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Solubility of Sodium Halides in Aqueous Mixtures with ε‑Increasing Cosolvents: Formamide, N‑Methylformamide, and N‑Methylacetamide at 298.15 K† Felipe Hernández-Luis,*,‡ Raquel Rodríguez-Raposo,‡ Héctor R. Galleguillos,§ and Jaime W. Morales⊥ ‡

Departamento de Química (U.D. Química-Física), Universidad de La Laguna, Tenerife, Spain Departamento de Ingeniería Química y Procesos de Minerales, CICITEM, Universidad de Antofagasta, Antofagasta, Chile ⊥ Escuela de Ingeniería Química, Pontificia Universidad Católica de Valparaíso, Valparaíso, Chile §

ABSTRACT: The solubility data for sodium halides (NaF, NaCl, NaBr, and NaI) have been determined at 298.15 K in the εincreasing mixtures of water−formamide, water−N-methylformamide, and water−N-methylacetamide. In all cases, the solubility of the electrolytes was decreased significantly in the presence of a cosolvent. The solubility of an electrolyte in a given solvent depends on both the nature of the solvent (mainly the polarity, dielectric constant, solvation, or preferential solvation if the solvent is a mixture of solvents, etc.) and the properties of the electrolyte (principally the size, the charge, and the possible association of its ions). If the molality of saturation is not sufficiently small, the ion−ion and ion−solvent interactions must be considered. Although it was not the objective of this work, the solubility values were qualitatively correlated with several properties of the solvent and solute. Finally, a simple justification, which was both qualitative and quantitative, was carried out.

1. INTRODUCTION Aqueous−organic electrolyte solutions are important for a large number of applications, and new data therefore are frequently required. The solubility of electrolytes in aqueous−organic media is a thermodynamic parameter of enormous importance.1−9 The addition of a new component such as an organic solvent, totally or partially miscible with an original solution (normally aqueous), usually decreases the solubility of the salt in the original solution (drowning-out process).10,11 This crystallization technique has a number of advantages compared with the usual evaporation or cooling procedures, including increased yields, occurrence of the process at ambient temperature (saving of energy), higher purity of the crystal, great selectivity, and others. Knowledge of the solubility of compounds in pure solvents or mixtures of solvents is also of great importance for the design and simulation of operations, including5−11 the construction of phase or solubility diagrams, liquid−liquid extraction, study and testing of models, correlation and prediction of solubility, vapor−liquid equilibria, checking of the activity coefficients or osmotic values, etc. The immense majority of experimental studies on the solubility of salts have been carried out in water,12,13 pure organic solvents, or aqueous−organic mixtures with cosolvents that have a dielectric constant lower than that of water (εdecreasing system). Among the systems of this kind, the most studied are those containing alcohols1−5,9−11,14−23 and ketones, esters and ethers, among other chemical compounds.6,7,14,16,19−34 Data on systems with cosolvents with dielectric constants greater than the dielectric constant of pure water (ε-increasing system) are somewhat more scarce and usually include some amides as the cosolvent.35−51 Aqueous mixtures containing amides (in particular, cyclic amides) constitute an essential tool in the understanding of the behavior of complex molecules of biological interest.38,40 Extensive work © XXXX American Chemical Society

has been published on water−amide systems to discover the mode in which water exercises kinetic and thermodynamic control of the chemical activities of polypeptides. The unusually high density of hydrogen bonds in water (strongly selfassociated) and the character of the acceptor−donor (−CO− NH−peptide bond) make these water−amide systems very interesting from a structural point of view. The presence of an extra electrolyte further complicates the picture (structurebreaking or -making effects). In this paper, we present the results of the study of the solubility of halides of sodium (NaF, NaCl, NaBr, and NaI) in aqueous mixtures containing the amides formamide (F), Nmethylformamide (NMF), and N-methylacetamide (NMA), whose structures are shown in Figure 1 and most notable

Figure 1. Molecular formulas of the amides studied.

properties in Table 1.52−54 Additionally, Figure 2 shows the effect of variation of three important properties (dielectric constant, density, and viscosity) on the basis of the composition of the water−amide system.52−55 Formamide is well-known54 as a highly ionizing polar liquid with a dielectric constant and a dipole moment higher than Received: December 2, 2015 Revised: December 19, 2015 Accepted: December 31, 2015

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DOI: 10.1021/acs.iecr.5b04614 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

Table 1. Some Physical Constants for Pure Solvent: Water (W), Formamide (F), N-Methylformamide (NMF), and NMethylacetamide (NMA) at 298.15 K mol wt molar volume boiling point melting point density viscosity surface tension dielectric constant refractive index dipole moment solubility parameter polarizability donor number acceptor number

Mw V (cm3 mol−1) bp (K) fp (K) ρ (g cm−3) η (mPa s) γ (N m−1) εr nD μ (D) δ (J1/2cm−3/2) α (×10−30 m3) DN (kcal mol−1) AN (kcal mol−1)

W

F

NMF

NMA

18.015 18.1 373.15 273.15 0.9971 0.890 0.0718 78.38 1.3325 1.82 48.1 1.46 18.0 54.8

45.041 39.9 483.65 275.70 1.1297 3.302 0.0582 109.57 1.4468 3.37 39.3 4.23 24.0 39.8

59.068 59.1 453.15 269.35 0.9988 1.650 0.0395 182.40 1.4300 3.86 20.3 6.05 27.0 32.1

73.094 77.0 479.15 303.70 0.950030 °C 3.65030 °C 0.033730 °C 191.332 °C 1.425335 °C 4.2730 °C 29.9 7.85

Figure 2. Composition dependences of the relative permittivity (relative dielectric constant εr), density (ρ), and viscosity (η) in amide−water mixtures at 298.15 K: water−F (black ●);52,53 water−NMF (red ■);52,53 water−NMA (green ▲). Three-dimensional fit of the extrapolated data.55

According to Marcus,52 in aqueous−F mixtures, the volumecorrected preferential solvation parameters show that this system is nearly ideal. In aqueous−NMF mixtures, water selfinteraction is not favored.52 The better compatibility of this cosolvent than that of F with water is unexpected. Finally, in the aqueous−NMA mixture, the behavior is practically indistinguishable from ideal behavior, so no preferential solvation takes place.52 The properties of the sodium halides studied are well-known. Perhaps the most significant difference is their solubility in water at 298.15 K (0.983 mol kg−1 for NaF,57 6.146 mol kg−1 for NaCl,58 9.186 mol kg−1 for NaBr,59 and 12.340 mol kg−1 for NaI49). In this first study, together with an extensive bibliographic review and a description and verification of the method used, we present solubility values at 298.15 K of NaF, NaCl, NaBr, and NaI in the above-mentioned aqueous−organic mixtures (because there are very limited data available for the physical constants of these mixed solvents at other temperatures). In addition, a simple qualitative and quantitative justification is carried out.

those of water, with which it is completely miscible at 298.15 K throughout the complete composition range. In the water−F mixture, both the density and viscosity increase uniformly with the cosolvent content. The dielectric constant reaches a maximum value at 80−90%. NMF54 is also completely miscible with water at 298.15 K throughout the complete composition range. Like F, NMF is a highly ionized polar liquid with a dipole moment higher than the dipole moment of water and a very large dielectric constant (NMF has the highest dielectric constant at room temperature of any known liquid). NMF is used mainly as a reagent in various organic syntheses and in the production of some pharmaceutical compounds. Certain antitumor activities of NMF have been estimated. An NMF−water mixture exhibits an increase in both the density and viscosity, increasing more smoothly than those of F, reaching maximum values of 60− 70%, and then decreasing between 70% and 100%. Finally, like the other two amides studied, NMA54,56 has extensive applications as a solvent because it is effortless to prepare and purify. However, the miscibility with water at 298.15 K reaches only approximately 80%. NMA is characterized by a broad solubility and dissociating ability.52,56 NMA is also characterized by a high dielectric constant (165.5 at 40 °C) as well as a large dipole moment. Figure 2 shows an increase in the dielectric constant with lower density values in comparison with the other two amides studied, with a maximum up to approximately 50% and a large increase in the viscosity up to 80% of the weight of composition.

2. EXPERIMENTAL SECTION 2.1. Materials. NaF (99.5%), NaCl (99.5%), and NaBr (99.0%) from Merck (pro analysis) and NaI (99.0%) from Fluka were vacuum-dried at 383 K for 72 h before being used and then stored in desiccators over silica gel, anhydrous CaCl2, and P2O5. Formamide (F, 99.0%), N-methylformamide (NMF; 99.0%), and N-methylacetamide (NMA; 99.0%) from SigmaB


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Industrial & Engineering Chemistry Research Aldrich were used without any prior treatment. Correction of the concentration for the very small water content of the original product was considered unnecessary. For the experiments, double-distilled water (κ < 10−5 S cm−1) was used. The solvents were prepared by the direct weighing of water and amide on a balance (Mettler PM2000) with a resolution of 0.01 g. 2.2. Procedure. Oversaturated solutions of solute in the different solvents were shaken in 100 mL bottles with Vibromatic-384 Selecta for 48 h in a laboratory where the temperature was controlled at 298.15 ± 0.50 K. Subsequently, solutions were decanted into a thermostatted vessel at 298.15 ± 0.02 K for 72 h until clear and transparent supernatant solutions were obtained. The method to estimate the saturation molality of the salts in each of the mixtures was as follows: (a) Approximately 2 g of the sample was carefully taken from the supernatant solution with a Pasteur micropipette and deposited in small beakers of 10 mL previously weighed on a balance (Mettler AE163) with a resolution of 0.1 mg. (b) The samples were then introduced into an oven and gently heated to avoid weight loss by crackling to a higher temperature of approximately 10−15 K to the boiling point of the cosolvent. (c) Then, the samples were left to cool to room temperature inside the oven and weighed. The process was repeated until a constant weight was obtained (2− 3 times was sufficient). (d) On the basis of the final weight and the weight of the initial sample, the saturation molality was then calculated. To obtain a good estimation, the process was quadrupled and the average value subsequently calculated. The relative error, in water, of the saturation molality is less than approximately 0.4%. In the pure cosolvents (F and NMF), when our data were compared with those of the bibliography,49 the maximum relative error was less than 4%, in the worst case. NMA is a solid at 298.15 K, and a comparison was not possible because of a lack of bibliographic data for soluble mixtures of water−NMA at 298.15 K. We can say that, in principle, our method is validated, within the usual error encountered in such measures.

Table 2. Solubilities of NaF, NaCl, NaBr, and NaI in Water− F, Water−NMF, and Water−NMA Mixtures at 298.15 Ka

a

3. RESULTS AND DISCUSSION Table 2 summarizes the solubility of sodium halides in the different aqueous−organic mixtures studied. Figure 3 shows a plot of the normalized saturation molality (molality in the mixture divided by the corresponding value in water). We used the normalized values so that the Y axis would be the same for all of the electrolytes studied because these electrolytes have very different solubilities in water. For NaF and NaCl, the curves are similar and well-defined, with an order of solubilities of water−NMA < water−NMF < water−F. For NaBr, the order is the same that as above, but the curve corresponding to the water−F mixture is different from the others (concave down). Finally, for NaI, the profile is something more anomalous, and the order of solubilities is water−NMA < water−F < water− NMF. In the case of NaF, NaCl, and NaBr, the sequence of the solubilities, for a given weight percent of cosolvent, is similar to the increase in both of the dielectric constants with the dipole moments of the corresponding amides. This sequence is not observed for NaI. The values that appear in parentheses in Table 2 should be taken with some prudence. On the one hand, there are the compounds corresponding to NaBr in water−F; we have no explanation for the change of concavity mentioned above with regard to the other curves of this salt. On the other hand, all of

wt % of cosolvent

mS(NaF)/ mol kg−1

0 10 20 30 40 50 60 70 80 90 100

0.9851 0.7145 0.5228 0.3814 0.2825 0.2026 0.1363 0.0676 0.0345 0.0109 0.0020

0 10 20 30 40 50 60 70 80 90 100

0.9851 0.6894 0.4849 0.2966 0.1989 0.1334 0.0803 0.0560 0.0370 0.0109 0.0020

0 10 20 30 40 50 60 70 80

0.9851 0.6924 0.3968 0.2335 0.1288 0.0683 0.0248 0.0136 0.0055

mS(NaCl)/ mol kg−1 Water−F 6.1419 5.6250 5.0310 4.5090 4.0260 3.5810 3.1690 2.7620 2.3850 1.9880 1.5550 Water−NMF 6.1419 5.4085 4.4650 3.7100 3.0222 2.3923 1.8495 1.4095 1.0645 0.8077 0.5500 Water−NMA 6.1419 4.8686 4.0836 3.1426 2.4883 1.6945 1.2080 0.8494 0.5899

mS(NaBr)/ mol kg−1

mS(NaI)/ mol kg−1

(9.1941) (9.1734) (8.6921) (8.3007) (8.0138) (7.6162) (7.1066) (6.4629) (5.6721) (4.6283) (3.3134)

(12.3239) (10.3781) (10.5011) (9.6505) (8.8050) (7.9572) (7.1176) (6.2790) (5.4433) (4.6093) (3.7767)

(9.1941) 8.4619 7.5308 6.7606 6.1053 5.6142 5.3039 4.6246 3.9469 3.3969 2.9286

(12.3239) (11.1845) (10.5163) (9.7535) (9.0937) (8.7887) (8.4343) (7.9080) (7.2296) (6.4908) (5.1973)

(9.1941) 7.9734 6.9556 5.9542 5.2165 4.4861 3.8791 3.2420 2.8007

(12.3239) (11.2575) (10.1022) (9.0042) (8.2297) (6.8347) (6.3598) (5.7742) (5.0272)

The values in parentheses are explained in the text.

the values of systems containing NaI are considered. In the case of NaI, the solids obtained after evaporation of the solvent always had a brownish-yellow color; however, we know that pure NaI, like other halides of sodium, is white. For this reason, with this electrolyte, we made some modifications to the cosolvent evaporation process: (a) Prior to shaking, we used a vacuum to avoid the presence of air and therefore oxygen, which could promote any reactions of oxidation−reduction that involve I2 or I−. (b) Additionally, we tried to eliminate the air by bubbling argon through the oversaturated solution for approximately 30 min. (c) We worked in the dark (topazcolored flask, coated aluminum paper, and dark laboratory) to prevent any reactions favored by light. (d) Finally, we used methods b and c simultaneously. In any case, we could avoid the appearance of the brownish-yellow color in the solids, and the results were very similar. New studies must be carried out to explain the appearance of color.60−63 Strack et al.49 described this same effect not only for NaI but also for KI. The explanation involves the spontaneous formation of iodine, which also increases with time. They also eliminated oxygen bubbles in the nitrogen. C


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Figure 3. Plot of normalized solubility (ms,S/ms,W) versus weight percent of cosolvent: water−F (black ●), water−NMF (red ■), and water−NMA (green ▲) at 298.15 K.

Table 3. Parameters To Be Used with Equation 1 To Interpolate the Solubilities of Sodium Halides in Each Mixture of Solvents Studied at 298.15 K and Fit Standard Error a (mol kg−1)

a

102b(mol kg−1)

104c(mol kg−1)

NaF NaCl NaBr NaI

0.9830a 6.1460a 9.1860a 12.3400a

−3.2388 −4.4043 4.6540 −1.0396

3.7535 −13.1444 −71.4088 10.0401

NaF NaCl NaBr NaI

0.9830a 6.1460a 9.1860a 12.3400a

−3.2389 −5.4787 −4.5118 −1.3072

3.7535 −31.8143 −39.9570 15.0925

NaF NaCl NaBr NaI

0.9830a 6.1460a 9.1860a 12.3400a

−1.8415 −19.0104 −1.4553 −4.9306

−0.0199 88.8310 36.0728 −98.9716

106d(mol kg−1) Water−F −2.1101 52.4552 256.4661 −26.6242 Water−NMF −2.1101 127.7161 142.5110 13.2261 Water−NMA 113.4950 319.7071 −153.0085 533.9708

108e(mol kg−1)

1010f(mol kg−1)

1012g(mol kg−1)

σ(mS)/√n

3.4711 −87.6094 437.0716 37.5004

−5.3740 69.9273 348.5585 −26.4146

2.5170 −21.8172 −107.0756 7.3310

0.0001 0.0001 0.0003 0.0001

3.4711 −223.8752 −199.4757 −69.4285

−5.3740 187.6329 118.0651 77.3962

2.5170 −60.7799 −22.7861 −30.3512

0.0001 0.0003 0.0003 0.0006

−248.7254 551.7237 366.9308 −1267.0901

251.6693 −436.7709 −416.5603 1394.8442

−97.066 128.2664 178.1976 −577.8506

0.0001 0.0004 0.0002 0.0010

Fixed values.57−59,49

In spite of these problems, the main conclusion from these studies of the solubility is that the addition of a cosolvent (either F, NMF, or NMA) to an aqueous sodium halide solution produces a significant decrease in the solubility of the salt. This decrease is most important in the NaF > NaCl > NaBr > NaI series and usually in the order NMA > NMF > F. The following polynomial equation, together with the parameters summarized in Table 3 for each system studied, can be used to interpolate values of the solubility to other values of weight percent.

18, 19, 22−25, 49, 34, and 64), almost always on the theoretical basis of the model of Born.65,66 Although it is not the main objective of this work, we tried to qualitatively test some of these correlations. For example, Izmailov et al.64 proposed that the activity coefficients of the ions at infinite dilution in a given solvent, referred to as an infinitely diluted solution in a standard solvent, can be represented by

mS = a + b(wt %) + c(wt %)2 + d(wt %)3 + e(wt %)4

where log γ(S) is determined by the chemical nature of the solvent and log γ(electrostatic) is a function of the radius of the ions and the dielectric constants of the media. In the absence of strong interactions of both ion−ion and ion−solvent and for solvents of similar characteristics, log γ(S) is approximately zero. When water is chosen as a reference, the following expression can be written:

+ f (wt %)5 + g (wt %)6

log γ = log γ(S) + log γ(electrostatic)

(1)

The solubility of an electrolyte in a given solvent depends on both the nature of the solvent and its properties (mainly the polarity, dielectric constant, solvation, or preferential solvation if the solvent is a mixture of solvents, etc.) as the electrolyte (principally the size, the charge, and the possible association of its ions). Both the ion−ion and ion−solvent interactions must be considered when the saturation molality is not very small. Therefore, it is not easy to find an equation unequivocally describing the solubility as a function of all of these parameters. Many authors have tried to deduce an equation that considers all of these aspects at the same time (see, for example, refs 7, 9,

log

⎛1 1 ⎞ = K⎜ − ⎟ εW ⎠ ms,W γs,W(electrostatic) ⎝ εS

(2)

ms,Sγs,S(electrostatic)

(3)

with K a constant of proportionality that depends on the charge and size of the ions and s, S, and W meaning the solubility, mixed solvent, and water, respectively. If it is verified that D


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Industrial & Engineering Chemistry Research γs,S(electrostatic) ≅ γs,W(electrostatic), the following expression can be written: log

ms,S ms,W

⎛ 1 1 ⎞ ⎟⎟ ≅ K ⎜⎜ − εr,W ⎠ ⎝ εr,S

linear only up to 60−70 wt %, with the deviation from linearity very marked especially for NaF. In all systems, the slopes are positive and the straight lines pass through the origin because the three mixes contain cosolvents with ε increasing. For εdecreasing cosolvent mixtures, the straight lines pass through the origin but the slopes are negative.18,36,37,64 The Izmailov et al. 64 relationship is only a valid approximation when γs,S(electrostatic)/γs,W(electrostatic) is approximately 1. In addition, if the molality of saturation does not vary much with the composition of the mixed solvent, Walden’s rule, mSε1/3 = constant, must be followed,67 at least in the first approximation. However, Figure 5 shows that this rule is not followed because the molality of saturation varies greatly with increasing cosolvent (see Table 2). To correlate the solubility of pharmaceuticals in mixed water−alcohol, Zhang et al.3 use the following equation:

(4)

Figure 4 shows a plot of log(ms,S/ms,W) versus 1/εr,S − 1/εr,W for the three mixtures studied. For the systems water−NMF

⎛ M⎞ x = k exp⎜ − ⎟ ⎝ Rεr ⎠

(5)

where x is the solubility of the pharmaceutical, εr is the dielectric constant of the mixed solvent, R is the gas constant, k is the preexponential factor, and M is a solution characteristic constant. According to Zhang et al.,3 this equation is formally similar to the Arrhenius equation ⎛ E ⎞ ⎟ x = k exp⎜ − ⎝ RT ⎠

(6)

with two differences: the absolute temperature, T, has been replaced by εr, and the activation energy, E, has been replaced by the constant M. Because E represents the energy of activation of the process of dissolution, the value of M could represent the level of resistance or difficulty of dissolution of the electrolyte in the mixed solvent.3 When logarithms are applied to eq 2, the following expression is obtained: ln x = ln k −

M 1 R εr

(7)

Equation 7 shows that the value of M can be calculated from the slope of the plot of ln x versus 1/εr shown in Figure 6. The linearity is good except for the system containing F as the cosolvent and, in particular, for NaF. Table 4 summarizes the values of various parameters obtained as well as the corresponding correlation coefficient. According to Zhang et al.,3 we can conclude that the higher the value of the parameter M (in absolute value), the greater the difficulty of dissolving the electrolyte. Our systems are ε-

Figure 4. Plot of log(ms,S/ms,W) versus 1/εr,S − 1/εr,W for the three water−amide mixtures studied: (black ●) NaF; (red ■) NaCl; (green ▲) NaBr; (yellow ▼) NaI.

and water−NMA, good straight lines are obtained, as suggested by Izmailov et al.,64 over the entire range of percentages of cosolvent. However, for the water−F system, the system is

Figure 5. Plot of msεr1/3 versus weight percent of cosolvent for the three water−amide mixtures studied: (black ●) NaF; (red ■) NaCl; (green ▲) NaBr; (yellow ▼) NaI. E


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Figure 6. Plot of ln x versus 1/εr for the three water−amide mixtures studied: (black ●) NaF; (red ■) NaCl; (green

Table 4. Calculated Values of Both lnk and M for Different Studied Systems ln k NaF NaCl NaBr NaI

−9.761 −3.089 −1.331 −1.020

NaF NaCl NaBr NaI

−13.590 −5.327 −2.098 −1.095

NaF NaCl NaBr NaI

−23.041 −9.079 −3.932 −2.002

slope Water−F 512.037 137.813 47.736 46.045 Water−NMF 812.171 313.223 107.834 51.991 Water−NMA 1552.966 607.333 251.649 123.065

−103M

r

4.257 1.146 0.397 0.383

0.989 0.987 0.924 0.993

6.753 2.604 0.897 0.432

0.997 0.998 0.996 0.995

12.912 5.050 2.092 1.023

0.999 0.999 0.999 0.999

▲)

NaBr; (yellow

▼)

NaI.

(a) In all the cases, decreasing the solubility of the salts increases the percentage by weight of the cosolvent. This decrease is most important for NaF > NaCl > NaBr > NaI. (b) Usually, the order of decrease with the cosolvent added to the aqueous solution is NMA > NMF > F. (c) In spite of failing to consider the ion−ion and ion− solvent interactions, the Izmailov et al.64 equation test is obeyed over the entire range of miscibility for both water−NMF and water−NMA mixtures and almost to 60−70% for mixing water−F. (d) Walden’s rule, mSε1/3 = constant, is not followed because the molality of saturation varies greatly with the increase in the cosolvent. (e) An increase in the solubility with an increase in the size of the anion is true for all of the mixtures studied (the cation is always the same). (f) Finally, the application of the model of Zhang et al.3 has been confirmed quantitatively, as stated previously.

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increasing, and those used by Zhang et al.3 are ε-decreasing. For this reason, the slopes have different signs. This model confirms what was said above: for the three mixtures studied, the solubility decreases in the order NaF < NaCl < NaBr < NaI and NMA < NMF < F. The electrolytes studied are practically dissociated in these media,46,47 and the charges of its ions are 1. In Figure 7, we plot ms versus anions of the crystallographic radius, r−68 (the cation is always the same, Na+). For the three mixtures studied, the solubility decreases with the weight percent of cosolvent, and the order is NaF < NaCl < NaBr < NaI.47,49−51

AUTHOR INFORMATION

Corresponding Author

*E-mail: ff[email protected]. Fax: +34 922 318514. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors of this work want to thank CONICYT-Chile and the ORI (International Relations Office) of the Universidad de La Laguna for the collaboration that made the stay in La Laguna of J.W.M. and H.R.G. possible.

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4. CONCLUSIONS After determining the molality of saturation for sodium halides (NaF, NaCl, NaBr, and NaI) in the three ε-increasing systems studied, we can conclude the following:

DEDICATION Dedicated to the memory of Professor Emeritus Dr. Agustin Arévalo Medina and Dr. José Acosta Rubio, recently deceased. †

Figure 7. Plot of ms versus r− for the three water−amide mixtures studied: (black ●) 0%; (red ■) 20%; (green ▲) 40%; (yellow ▼) 60%; (blue ◆) 80%; (purple ⬣) 100%. F


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