Studies of volumetric and activity behaviors of binary and ternary aqueous solutions containing β-cyclodextrin and glucose

Journal of Molecular Liquids 178 (2013) 185–191

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Studies of volumetric and activity behaviors of binary and ternary aqueous solutions containing β-cyclodextrin and glucose Rahul Kolhapurkar 1, 2, Kesharsingh Patil ⁎ Department of Chemistry, Shivaji University, Kolhapur 416 004, Maharashtra State, India

a r t i c l e

i n f o

Article history: Received 14 September 2012 Accepted 31 October 2012 Available online 5 December 2012 Keywords: β-CD Glucose Activity and activity coefficients Gibbs free energy of transfer Hydrophobic interactions Thermodynamic equilibrium constant

a b s t r a c t Osmotic vapor pressure and density measurements have been carried-out for binary aqueous solutions of glucose and ternary aqueous solutions containing glucose and a fixed concentration of β-cyclodextrin (β-CD) [0.01262 mol kg−1] using techniques of vapor pressure osmometer and digital densitometer at 298.15 K. The measured water activities were used to estimate the activity coefficients and free energy changes in the binary system. The activity data of binary aqueous glucose solutions are further used to obtain the hydration number of glucose. The free energy change shows that at low concentration glucose forms almost an ideal solution in water. In ternary solutions the activity coefficient and transfer free energies of β-CD and glucose are studied as a function of concentration of glucose. Using earlier reported data for binary aqueous β-CD system and these newly studied systems, transfer free energies of β-CD from aqueous to aqueous glucose solutions are determined. These results are further subjected to McMillan–Mayer theory of solutions which yield the value of thermodynamic equilibrium constant (K) for β-CD:glucose complex in aqueous solutions after evaluation of salting constant and pair and triplet interaction parameters. It has been found that log K values are positive and high. These results are discussed in terms of hydrogen bonding, replacement of water molecules from β-CD cavity and solute–solute association equilibria involving pair and triplet interactions. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Several systematic studies of bulk and molecular properties of dilute aqueous solutions of monofunctional solutes such as alcohols, amines and ethers are now available [1–5]. These led to the concept of water structure making and breaking by the solutes as well as hydrophobic hydration and interaction [6–8]. However, the situation is more complex for solutions of solutes possessing several functional groups capable of hydrogen bonding to solvent water. Hydration effects due to solute–solvent H-bonding would be expected to be highly orientation dependent and conformation of solute might be the important factor determining hydration effects. It is known that thermodynamic properties like enthalpy and entropy decide the classification of solutes. Monofunctional solutes exhibit the type of behavior, |ΔHE| > T|ΔS E|, more commonly occurring in nonaqueous systems.

⁎ Corresponding author at: School of Chemical Sciences, North Maharashtra University, Jalgaon-425001, Maharashtra, India. Tel.: +91 8975012226 (mobile). E-mail addresses: [email protected] (R. Kolhapurkar), [email protected] (K. Patil). 1 This reported work is taken from the Ph.D. thesis submitted by Rahul Kolhapurkar to Shivaji University, Kolhapur in 2006. 2 Present address: Department of Chemistry, Tuljaram Chaturchand College, Baramati413102, Pune affiliated to University of Pune, Pune, Maharashtra, India. Tel.: +91 9833439402 (mobile). 0167-7322/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.molliq.2012.10.044

Taylor and Rowlinson [9] suggested that thermodynamic behavior of solutions of glucose and sucrose resembles hydrogen peroxide– water mixtures. Stokes and Robinson [10], using the concept of ideal solution, have suggested that the observed concentration dependence of the thermodynamic quantities is due to solute–water interactions in terms of hydration equilibria. They could fit the experimental activity data (at high concentration ≈ 7.5 mol kg−1) with a hydration number value of 6 and equilibrium constant K = 0.789. It has been suggested that for carbohydrate–water systems, free energies, as a result of enthalpy–entropy compensation effects [11,12], may be fairly insensitive to the hydration model chosen [13]. Franks et al. [13] have developed specific hydration model for aqueous–carbohydrate solutions using calorimetry and dielectric measurements. According to them the nature and extent of hydration is a function of relative compatibility of the solute conformation with the aqueous environment as well as by the intermolecular order through hydrogen bonding interaction. Warner [14] has discussed the correlation between the stereochemistry of sugars and their hydration behavior with reference to biological specificity and cryoprotective action. It is to be noted that no reliable data of free energy changes are available in dilute concentration range maybe because of complications due to mutarotation. Cyclohexaamylose and cycloheptaamylose, also known as α- and β-cyclodextrin, respectively have been the subject of much intensive research [15,16]. An important reason for the interest in cyclodextrins is their ability to form inclusive complexes with molecules and anions in

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aqueous solutions. These complexes serve as a model for the study of enzymes [17]. Possible binding forces within these inclusion complexes may involve hydrophobic, van der Waals, hydrogen bonding or dipole interactions. During the last about ten years, we reported the measurements of osmotic coefficients, densities, activity and activity coefficient properties of electrolytes, amino-acids, non-electrolytes as well as of surfactants in aqueous solutions of 18-crown-6 as well as of α- and β-cyclodextrin solutions [18–25]. It has been shown that the conformations of these host molecules, their H-bonding with water molecules in the cavities, as well as hydrophobic interaction along with water structure making effect govern equilibrium properties into solution phase. The observed enthalpy–entropy compensation effect and the application of McMillan–Meyer [26] and Kirkwood–Buff theories [27] for solutions to obtain second virial coefficient values for solute as well as equilibrium constant values along with pair and triplet interaction parameters in the case of ternary solutions have helped us to understand solute–solvent, solute–solute and host–guest interactions in such solutions. Recently, we came across two conflicting publications in which complexation reactions of carbohydrates with β-cyclodextrin [β-CD] as a host have been studied [28,29]. According to Hirsch et al. [28], concentration of glucose can be monitored using β-CD as a complexing host. They suggested a possibility of hydrogen bond formation on the basis of large negative enthalpies. In another work by Sebestyén et al. [29], it is reported that no significant interaction exists between glucose and β-CD. Therefore, we were prompted to investigate the possibility of complexation between glucose (as a solute) and β-CD as host in aqueous solutions. We report in this article, the volumetric and osmotic coefficient studies in very low concentration region (0.03 mol kg−1 to 0.1 mol kg−1) of aqueous solutions of glucose at 298.15 K. The activity, activity coefficient and hydration number values are calculated and examined. The measurements were extended to ternary aqueous solutions containing fixed concentration of β-CD (0.01262 mol kg−1) near to its solubility in water medium and varying concentration of glucose (0.005 mol kg−1 to 0.2 mol kg−1) in water at 298.15 K. The data of water activities in ternary aqueous solutions have been obtained and processed to estimate activity coefficients of the individual components in the solution phase. These are further utilized to calculate the Gibbs free energies of transfer of a glucose molecule from binary to ternary solutions containing β-CD. The estimated values of Gibbs free energies of transfer are used to evaluate pair and triplet interaction parameters by means of virial expansion of transfer Gibbs free energy as a function of β-CD and glucose molality. The resulting salting-in parameter, equilibrium constant value along with pair and triplet interaction parameters of McMillan Mayer theory for the complexation reaction are reported and discussed.

The water activities (aw) in binary and ternary systems were determined using a KNAUER K-7000 vapor pressure osmometer at 298.15 K. The reproducibility of the temperature changes, as monitored by thermisters, is better than ±0.001 K. The instrument was kept in specially fabricated refrigerated thermostat operating between 273 K and 313 K. The osmometer was calibrated using aqueous NaCl solutions and taking water as the reference. The osmotic coefficient data for aqueous NaCl solutions were obtained from the tabulation given by Robinson and Stokes [30]. The estimated accuracy in activity and activity coefficient data are of the order 5 × 10−5 and 5 × 10−4, respectively. In binary glucose+ H2O system, the concentration of glucose varied between 0.03 mol kg−1 and 0.1 mol kg−1 while the measurements in ternary glucose+ β-CD+ H2O system the molality of β-CD was kept constant at 0.01262 mol kg−1 and the molality of glucose was varied between 0.005 mol kg−1 and 0.2 mol kg−1. 3. Results The density data for binary aqueous solutions of glucose and ternary aqueous solutions containing glucose and fixed concentration of β-CD (0.01262 mol kg−1) were obtained at 298.15 K. The density data were used to obtain the apparent molar volumes (ϕV) using the equation given below [31]: φV ¼ ðM 2 =ρÞ þ fn1 M 1 ðρ0 −ρÞ=mρρ0 g

ð1Þ

where n1 is the number of moles of water in 1 kg of water, M1 is the molar mass of the solvent, M2 is the molar mass of the glucose and m is the molality, whereas ρ0 and ρ represent the density values for the solvent (ρ0 = 997.047 kg m −3) and the solution, respectively. The estimated uncertainty in ϕV values at lowest concentration studied are found to be of the order ±0.5× 10−6 m3 mol−1 for glucose+ H2O and ±1.0 × 10−6 m 3 mol−1 for glucose+ 0.01262m β-CD + H2O, respectively. The variation of ϕV as a function of concentration is depicted in Fig. 1 for binary glucose + H2O and ternary glucose+ 0.01262m β-CD + H2O.

2. Experimental β-CD (purity of about 0.98 mass fraction) procured from Merck– Schushardt was dried at 100 °C under vacuum for about 48 h and used without further purification. A.R. grade glucose from BDH was used without further purification. All of the solutions were prepared on molality basis (mol kg −1) using freshly prepared doubly quartz distilled deionized water. A Mettler Toledo AB 204-S balance, having a readability of 0.1 mg, was used to determine sample masses. The densities of binary and ternary solutions were measured at 298.15 K using a high precision Anton PAAR (DMA 60/602) digital readout densimeter. The temperature constancy of the vibrating tube was controlled to better than ±0.02 K and was maintained constant by circulating water through Julabo cryostat having a temperature stability of ± 0.01 K. After applying humidity and air pressure corrections, the accuracy in density values was found to be of the order ± 5 × 10 −3 kg m −3.

Fig. 1. Apparent molar volume (ϕV) of glucose as a function of molality (m) of glucose in binary and ternary aqueous solutions at 298.15 K: ○, in glucose + H2O; ●, in glucose + 0.01262m β-CD + H2O.

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Table 1 Density (ρ), apparent molar volumes (ϕV), partial molar volumes (V 2 , V 1 ), water activity (aw), osmotic coefficients (ϕ), activity coefficients (γ0, γ20 ) and excess Gibbs free energy change (ΔGE) data for aqueous glucose solutions, at 298.15 K. m (mol kg−1)

ρ (kg m−3)

106 × ϕV (m3 mol−1)

106 × V 2 (m3 mol−1)

106 × V 1 (m3 mol−1)

0.00000 0.00384 0.00817 0.01048 0.01205 0.01584 0.01998 0.03487 0.05012 0.06582 0.07951 0.09995

997.047 997.302 997.611 997.746 997.859 998.125 998.398 999.423 1000.456 1001.496 1002.432 1003.809

113.0a 113.9 111.2 113.5 112.8 112.1 112.5 111.9 111.9 112.2 112.0 111.9

113.0a 113.8 111.0 113.3 112.5 111.8 112.2 111.3 111.2 111.4 111.1 111.1

18.068 18.068 18.068 18.068 18.068 18.068 18.068 18.069 18.069 18.069 18.069 18.070

a

ϕ

aw

γ0

γ20

ΔGE (J mol−1)

1.0016 1.0019 1.0021 1.0022 1.0025 1.0028 1.0039 1.0050 1.0061 1.0071 1.0086

0.9999 0.9999 0.9998 0.9998 0.9997 0.9996 0.9994 0.9991 0.9988 0.9986 0.9982

1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999

1.0012 1.0025 1.0032 1.0036 1.0046 1.0057 1.0089 1.0117 1.0142 1.0163 1.0199

0.000 0.001 0.001 0.001 0.002 0.003 0.010 0.020 0.032 0.046 0.070

Extrapolated values at infinite dilution.

where x1 and x2 are the mole fractions of solvent and solute respectively and aw is the solvent activity. From the solvent activities, solvent activity coefficients (γ0) can be estimated very easily [32]. The activity coefficient of glucose (γj0) has been calculated using following equation

The partial molar volume (V 2 ) of solute in binary aqueous solutions of glucose at various concentrations was estimated using following equation:
 dϕV : V 2 ¼ ϕV þ m dm

ð2Þ

m

0

lnγ j ¼ ðϕ−1Þ þ ∫ ðϕ−1Þd lnm:

The calculations of partial molar volumes (V 1 ) of water at different concentrations of glucose were made using Eq. (3): 
 1 0 2 dϕV V1 ¼ n1 V 1 −m n1 dm

Hereafter, j = 1 refers to β-CD and j = 2 refers to glucose throughout the text, to eliminate the ambiguity that may arise while assigning symbols for various properties for additional term due to β-CD in ternary system. The osmotic coefficients, ϕj0 of glucose in binary aqueous system can be expressed as a power series on the molality by the equation

ð3Þ

where, n1 is number of moles of water and V10 is the molar volume of water at 298.15 K. The data of density, ϕV, V 2 and V 1 for binary glucose + H2O systems are collected in Table 1 while the data of density and ϕV for glucose+ 0.01262m β-CD+ H2O are collected in Table 2. The osmotic coefficients (ϕ) of glucose in aqueous solutions were determined over the concentration range of 0 to ~ 0.1 mol kg −1 at 298.15 K. The data of osmotic coefficients for glucose are represented by following equation: 2

3

ϕ ¼ 1 þ 0:1655m−1:739m þ 9:4971m :

0

ϕj ¼ 1 þ

n X

i

Ai m :

ð7Þ

i¼1

The coefficient Ai can be obtained by the method of least squares. The equation ln γj0 takes the form, after solving the integral, 0

ln γj ¼

ð4Þ

The solvent activities (aw i.e., for water) were calculated from the experimental osmotic coefficient data by making use of following expression
 x lnaw ¼ −ϕ 2 x1

ð6Þ

0

 n
X iþ1 i Ai m : i i¼1

ð8Þ

The dependence of solute activity coefficient on the solute concentration is shown in Fig. 2 for glucose in aqueous solutions. The activity data in mole fraction scale have been used to estimate excess Gibbs free energy change (ΔG E) by using standard thermodynamic equation ΔG E = ∑xi ln γi for binary system [33]. The data for ϕ, aw, γ20 and ΔGE for glucose+ H2O are collected in Table 1.

ð5Þ

Table 2 Density (ρ), apparent molar volumes (ϕV), water activity (aw), values of (Δ/m1m2), activity coefficients (γ0, γ1, γ2 ) excess Gibbs free energy change (ΔGE), and Gibbs energies of G β-CD transfer (ΔGtra , ΔGtra ) data for ternary glucose + 0.01262m β-CD + H2O system at 298.15 Ka. m2 (mol kg−1)

ρ (kg m−3)

106 ϕV (m3 mol−1)

aw

Δ/m1m2

γ0

γ1

γ2

ΔGE (J mol−1)

G ΔGtra (J mol−1)

β-CD ΔGtra (J mol−1)

0.00000 0.00512 0.00759 0.01025 0.01264 0.01545 0.01753 0.02017

1002.438 1002.781 1002.952 1003.129 1003.288 1003.483 1003.629 1003.824

111.8b 111.5 110.6 110.9 111.0 110.5 110.1 109.3

0.9998 0.9997 0.9996 0.9996 0.9995 0.9995 0.9994 0.9994

– −5.3063 0.2909 2.4003 2.8930 2.6378 2.1099 1.1783

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.1614 1.1063 1.0994 1.1012 1.1080 1.1185 1.1255 1.1306

0.8240 0.9474 0.9920 1.0265 1.0442 1.0479 1.0387 1.0127

0.108 0.019 0.008 0.011 0.020 0.030 0.033 0.024

−479.3 −137.9 −25.8 56.7 97.2 103.9 80.5 15.8

−1.2 −121.4 −136.7 −132.6 −117.3 −93.8 −78.0 −66.9

a b

Note: Units of m1 and m2 are moles of β-CD and glucose per kg of water respectively. Extrapolated value at infinite dilution.

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our earlier reported data [20] which are represented by the following equation: 2

3

ϕ ¼ 1 þ 7:1651m−251:97m þ 11114m :

ð13Þ

The parameter −RTΔ is the free energy change of the water in the following process: a solution of m1 moles of β-CD in 1 kg of water is mixed isothermally with a solution of m2 moles of glucose in 1 kg of water; the 1 kg of liquid water is isothermally separated via an osmotic membrane. Applying Gibbs–Duhem relation it has been shown that: 2

2

Δ=ðm1 m2 Þ ¼ A01 þ A11 m1 þ 2A02 m2 þ 3A03 m2 þ A21 m1 þ ð3=2ÞA12 m1 m2 :

ð14Þ

Since the concentration of β-CD (m1) is kept constant in this study, it can be incorporated into the coefficient Aij so that Eq. (14) can be expressed in terms of concentration of glucose (m2) as Δ=ðm1 m2 Þ ¼ B0 þ B1 m2 þ B2 m22

ð15Þ

i

i:e:Δ=ðm1 m2 Þ ¼ ∑ Bi m2 i¼0

where Fig. 2. Variation of activity coefficient (m) of glucose at 298.15 K.

(γ20)

of glucose in water as a function of molality

The molal activity coefficient of β-CD (γ1) in the ternary system may be expressed as a function of concentration of the β-CD (m1) and glucose (m2) using the equation [33–36]: ln γ1 ¼

∞ X ∞ X

i

j

Aij m1 m2 ðA00 ¼ 0Þ:

ð9Þ

i¼0 j¼0

If we neglect all terms higher than third order in the power series, Eq. (9) may be written as 0

2

þ

þ

3 A03 m2

0

∞ X ∞  X i¼0 j¼0

 j ðiþ1Þ ð j−1Þ A m m2 ðA00 ¼ 0Þ ð j þ 1Þ ij 1

ð11Þ

where γ2 is the activity coefficient of solute in ternary aqueous system and γ20 is that in binary aqueous system. The coefficient Aij in Eq. (9) may be related to an experimental quantity Δ, which is defined as [37] Δ¼

0 0 −55:51 lnaw −m1 ϕ1 −m2 ϕ2

ð12Þ

where, ϕ10 and ϕ20 are the osmotic coefficients of β-CD and glucose respectively in a binary system containing water. The values for ϕ10 i.e. osmotic coefficients of β-CD in aqueous solution are taken from

∞ X

 1 ðiþ1Þ Bm ði þ 1Þ i 2

ð17Þ

ðiÞ

ðm1 ÞBi m2 :

ð18Þ

i¼0

The values of Δ/(m1m2) are recalculated using the least square fit coefficient Bi and were further used for the recalculation of water activity. The reliability of the data is expressed in terms of percentage error given by [22]. %error in aw ¼

The result is

∞
X i¼0

ð10Þ



 ∂ ln γ1 ∂ ln γ2 ¼ : ∂m2 m1 ∂m1 m2

ln γ 2 ¼ ln γ2 þ

0

ln γ1 ¼ ln γ1 þ

ln γ2 ¼ ln γ2 þ

where, γ10 is activity coefficient of non electrolyte β-CD in binary aqueous solutions represented as ln γ10 = A10m1 + A20m12 + A30m13. It is taken from the data of binary aqueous solutions reported earlier [20]. A similar expression for activity coefficient of solute (ln γ2) may be obtained by applying the cross differentiation relation

ð16Þ

The values of Δ/(m1m2) have been obtained using the measured water activity for ternary system with the help of Eq. (12) and are given in Table 2. The coefficient Bi in Eq. (16) have been estimated by the method of least squares and are given in Table 3 for the ternary system. Using Eq. (16), Eqs. (10) and (11) have been derived in terms of Bi coefficients and are given below.

0

2

ln γ 1 ¼ ln γ1 þ A01 m2 þ A11 m1 m2 þ A02 m2 þ A21 m1 m2 2 A12 m1 m2

B0 ¼ A01 þ A11 m1 þ A21 m21 B1 ¼ 2A02 þ ð3=2ÞA12 m1 B2 ¼ 3A03 :

aw ðcaldÞ−aw ðobsdÞ  100: aw ðobsdÞ

ð19Þ

The observed and calculated values of Δ/(m1m2) along with the percentage error in water activity are produced in Table 4. From the measured water activity, the activity coefficients of solvent (γ0) i.e. of water, have been calculated for ternary system and are incorporated in Table 2. The molal activity coefficients for β-CD (γ1) and molal activity coefficients for glucose (γ2) in ternary system are calculated using Eq. (17) and Eq. (18) respectively with help of coefficients Bi of Eq. (15). These data are collected in Table 2 for ternary system. Table 3 Coefficients Bi in Eq. (15). Bi coefficients

Glucose + 0.01262m β-CD + H2O

B0 B1 B2

−15.3419 2591.2053 −89,414.8789

R. Kolhapurkar, K. Patil / Journal of Molecular Liquids 178 (2013) 185–191 Table 4 Water activity for ternary system, glucose + 0.01262m β-CD + H2O. m2 (mol kg−1)

aw

Δ/m1m2 (observed)

Δ/m1m2 (calculated)

% Error in aw

0.00512 0.00759 0.01025 0.01264 0.01545 0.01753 0.02017

0.9997 0.9996 0.9996 0.9995 0.9995 0.9994 0.9994

−5.3063 0.2909 2.4003 2.8930 2.6378 2.1099 1.1783

−4.4189 −0.8257 1.8238 3.1252 3.3487 2.6047 0.5462

−0.0001 0.0002 0.0001 −0.0001 −0.0002 −0.0002 0.0003

The variation of molal activity coefficients of solvent water (γ0), 0.01262 mol kg−1 β-CD (γ1) in aqueous glucose solutions and glucose (γ2) in 0.01262 mol kg−1 aqueous β-CD solutions as a function of glucose concentration are shown in Fig. 3.

189

increase is found to be more pronounced in the case of binary aqueous β-CD solutions, as per earlier reported data [20], compared to glucose. It is observed that the increase in γj0 as a function of concentration is similar to aqueous solutions of monofunctional nonelectrolytes [39] and can be attributed to solute–solute association [40]. The water activity decreases slightly (or remain more or less constant within the limits of experimental accuracy) as a function of mole fraction of solute. For studied concentration range ΔG E values are found to be very negligible in binary aqueous glucose solutions, which show that at low concentration glucose forms almost an ideal solution in water. Similar type of results were obtained in the case of β-CD and 18C6 in aqueous medium, however the values obtained for ΔG E in the case of 18C6 in CCl4 medium are negative [18,20]. The data of aqueous solutions of glucose is used to obtain the hydration number of glucose using the water activity of the solutions. The water activity is related with hydration by the following expression [35]:

4. Discussion It is observed from Table 1 and Table 2 that the densities of binary aqueous solutions of glucose increase as a function of glucose concentration while in ternary aqueous solutions containing fixed concentration of β-CD, the addition of glucose also results in an increase in density of solutions. The apparent molar volume of glucose (ϕV) is positive which decreases as a function of the glucose concentration in binary as well as in ternary solutions but the decrease is found to be predominant in the case of ternary system. The ϕV0 value obtained for glucose in aqueous solutions is about 113.0× 10−6 m3 mol−1 which is in agreement with the literature value of 111.9 × 10 −6 m 3 mol−1 [38]. The limiting ϕV0 value for glucose is 111.8 × 10 −6 m3 mol−1 in 0 ternary solutions and gives volume of complexation ΔϕVcomp as −6 3 −1 − 1.2 × 10 m mol which indicates that small volume loss occurs during complexation process between β-CD and glucose dϕV but negative certainly indicates the presence of strong glucose– dm glucose interactions in the presence of β-CD. The osmotic coefficient and activity coefficient values of glucose in aqueous medium increase as a function of concentration but the

Fig. 3. Variation of molal activity coefficients of solvent water, β-CD and glucose in ternary aqueous solutions as a function of molality of glucose at 298.15 K: ○, for glucose; □, for β-CD; Δ, for water.

aw ¼

1−0:018hm 1−0:018ðh−1Þm

ð20Þ

where aw is the water activity and h is the hydration number. By expanding ln aw in a series and converting to the osmotic coefficient, the equation has the form

 1 1 2 2 2 m : ϕ ¼ 1 þ 0:018 h− m þ 0:018 h −h þ 2 3

ð21Þ

Comparing Eq. (4) and Eq. (21) the hydration number h was found to be 9.7 (~ 10) for glucose. Stokes and Robinson [10] have proposed an associative equilibrium in glucose + H2O system based upon isopiestic vapor pressure measurements in the range of 1 mol kg −1 to 7.5 mol kg −1 of concentration. They could fit their data having equilibrium constant (K = 0.786) and hydration number of 6. However, our measurements have been extended to very dilute solution and the calculated hydration number is reasonable considering the near ideality of solutions and without evoking any value for the equilibrium constant. The enthalpy and entropy parameters are not available at such a low concentration and hence we do not venture to advance deeper interpretation. In ternary glucose+ 0.01262m β-CD + H2O system, the activity of solvent (aw) decreases with an increase in concentration of glucose. The results of variation of individual activity coefficients i.e. for water (γ0), for β-CD (γ1) and for glucose (γ2) in ternary solutions are depicted in Fig. 3. The study of variation of γ1 reveals that γ1 decreases as a function of glucose concentration showing a shallow minimum around the stoichiometric concentration while γ2 increases and passes through a maximum at the same stoichiometric concentration as a function of glucose concentration while γ0 remains almost constant. The extrema observed at stoichiometric concentration for the values of γ1 and γ2 are attributed to the complexation between β-CD and glucose in solution. The excess total Gibbs energy change (ΔGE) for ternary systems has been computed using the standard expression ΔGE = RT ∑ xi ln γi. The β-CD Gibbs energies of transfer of β-CD (ΔGtr ), i.e. 0.01262 mol kg−1 of β-CD, from water to aqueous glucose solutions and that of glucose G (ΔGtr ) from water to 0.01262 mol kg−1 aqueous β-CD solution are calculated using the method described by Wen and Chen [36] using mole fraction statistics. The corresponding data are collected in Table 2 and are shown in Fig. 4 as a function of glucose concentration. The transfer free energies are negative for both components but for glucose it increases and becomes positive; however, glucose transfer shows a maximum around stoichiometric complex concentration of 0.01262 mol kg−1. We will not discuss this at length as the corresponding enthalpy and entropy data are not available.

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At low G and C concentrations, all triplet and higher order interaction terms can be neglected and the pair interaction parameter gCG can be related to the familiar salting coefficient ks by RTks ¼ 2g CG :

β-CD Fig. 4. Gibbs energies of transfer of 0.01262 mol kg−1 β-CD (ΔGtr ) from water to aqueG ous glucose solutions and glucose (ΔGtr ) from water to aqueous 0.01262 mol kg−1 β-CD solutions plotted against molality of glucose at 298.15 K: ○, for glucose; □, for β-CD.

We adopt the methodology given by Desnoyers et al. [41] to account the observed transfer free energies for ternary systems by applying McMillan–Mayer's approach. With this approach, total Gibbs free energy of a solution containing 1 kg of water (W), mC moles of β-CD (C), and mG moles of glucose (G), is given by 0

0

0

GðmC ; mG Þ ¼ GW −RT ðmC −mG Þ þ  mCμ C þ RTm  C lnm  C þ mG μ G þ RTmG lnmG þ

þ 3m2C mG g CCG þ

3=2 m2C þ GEX G mC 2 3mC mG g CGG þ …

GEX G

þ 2mC mG g CG ð22Þ

0 where GW is the free energy of 1 kg of pure water, μC0 and μG0 are the standard chemical potentials of the β-CD and glucose respectively, GCEX(mC2) and GGEX(mG3/2) are the excess free energies of binary β-CD + H2O and glucose+ H2O systems, and gCG, gCCG, …etc., are pair and triplet interaction parameters which take into account all sources of nonideality in the ternary system. For example, gCG is a measure of the new interactions between C and G and of the corresponding decrease in the β-CD+ H2O and glucose + H2O interactions. An equation for binary system G(mC) and G(mG) can readily be derived from Eq. (22) by setting mC or mG equal to zero. The free energies of transfer of β-CD from water to aqueous glucose solution are given by the difference between the chemical potentials of β-CD in glucose solution and in water. At constant temperature and pressure, from Eq. (22)

ΔGC ðW→W þ GÞ ¼ ð∂GðmC ; mG Þ=∂mC ÞmG −ð∂GðmC Þ=∂mC ÞmG ¼0 ¼ 2mG g CG þ 6mC mG g CCG þ 3m2G g CGG þ …

ð23Þ

It immediately follows that



 ∂ΔGC ðW→W þ GÞ ∂ΔGG ðW→W þ C Þ ¼ ∂mG ∂mC mC mG ¼ 2g CG þ 6mC g CCG þ 6mG g CGG þ …

It is obvious from this relation that, at low concentrations, the change in free energy of β-CD by glucose (salting-out or salting-in) is exactly equal to the change in free energy of glucose by a β-CD. Transfer Gibbs energies for transfer of β-CD from water to aqueous glucose solutions have been used to calculate pair and triplet interaction parameters using Eq. (23). The values gCG, gCCG and gCGG are found to be −12,107 J kg mol−2, −446 J kg mol−3 and 364,268 J kg mol−3 respectively. These clearly indicate that complexation is favored but with hydrophobic interactions (both gCG and gCCG are negative) while the triplet interactions gCGG (positive) are not operative. The salting constant in Eq. (25) is negative; one can equate ks with −ln K (negative of the natural logarithm of thermodynamic equilibrium constant for complexation) [12,22]. Using the value of ks, we obtained the log K value for β-CD:glucose complex as 4.24. Hirsch et al. [28] have reported the value of 2.62 for log K while the value of −0.22 for log K is reported by Hacket et al. [42] from spectroscopic, fluorimetry and calorimetric measurements respectively. Considering the drawbacks of these techniques, we feel that our value of log K for β-CD:glucose complexation is much reliable. Thus our activity data and the analysis indicate strong complexation between β-CD and glucose and the process is being assisted by β-CD–β-CD–glucose hydrophobic interaction. The positive magnitude for β-CD–glucose–glucose interaction parameter certainly points-out the encapsulation of glucose molecules singly in the β-CD cavities in solution phase. 5. Conclusions The volumetric, activity and activity coefficient behaviors for aqueous glucose solutions are examined. The hydration number of ~10 is estimated from the activity data for glucose. The free energy change shows that at low concentration glucose forms almost an ideal solution in water. Similar studies of osmotic and activity coefficients were made for various concentrations of glucose in aqueous solutions containing fixed concentration of β-CD. The results are examined from the point of view of complexation (incorporation of glucose in β-CD molecule cavities). The lowering of activity coefficient of β-CD indicates the formation of complexed species in solution. The calculation of equilibrium constant value (log K) for the complexation was made and is positive and high. The results are discussed in terms of H-bonding, replacement of water molecules from β-CD cavity and solute–solute association equilibria involving pair and triplet interactions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

ð24Þ

ð25Þ

[16] [17] [18]

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