{3,1+2}^{\mathrm{cal}}-{\mathrm{x}}_{3,1+2}^{\mathrm{Exp}}\right|}{{\mathrm{x}}_{3,1+2}^{\mathrm{Exp}}}$$

(8)

where N is the number of experimental data points, and ${\mathrm{x}}_{3,1+2}^{\mathrm{cal}}$ and ${\mathrm{x}}_{3,1+2}^{\mathrm{Exp}}$ are the calculated and experimental solubility values.

Thus, amygdalin solubility was estimated employing Equations (3), (6), and (7) and then, the MPD values were calculated using Equation (8).

The MDP values show that the model that best predicts the experimental data is the van’t Hoff model (3.3%), followed by the Buchowski–Ksiazczak model (4.3%), and finally, the Jouyban–Acree–van’t Hoff mode presents a MDP of 22.8%.

Figure 3 shows the calculated solubility versus observed solubility data of amygdalin in {ethanol (1) + water (2)} cosolvent mixtures, using the van’t Hoff, Jouyban–Acree–van’t Hoff and Buchowski–Ksiazczak models. A relatively low determination coefficient was observed (R² = 0.76) indicating a poor prediction accuracy for the Jouyban–Acree–van’t Hoff model; however, the van’t Hoff and Buchowski–Ksiazczak models present correlation coefficients close to one, indicating a good correlation of the data calculated with these models and the experimental data.

Therefore, in general terms, the Jouyban–Acree–van’t Hoff model does not predict the solubility of amygdalin in {ethanol (1) + water (2)} cosolvent mixtures adequately; however, the van’t Hoff and Buchowski–Ksiazczak models show very good precision, as demonstrated with the MDP values.

2.3. Thermodynamic Functions of Dissolution

From the experimental solubility data (Table 1), the thermodynamic functions of dissolution (

Table 3. Thermodynamic functions of dissolution processes of amygdalin in {ethanol (1) + water (2)} cosolvent mixtures at T_hm = 310.22 K.

x1a	ΔsolnGo/ kJ mol−1	ΔsolnHo/ kJ mol−1	ΔsolnSo/ J mol−1 K−1	TΔsolnSo/ kJ mol−1	ζH b	ζTS b
0.00	13.31	34.10	67.03	20.79	0.621	0.379
0.10	13.02	35.34	71.93	22.31	0.613	0.387
0.20	12.71	35.86	74.62	23.15	0.608	0.392
0.30	12.41	36.77	78.52	24.36	0.601	0.399
0.40	12.15	37.04	80.21	24.88	0.598	0.402
0.50	13.07	29.92	54.31	16.85	0.640	0.360
0.60	14.13	29.10	48.26	14.97	0.660	0.340
0.70	15.12	28.14	41.95	13.01	0.684	0.316
0.80	16.32	25.36	29.14	9.04	0.737	0.263
0.90	17.48	20.26	8.96	2.78	0.879	0.121
1.00	18.66	17.03	−5.25	−1.63	0.913	0.087

^ax₁ is the mole fraction of ethanol (1) in the {ethanol (1) + water (2)} mixtures free of amygdalin (3). Standard uncertainty in T is u(T) = 0.10 K. Average relative standard uncertainties in apparent thermodynamic quantities of real dissolution processes are u_r(∆_solnG°) = 0.02, u_r(∆_solnH°) = 0.02, u_r(∆_solnS°) = 0.03, and u_r(T∆_solnS°) = 0.03. ^b ζ_H and ζ_TS are the relative contributions by enthalpy and entropy toward apparent Gibbs energy of dissolution.

) were calculated using the van’t Hoff and Gibbs equations, under Krug modifications [47,48]:

$${∆}_{\mathrm{soln}}{H}^o=-R\left(\partial \ln x_3/\partial \left(T^{-1}-T_{\mathrm{hm}}^{-1}\right)\right)$$

(9)

$${∆}_{\mathrm{soln}}{G}^o=-R\times T_{\mathrm{hm}}\times \mathrm{intercept}$$

(10)

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

(11)

where ${∆}_{\mathrm{soln}}{H}^o$ represents the solution standard enthalpy, ${∆}_{\mathrm{soln}}{S}^o$ represents the solution standard entropy, ${∆}_{\mathrm{soln}}{G}^o$ represents the solution standard Gibbs energy, R represents the constant of gases, and T_hm represents the mean harmonic temperature defined as: T_hm = n/Σ(1/T), where n is the number of studied temperatures (the harmonic mean temperature for this investigation is 310.22 K).

Upon graphing ln x₃ vs. (T⁻¹ − T_hm⁻¹), the slope (∂ln x₃/∂ (T⁻¹ − T_hm⁻¹)) and intercept used in Equations (10) and (11) are obtained.

Table 3 shows the data for the apparent thermodynamic functions of solution for amygdalin, Δ_solnH^o, Δ_solnG^o,and Δ_solnS^o. The values of the slope and intercept with their respective standard deviations were calculated using the TableCurve 2D program. The resulting graphs are linear for each of the EtOH + W cosolvent mixtures, obtaining correlation coefficients very close to 1 for first order linear regressions (y = a + bx) (Figure 4).

The standard Gibbs energy Δ_solnG° (Table 3) is positive over the whole composition range and decreases from neat water to the cosolvent mixture x₁ = 0.4. From this solvent composition to pure EtOH, Δ_solnG^° increases. The Δ_solnH^° is positive in every case indicating that the process of dissolution of amygdalin powder in solvents is endothermic [49,50]. The enthalpic values increase nonlinearly from neat water up to 40% in volume of EtOH, presumably because, by increasing ethanol content, the interaction of this solvent with the solute promotes the breaking of the structured water molecules (hydrogen bonds) around the non-polar group of amygdalin [33,51]. As for the standard entropy of solution (Δ_solnS^°), it is negative for pure ethanol, while it is positive for water-rich mixtures, suggesting an overall entropy-driven process for the latter mixtures. The relative contributions by enthalpy (ζ_H) and by entropy (ζ_TS) toward standard Gibbs free energy of solution are given by Equations (12) and (13), respectively:

$${\zeta }_H=\left|{\mathsf{\Delta }}_{\mathrm{soln}}{H}^o\right|{\left(\left|{\mathsf{\Delta }}_{\mathrm{soln}}{H}^o\right|+\left|T_{\mathrm{hm}}{\mathsf{\Delta }}_{\mathrm{soln}}{S}^o\right|\right)}^{-1}$$

(12)

$${\zeta }_{TS}=\left|T_{\mathrm{hm}}{\mathsf{\Delta }}_{\mathrm{soln}}{S}^o\right|{\left(\left|{\mathsf{\Delta }}_{\mathrm{soln}}{H}^o\right|+\left|T_{\mathrm{hm}}{\mathsf{\Delta }}_{\mathrm{soln}}{S}^o\right|\right)}^{-1}$$

(13)

It may be seen from

Table 4. Coefficients of the Equation (22) to Gibbs energy of transfer of amygdalin (3) at several temperatures.

Coefficient	293.15 K	298.15 K	303.15 K	308.15 K	313.15 K
a	−0.0022	0.0356	0.0446	0.0879	0.0706
b	−0.3574	−1.2782	1.8586	−5.1881	−3.8227
c	−20.48	−19.625	−18.477	−1.9751	−8.7567
d	53.775	55.874	57.349	32.26	43.505
e	−29.053	−30.735	32.277	−19.838	−25.332
R2	0.998	0.994	0.995	0.993	0.994

that, in all cases, the main contributor to the (positive) standard Gibbs energy of dissolution is the (positive) enthalpy term (ζ_H > 0.59).

2.4. Enthalpy–Entropy Compensation

The study of enthalpy–entropy compensation effects for solute dissolution has been used to identify the main mechanism involved in the cosolvent behavior on dissolution processes [52,53]. Plots of ∆_solnH^o as a function of ∆_solnG^o or T∆_solnS^o at the harmonic temperature are employed for this purpose.

Thus, when plotting ∆_solnH^o vs. ∆_solnG^o, a positive slope will indicate an enthalpy-driven dissolution process, while a negative one will indicate an entropy-driven dissolution process [39].

Similarly, when plotting ∆_solnH^o vs. T∆_solnS^o, a slope greater than one will indicate an enthalpy-driven dissolution processes while a slope of less than one indicates entropy-driven dissolution processes [7,52,53,54,55,56].

Figure 5 shows that amygdalin in {ethanol (1) + water (2)} mixture solvents exhibits two trends, both with a negative slope, suggesting that the whole dissolution process is driven by entropy.

When plotting ∆_solnH^o vs. T∆_solnS^o (Figure 6), a linear trend is observed, described by the following equation: ∆_SolnH^o = 0.758 ± 0.014T∆_SolnS^o + 18.15 ± 0.25. The slope is inferior to 1, which corroborates the previous results.

2.5. Preferential Solvation

The preferential solvation model suggested by Ben Naim, called the inverse Kirkwood–Buff Integral (IKBI), allows determining, at the molecular level, the arrangement of the solvent molecules that make up the cosolvent mixture around a dissolved solute molecule [57,58,59].

This model allows to obtain the preferential solvation parameter of amygdalin (3) by ethanol molecules ($\delta x_{1,3})$ according to [60,61,62]:

$$\delta x_{1,3}=\left[x_1\left(1-x_1\right)\left(G_{1,3}-G_{2,3}\right)\right]{\left[x_1{G}_{1,3}+\left(1-x_1\right)G_{2,3}+V_{cor}\right]}^{-1}$$

(14)

where x₁ is the molar fraction of ethanol-free amygdalin, G_1,3 and G_2,3 are the Kirkwood–Buff integrals (cm³ mol⁻¹), and V_cor is the correlation volume (cm³ mol⁻¹).

Thus, $G_{1,3}$ and $G_{2,3}$ are calculated as [63,64]:

$$G_{1,3}=RT{\kappa }_T-V_3+(1-x_1)V_{2}D{Q}^{-1}$$

(15)

$$G_{2,3}=RT{\kappa }_T-V_3+x_1{V}_{1}D{Q}^{-1}$$

(16)

where κ_T is the isothermal compressibility of ethanol + water mixtures (GPa⁻¹), V₁ and V₂ are the molar volumes of ethanol and water, respectively, in the mixtures (cm³ mol⁻¹), and V₃ is the molar volume of amygdalin in the mixed solvent (cm³ mol⁻¹)

V_cor is defined as [65,66]:

$$V_{cor}=2522.5{\left[r_3+0.1363\sqrt[3]{x_{1,3}^L{V}_1+x_{2,3}^L{V}_2}-0.085\right]}^3$$

(17)

where x_1,3^L is the local molar fraction of ethanol (1) in the surrounding area of amygdalin (3) and r₃ is the amygdalin molecular radius (nm).

V_cor is calculated by iteration using Equations (15) and (18) [67]:

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

(18)

The functions D and Q (kJ mol⁻¹) are calculated using the following equations:

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

(19)

$$D={\left(\partial {∆}_{tr}{G}_{3,2\to 1+2}^o/\partial x_1\right)}_{T,P}$$

(20)

∆_trG^o_{3,2→1 + 2} is the standard molar Gibbs energy of transfer of the solute from pure water to each {ethanol (1) + water (2)} mixture and G^E_1,2 is the excess molar Gibbs energy of mixing of the two solvents free of amygdalin.

Figure 7 shows the behavior of the Gibbs energy of transfer of amygdalin (3) from pure water (2) to {ethanol (1) + water (2)} mixtures at several temperatures. The numerical values were computed from the experimental solubility data (Table 1), by using the following equation:

$${∆}_{tr}{G}_{3,2\to 1+2}^o=RT\ln \left(x_{3,2}{x}_{3,1+2}^{-1}\right)$$

(21)

$${∆}_{tr}{G}_{3,2\to 1+2}^o=a+b{x}_1+c{x}_1{}^2+d{x}_1{}^3+e{x}_1{}^4$$

(22)

Table 4 records the numerical values of the coefficients of Equation (22) at 293.15, 298.15, 303.15, 308.15, and 313.15 K.

On the other hand, Q is calculated according to Equation (19), where G_1.2^E is calculated as [64]:

$$G_{1,2}^{\mathrm{E}}={\mathrm{x}}_1{\mathrm{x}}_2\left[2907-777\left(1-2{\mathrm{x}}_1\right)+494{\left(1-2{\mathrm{x}}_2\right)}^2\right]$$

(23)

Once D and Q are calculated together with the isothermal compressibility (κ_T) for water (0.457 GPa⁻¹) [68] and ethanol (1.248 GPa⁻¹), in addition to the molar volumes of amygdalin and the solvents in the binary mixture reported by Jiménez et al. [69], the Kirkwood–Buff integrals are calculated, and from these, the preferential solvation parameters δx_1,3 of amygdalin in the binary solvent mixtures at the studied temperatures are calculated [70].

According to the literature, positive values of δx_1,3 indicate preferential solvation of amygdalin by ethanol. Conversely, negative values of δx_1,3 indicate preferential solvation of amygdalin by water.

The values of δx_1,3 are presented in

Table 5. The δx_1,3 values of amygdalin in {ethanol (1) + water (2)} mixtures at some temperatures.

x1 a	δx1,3
	293.15	298.15	303.15	308.15	313.15
0.000	0.000	0.000	0.000	0.000	0.000
0.050	−0.001	−0.001	−0.001	−0.002	−0.001
0.100	−0.001	−0.002	−0.002	−0.002	−0.002
0.150	−0.001	−0.002	−0.002	−0.002	−0.002
0.200	−0.001	−0.001	−0.001	−0.001	−0.001
0.250	0.000	0.000	0.000	0.000	0.000
0.300	0.001	0.001	0.000	0.000	0.000
0.350	0.000	0.000	−0.001	−0.002	−0.002
0.400	−0.002	−0.003	−0.004	−0.005	−0.005
0.450	−0.007	−0.008	−0.009	−0.010	−0.010
0.500	−0.014	−0.015	−0.017	−0.017	−0.017
0.550	−0.023	−0.024	−0.026	−0.026	−0.026
0.600	−0.033	−0.035	−0.037	−0.036	−0.036
0.650	−0.043	−0.046	−0.048	−0.046	−0.047
0.700	−0.051	−0.054	−0.056	−0.054	−0.056
0.750	−0.052	−0.055	−0.058	−0.057	−0.059
0.800	−0.044	−0.047	−0.050	−0.051	−0.054
0.850	−0.030	−0.033	−0.035	−0.037	−0.039
0.900	−0.016	−0.017	−0.018	−0.021	−0.022
0.950	−0.005	−0.005	−0.006	−0.008	−0.008
1.000	0.000	0.000	0.000	0.000	0.000

^ax₁ is the mole fraction of ethanol (1) in the {ethanol (1) + water (2)} mixtures free of amygdalin (3).

and the behavior of δx_1,3 is illustrated in Figure 8. Thus, from neat water to x₁ = 0.45, the absolute value of δx_1,3 is inferior to 0.01, indicating insignificant preferential solvation, probably because the values are within the error of the measurement [60]. From this composition to pure ethanol, the values of δx_1,3 are negative and greater than 0.01. The maximum negative δx_1,3 value is reached in the mixture x₁ = 0.75 (Figure 8). These results indicate the preferential solvation of amygdalin by water. Because of the availability of two sugar moieties in the molecular structure of amygdalin, this molecule can form hydrogen bonds with proton-acceptor solvents. At the same time, amygdalin can act as a proton-acceptor (base group) molecule due to the free electron pair of the oxygen atom in the OH group or nitrogen atom of the C≡N group. Thus, the tendency of amygdalin for water in ethanol-rich mixtures could be explained in the matter of the greater acidic character of water (1.17 for water and 0.86 for ethanol, as stated in the acid scale of Taft and Kamlet [71]) interacting with proton-acceptor groups of amygdalin.

3. Experimental procedures

3.1. Reagents

Amygdalin (purity 98%) and HPLC-grade ethanol (purity 99.9%) were acquired from Sigma-Aldrich (San Luis, WA, USA).

Doubly distilled and deionized water were used in all experiments. All chemicals were used without further purification.

3.2. Solubility Determination

The employed techniques to prepare ethanol–water binary solvent mixtures and to measure the solubility of amygdalin in these solvents were used as reported in different studies [72,73,74]. The solubility of amygdalin in pure and mixed solvents was investigated at different temperatures in the range of 298.15–328.15 K. The gravimetric method was used to measure the composition of the saturated solutions.

The solvent mixtures were prepared by mass using a Sartorius balance (CP225D) with an accuracy of ±0.01 mg. An excess of amygdalin powder was added to the liquid phase, and the saturated solutions were brought into a twofold jacketed reactor (Polystat Huber CC2) at T ± 0.1 K. The solutions are magnetically stirred at the desired temperature for at least 72 h to ensure the saturation equilibrium. Thereafter, they were allowed to settle for 2 h before sampling.

The supernatant solutions were withdrawn, filtered through a 0.45-µm syringe filter, and then dried in a vacuum oven at 328.15 K. The mass of the dried samples was periodically measured using an analytical balance until stability. All determinations were performed three times to check reproducibility, and then an average value was taken to determine the amygdalin solubility in all systems at each condition. The solubility of amygdalin was calculated by molar fraction ($x_A$) in pure and different binary ethanol–water mixtures using the Equations (1) and (2), respectively.

4. Conclusions

The solubilities of amygdalin in {ethanol (1) + water (2)} mixtures were determined at different temperatures. The maximum solubility was obtained in 0.4-mole fraction of ethanol at 328.15 K and the lowest one in pure ethanol at 293.15 K. The amygdalin solubility was calculated using the van’t Hoff, Jouyban–Acree-van’t Hoff, and Buchowski-–Ksiazczak models, the data obtained using the Jouyban–Acree–van’t Hoff model showing important deviations with respect to experimental solubility; however, the van’t Hoff and Buchowski–Ksiazczak models showed a good correlation with the experimental data. As for solution thermodynamics, an endothermic process was observed, with a pronounced enthalpic contribution, but with entropic conduction.

The IKBI approach demonstrated that amygdalin is preferentially solvated by water in ethanol-rich mixtures, which is consistent with the decrease in amygdalin solubility by the addition of ethanol. Whereas, in water-rich mixtures (0 < x₁ < 0.45), the solvent that will solvate the amygdalin molecule was not well defined.

In general terms, the data presented in this research expand the physicochemical information of amygdalin in binary aqueous-cosolvent mixtures, which are very useful, both for the pharmaceutical industry and for research processes related to this drug.