Density, Viscosity, and Refractive Index of Aqueous Solutions of

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Density, Viscosity, and Refractive Index of Aqueous Solutions of Sodium Lactobionate Shuo Han, Cun-Bin Du, Xu Jian, Long Meng, Xu Ren-Jie, Wang Jian, and Hong-Kun Zhao* College of Chemistry & Chemical Engineering, YangZhou University, YangZhou, Jiangsu 225002, PR China ABSTRACT: The density, viscosity, and refractive index of aqueous solutions of sodium lactobionate were measured as a function of molality m = 0.0251, 0.0502, 0.0753, 0.100, 0.126, 0.151, 0.176, and 0.201 mol·kg−1 and temperature T = 288.15, 293.15, 298.15, 303.15, 308.15, 313.15, 318.15, and 323.15 K. They all increase with decreasing temperature and with increasing concentration of sodium lactobionate. The Vogel−Tamman−Fulcher equation was used to correlate the dependence of density and viscosity on molality and temperature, and the refractive index was fitted by a polynomial model. The parameters were acquired by least-squares regression method for these models, and the calculated values showed good agreement with the experimental data for density, viscosity, and refractive index. The volumetric properties, including apparent molar volume, excess molar volume, and partial molar volume, among others, were determined according to the experimental values for sodium lactobionate mixtures. Furthermore, the relative viscosity and viscosity B coefficients were calculated. The molecule interactions and solution properties were studied on the basis of the evaluated thermodynamic property data. In the studied system, sodium lactobionate behaves as structure maker.

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identifying drug action.13,14 However, the density, viscosity, and refractive index of sodium lactobionate aqueous solutions have not been reported in previous publications. In the present paper, they were determined at various concentrations and temperatures. Furthermore, apparent molar volumes, standard partial molar volumes, excess molar volumes, and the viscosity B coefficient of lactobionate were calculated on the basis of the density and viscosity data.

INTRODUCTION Density, viscosity, and refractive index are very important thermodynamic data in the field of chemical engineering.1−3 Density can be used to calculate the volumetric properties of a solution, including apparent molar volume, partial molar volume, limiting partial molar volume, and excess molar volume, among others. These volumetric properties are of great importance in investigating molecule interaction and reactor design. The viscosity B coefficients, which are very important in analyzing the transport property, can be calculated through viscosity data. Lactose acid is a kind of advanced acid with a variety of biological functions. It is widely used as material in the production of antioxidant, humectant, and also used as additive in the food and medicine production.4−6 As one of the lactose acid salts, sodium lactobionate (CAS Reg. No. 27297-39-8; molecular weight, 398.3 g·mol−1) is generally employed in the preparing of erythromycin lactobionate.7 In addition, sodium lactobionate is also used for maintaining biological activities, for changing the properties of drugs, and as a cleaning agent.8−12 In view of the fact that many biochemical processes take place in aqueous media, the investigation of the volumetric and viscometric properties in the aqueous media gives valuable information in medicinal and pharmaceutical chemistry. For this reason, accurate prediction of physicochemical properties of the aqueous solution of sodium lactobionate is becoming increasingly important. Measuring the density, viscosity, and refractive index of an aqueous solution of sodium lactobionate at different temperatures and different concentrations has an extreme significance to the pharmacokinetic analysis of drugs in the body and the research and development of new drugs. The molecular interaction in a drug−water system play a key role in © XXXX American Chemical Society

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EXPERIMENTAL SECTION

Materials. Sodium lactobionate [C12H22O12Na], with a mass fraction purity of 0.984, was provided by Albert Wuhan Dahua Pharmaceutical Co., Ltd. It was purified three times in aqueous solutions of acetone by crystallization. The final purity of sodium lactobionate used in experiment was 0.997 in mass fraction, which was further confirmed by HPLC analysis. Solutions were prepared by using doubly distilled and deionized water. The sample information is given in Table 1. Apparatus and Procedure. The sodium lactobionate aqueous solutions were prepared using an analytical balance with a standard uncertainty of 0.0001 g at normal temperature. A certain amount of sodium lactobionate and water was charged into a 100 cm3 volumetric flask to obtain the desired concentrations m = 0.0251, 0.0502, 0.0753, 0.100, 0.126, 0.151, 0.176, and 0.201 mol·kg−1. The solutions were degassed with ultrasonic waves before use. The uncertainty of molality for all solutions is estimated to be ±0.0001 mol·kg−1. Received: May 11, 2015 Accepted: December 7, 2015

A

DOI: 10.1021/acs.jced.5b00407 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

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Table 1. Detailed Information of Sample chemical name

CAS NO.

molecular weight (g·mol−1)

source

mass fraction purity

purification method

sodium lactobionate

27297-39-8

398.3

Albert Wuhan Dahua Pharmaceutical Co., Ltd.

0.997

crystallization

Table 2. Density ρeand ρc, Apparent Molar Volume Vϕ, and Excess Molar Volume VE for Sodium Lactobionate Aqueous Solutions (m, mol·kg−1) at T = 288.15−323.15 K under p = 101.3 kPaa T (K) 288.15 293.15 298.15 303.15 308.15 313.15 318.15 323.15 m = 0.0753 288.15 293.15 298.15 303.15 308.15 313.15 318.15 323.15 m = 0.100 288.15 293.15 298.15 303.15 308.15 313.15 318.15 323.15 m = 0.126 288.15 293.15 298.15 303.15 308.15 313.15 318.15 323.15

ρe (g·cm−3)

ρc (g·cm−3)

Vϕ (cm3·mol−1)

VE (cm3·mol−1)

1.0105 1.0091 1.0076 1.0062 1.0045 1.0031 1.0016 0.9998

m = 0.0502 1.0109 1.0094 1.0078 1.0061 1.0044 1.0027 1.0009 0.9991

163.284 179.190 185.144 185.106 187.074 178.950 168.737 160.465

−0.0558 −0.0414 −0.0360 −0.0361 −0.0343 −0.0416 −0.0508 −0.0583

1.0152 1.0139 1.0126 1.0112 1.0097 1.0081 1.0065 1.0047

1.0155 1.0140 1.0124 1.0109 1.0093 1.0076 1.0059 1.0041

177.601 186.882 188.189 188.161 186.786 184.050 178.587 173.073

−0.0642 −0.0517 −0.0499 −0.0499 −0.0518 −0.0555 −0.0629 −0.0703

1.0201 1.0189 1.0177 1.0164 1.0149 1.0133 1.0115 1.0095

1.0200 1.0186 1.0171 1.0156 1.0141 1.0125 1.0108 1.0091

182.272 188.236 188.214 187.185 186.144 184.082 181.999 179.89

−0.0771 −0.0664 −0.0664 −0.0683 −0.0702 −0.0739 −0.0776 −0.0814

1.0245 1.0233 1.0221 1.0208 1.0194 1.0178 1.0159 1.0139

1.0246 1.0233 1.0219 1.0204 1.0189 1.0174 1.0158 1.0142

188.699 193.479 193.47 192.657 191.031 189.39 188.544 186.871

−0.0819 −0.0711 −0.0712 −0.0730 −0.0767 −0.0803 −0.0823 −0.0860

T (K) 288.15 293.15 298.15 303.15 308.15 313.15 318.15 323.15 m = 0.176 288.15 293.15 298.15 303.15 308.15 313.15 318.15 323.15 m = 0.201 288.15 293.15 298.15 303.15 308.15 313.15 318.15 323.15

ρe (g·cm−3)

ρc (g·cm−3)

Vϕ (cm3·mol−1)

VE (cm3·mol−1)

1.0291 1.0281 1.0269 1.0255 1.0239 1.0221 1.0205 1.0184

m = 0.151 1.0293 1.0279 1.0266 1.0252 1.0238 1.0223 1.0208 1.0193

191.348 194.004 193.998 193.990 193.979 193.967 191.247 190.537

−0.0911 −0.0839 −0.0839 −0.0839 −0.0839 −0.0840 −0.0913 −0.0933

1.0341 1.0330 1.0318 1.0303 1.0288 1.0272 1.0253 1.0232

1.0339 1.0326 1.0314 1.0301 1.0287 1.0273 1.0259 1.0244

190.702 193.547 193.541 194.106 193.523 192.356 191.756 191.148

−0.1083 −0.0993 −0.0993 −0.0975 −0.0994 −0.1030 −0.1050 −0.1068

1.0390 1.0377 1.0364 1.0351 1.0335 1.0318 1.03 1.028

1.0385 1.0374 1.0362 1.0349 1.0336 1.0323 1.0309 1.0295

190.481 193.969 194.465 193.959 193.951 193.437 192.409 191.369

−0.1245 −0.1119 −0.1101 −0.1119 −0.1120 −0.1138 −0.1175 −0.1213

Standard uncertainty: u(m) = 1 × 10−3 mol·kg−1, u(ρ) = 4 × 10−4 g·cm−3, u(Vϕ) = 1 × 10−3 cm3·mol−1, u(VE) = 1 × 10−4 cm3·mol−1, u(T) = 0.02 K, and u (p) = 350 Pa. ρe, experimental density; ρc, calculated density by eq 2. a

obtained values were compared with literature values.15 The relative standard uncertainty in kinematic viscosity determination is estimated to be about 0.58% in the temperature range studied. The sodium lactobionate aqueous solution was added to a cleaned and dried viscometer, and the viscometer was then placed vertically in the thermostatted water bath (model: DZKW-4; standard uncertainty, 0.01 K). When the system reached thermal equilibrium, the flow time of solution was recorded automatically by a computer connected to the viscometer. Each experiment was performed four times at a fixed temperature, and the average of the four readings was considered as final value within the deviation of ±0.2 s. The viscosity η of the sodium lactobionate solutions could be evaluated from eq 1.16

A vibrating tube densitometer (DMA4500 Anton Paar, Graz, Austria) was used to determine the densities of the sodium lactobionate aqueous solutions. The temperature was controlled with a standard uncertainty of 0.01 K. The densitometer was calibrated by means of doubly distilled water at different temperature under atmospheric pressure. The standard uncertainty in density determination is estimated to be about ±1.0 × 10−4 g·cm−3. The viscosities of the aqueous solutions of sodium lactobionate mixtures were determined through an Ubbelohde suspended-level capillary viscometer. Time was recorded using an electronic timer (type: Schott AVS 300; standard uncertainty, 0.01 s). In the present work, the doubly distilled water was also used to calibrate the Ubbelohde viscometer in the temperature ranging from 288.15 to 323.15 K, and the B

DOI: 10.1021/acs.jced.5b00407 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

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Table 3. Viscosity ηe and ηc and the Relative Viscosity Δη for Sodium Lactobionate Aqueous Solutions (m) at T = 288.15− 323.15 K under p = 101.3 kPaa m (mol·kg−1)

0.0251

0.0502

0.0753

0.100

T (K)

ηe (mPa·s)

ηc (mPa·s)

Δη

288.15 293.15 298.15 303.15 308.15 313.15 318.15 323.15 288.15 293.15 298.15 303.15 308.15 313.15 318.15 323.15 288.15 293.15 298.15 303.15 308.15 313.15 318.15 323.15 288.15 293.15 298.15 303.15 308.15 313.15 318.15 323.15

1.1570 1.0180 0.9041 0.8091 0.7289 0.6612 0.6029 0.5523 1.1770 1.0360 0.9201 0.8233 0.7415 0.6729 0.6136 0.5625 1.2110 1.0660 0.9457 0.8464 0.7623 0.6920 0.6313 0.5789 1.2570 1.1060 0.9824 0.8796 0.7921 0.7187 0.6556 0.6009

1.1237 1.0033 0.8989 0.8080 0.7287 0.6591 0.5978 0.5438 1.1606 1.0357 0.9275 0.8334 0.7512 0.6791 0.6158 0.5599 1.1987 1.0692 0.9570 0.8595 0.7744 0.6998 0.6342 0.5764 1.2381 1.1038 0.9875 0.8864 0.7983 0.7211 0.6532 0.5935

0.0146 0.0129 0.0116 0.0105 0.0089 0.0079 0.0069 0.0053 0.0321 0.0309 0.0295 0.0282 0.0263 0.0258 0.0247 0.0238 0.0619 0.0607 0.0582 0.0571 0.0551 0.0549 0.0543 0.0537 0.1022 0.1005 0.0993 0.0985 0.0963 0.0956 0.0949 0.0937

m (mol·kg−1)

0.126

0.151

0.176

0.201

T (K)

ηe (mPa·s)

ηc (mPa·s)

Δη

288.15 293.15 298.15 303.15 308.15 313.15 318.15 323.15 288.15 293.15 298.15 303.15 308.15 313.15 318.15 323.15 288.15 293.15 298.15 303.15 308.15 313.15 318.15 323.15 288.15 293.15 298.15 303.15 308.15 313.15 318.15 323.15

1.2860 1.1321 1.0060 0.9001 0.8111 0.7357 0.6711 0.6151 1.3320 1.1710 1.0410 0.9321 0.8399 0.7622 0.6951 0.6372 1.3740 1.2080 1.0740 0.9614 0.8655 0.7851 0.7157 0.6565 1.4320 1.2600 1.1190 1.0020 0.9029 0.8191 0.7469 0.6849

1.2788 1.1395 1.0190 0.9142 0.8229 0.7430 0.6728 0.6110 1.3209 1.1764 1.0514 0.9429 0.8483 0.7656 0.6930 0.6291 1.3643 1.2144 1.0849 0.9725 0.8745 0.7889 0.7138 0.6477 1.4091 1.2537 1.1194 1.0030 0.9016 0.8129 0.7352 0.6668

0.1277 0.1264 0.1257 0.1241 0.1226 0.1215 0.1207 0.1196 0.1680 0.1652 0.1648 0.1641 0.1625 0.1619 0.1608 0.1598 0.2048 0.2020 0.2018 0.2007 0.1979 0.1968 0.1952 0.1949 0.2557 0.2537 0.2521 0.2514 0.2497 0.2486 0.2473 0.2466

a Standard uncertainty: u(m) = 1 × 10−3 mol·kg−1, u(T) = 0.02 K, and u(p) = 350 Pa. Relative standard uncertainty: ur(η) = 0.58% and ur(Δη) = 0.59%. ηe, experimental viscosity; ηc, calculated viscosity by eq 2.

η ρt = ηw ρw tw

T = 293.15, 298.15, 303.15, 308.15, 313.15, 318.15, and 323.15 K. The molality of sodium lactobionate aqueous solutions was m = 0.0251, 0.0502, 0.0753, 0.100, 0.126, 0.151, 0.176, and 0.201 mol·kg−1. The experimental values are presented in Tables 2−4, respectively. The dependence of the densities, viscosities, and refractive index on the temperature (T) and the molality of sodium lactobionate (m) for sodium lactobionate + water binary mixture are shown graphically in Figures 1−3. They illustrate that the density, viscosity, and refractive index decrease with increasing temperature for constant composition of sodium lactobionate and increase with increasing sodium lactobionate concentration at a fixed temperature. The results can be explained as follows. As the solute concentration increases in the electrolyte solution, the numbers of collisions between different molecules also increase. As the kinetic energy of the molecules decreases, molecules have a tendency to stick together, so the density, viscosity, and refractive index of the solution increase.17 Correlation Models. The dependence of density and viscosity on molar mass fraction (w) of sodium lactobionate and temperature (T) under the studied conditions can be correlated by the Vogel−Tamman−Fulcher equation:18,19

(1)

where η, ρ, and t denote viscosities, densities, and flow times of the solutions, respectively, and ηw, ρw, and tw, denote the same for pure water. The relative standard uncertainty of calculated viscosity values was approximately 0.58%. The refractive index (n) for sodium lactobionate solutions was determined with an Abbe refractometer (ATAGO), which was connected with the same thermostatted water bath mentioned above. Anhydrous ethanol and doubly distilled water were employed as reference substances to calibrate the Abbe refractometer at normal pressure. The standard uncertainty of measurement is estimated to be 0.0001. The refractive index data is the average value of triplicate independent measurements for each solution at each temperature T = 288.15, 293.15, 298.15, 303.15, 308.15, 313.15, 318.15, and 323.15 K.

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RESULTS AND DISCUSSION The density, viscosity, and refractive index of the sodium lactobionate solutions were acquired at different temperatures T = 288.15, 293.15, 298.15, 303.15, 308.15, 313.15, 318.15, and 323.15 K, and the refractive index was obtained at temperatures C

DOI: 10.1021/acs.jced.5b00407 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

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Table 4. Refractive Index n for Sodium Lactobionate Aqueous Solutions (m) for T = 288.15−323.15 K under p = 101.3 kPaa m (mol·kg−1)

m = 0.0251

m = 0.0502

m = 0.0753

m = 0.100

a

m (mol·kg−1)

T (K)

n

nc

293.15 298.15 303.15 308.15 313.15 318.15 323.15 293.15 298.15 303.15 308.15 313.15 318.15 323.15 293.15 298.15 303.15 308.15 313.15 318.15 323.15 293.15 298.15 303.15 308.15 313.15 318.15 323.15

1.3331 1.3326 1.3321 1.3316 1.3310 1.3304 1.3297 1.3344 1.3339 1.3334 1.3328 1.3322 1.3316 1.3309 1.3360 1.3354 1.3348 1.3342 1.3335 1.3329 1.3322 1.3376 1.3370 1.3364 1.3357 1.3350 1.3343 1.3336

1.3331 1.3325 1.3320 1.3315 1.3309 1.3304 1.3299 1.3346 1.3340 1.3334 1.3328 1.3322 1.3316 1.3310 1.3361 1.3355 1.3348 1.3342 1.3335 1.3328 1.3322 1.3376 1.3370 1.3363 1.3356 1.3349 1.3342 1.3335

m = 0.126

m = 0.151

m = 0.176

m = 0.201

T (K)

n

nc

293.15 298.15 303.15 308.15 313.15 318.15 323.15 293.15 298.15 303.15 308.15 313.15 318.15 323.15 293.15 298.15 303.15 308.15 313.15 318.15 323.15 293.15 298.15 303.15 308.15 313.15 318.15 323.15

1.3392 1.3385 1.3378 1.3371 1.3364 1.3357 1.3349 1.3406 1.3400 1.3393 1.3387 1.3380 1.3373 1.3365 1.3421 1.3415 1.3409 1.3403 1.3397 1.3391 1.3384 1.3435 1.3430 1.3425 1.3420 1.3415 1.3409 1.3404

1.3391 1.3385 1.3378 1.3371 1.3364 1.3357 1.3350 1.3406 1.3400 1.3393 1.3387 1.3380 1.3373 1.3367 1.3421 1.3415 1.3409 1.3403 1.3397 1.3391 1.3385 1.3436 1.3430 1.3425 1.3420 1.3414 1.3409 1.3404

Standard uncertainty: u(m) = 1 × 10−3 mol·kg−1, u(n) = 1 × 10−4, u(T) = 0.02 K, u (p) = 350 Pa. nc, calculated refractive index by eq 5.

Figure 1. Density ρ of aqueous solutions of sodium lactobionate as a function of temperature and concentration.

⎛ P + P3m ⎞ Y = P1 exp⎜ 2 ⎟ ⎝ T − P4 ⎠

Figure 2. Viscosity η of aqueous solutions of sodium lactobionate as a function of concentration and temperature.

ARD =

(2)

where Y represents density (ρ, g·cm−3) or viscosity (η, mPa·s) of the sodium lactobionate solutions, m (mol·kg−1) is for the molality of sodium lactobionate in the solutions, T (K) is the absolute temperature, and P1, P2, P3, and P4 are the equation parameters that can be obtained from the experimental data. The values of parameters for the models of density, viscosity, and refractive index are acquired by least-squares regression method. The regressed values together with the average relative deviations (ARD) and the standard deviations (SD) are shown in Table 5. The ARD and SD are described by eqs 3 and 4:

1 N

∑

|Y cal − Y exp| Y exp

1/2 ⎡ (Y exp − Y cal)2 ⎤ ⎥ SD = ⎢∑ N−n ⎦ ⎣ exp

(3)

(4)

cal

where Y and Y are the experimental and calculated values, respectively. N denotes the number of experimental data points, and n denotes the number of parameters. The densities, viscosities, and refractive index of the binary sodium lactobionate + water mixture were correlated according to eq 2; the regressed model parameters are listed in Table 5. To compare the calculated values with experimental ones, the D

DOI: 10.1021/acs.jced.5b00407 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

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Figure 3. Refractive index n of aqueous solutions of sodium lactobionate as a function of temperature at different concentration.

Figure 4. Apparent molar volume Vϕ of aqueous solutions of sodium lactobionate as a function of concentration and temperature.

calculated results are also graphically shown in Figures 1−3. It can be seen that the agreement between the experimental and evaluated values for density and viscosity are very good. For density, the ARD and SD values are 0.044 and 0.059%, respectively. For viscosity the ARD is 0.87%, and the SD is 0.97%. Thus, the eq 2 can be used to predict the properties of the aqueous solutions of sodium lactobionate. The temperature (T) and concentration (m) dependence of the refractive index (n) of the binary mixtures are correlated to a polynomial of the form (eq 5)20 1

ln n =

volume (Vϕ) increase with increasing concentration of sodium lactobionate at a constant temperature; however, they increase at first and then decrease with the increase in temperature. These apparent molar volume values are very important in identifying the interactions of molecules. The relationship between the apparent molar volume and solute concentration reveals the interactions among solute molecules. When solute concentration tends to zero, the limiting apparent molar volume reflects the interactions between solute and solvent. The value of limiting apparent molar volume is equal to the standard partial molar volume in value. The dependence of the apparent molar volume (Vϕ) of sodium lactobionate in water on its molality at a certain temperature can be expressed as22

2

∑ ∑ cijT ix1j (5)

i=0 j=0

The six disposable parameters of eq 5 are obtained by means of the multilinear regression method. x is the concentration of sodium lactobionate in water in mole fraction. The resulting parameters cij are listed in Table 5. Equation 5 reproduces the experimental data with an ARD of 3.87 × 10−5 and a maximum deviation of 9.19 × 10−5. Volumetric Properties. The density data can be employed to evaluate the apparent molar volumes of mixed solutions, which can explicate the interactions of solvent−solvent and solute−solvent. The apparent molar volumes (Vϕ) of sodium lactobionate in water are calculated by the following equation21 Vϕ =

103(ρ − ρ0 ) M − ρ mρρ0

Vϕ = V ϕ0 + A v m1/2 + Bv m

(7)

−1

where (cm ·mol ) stands for the apparent molar volume of sodium lactobionate at infinite dilution, which equals the limiting partial molar volume of sodium lactobionate; Av (cm3· kg1/2·mol−3/2), which shows the solute−solute interaction, represents the slope of Debye−Huckel limiting law for the apparent molar volume; and Bv is an empirical constant that relates to solvent, solute, and temperature. Table 6 shows the values of V0ϕ, Av, and Bv regressed by least-squares analysis method. It can be seen from Table 6 that the values of limiting partial molar volume V0ϕ are all positive, which indicates that the interactions between sodium lactobionate and water molecules are weak. The solvent molecules sticking to the solute molecules expand with an increase in temperature below 308.15 K; however, they shrink above 308.15 K. The same conclusion can also be made according to the values of Bv. The Av values first decreases and then increases with an increase in temperature. The values of Av are all positive except for the V0ϕ

(6) −3

in which ρ0 is the density of pure water in kg·m and M is the molecular weight of sodium lactobionate. The evaluated apparent molar volumes of sodium lactobionate in water are presented in Table 2. Figure 4 is a stereogram of apparent molar volume as a function of temperature and concentration. The values of apparent molar volume (cm3·mol−1) are between 160.5 and 194.5. It can also be found that the apparent molar

3

Table 5. Parameters of Model for Density (ρ), Refractive Index (n), and Viscosity (η) as Functions of Quality Molarity Fraction (m) for Sodium Lactobionate Aqueous Solutions between 288.15 and 323.15 Ka ρ (g·cm−3) η (Pa·s) ij n quantity cij a

P1

P2

P3

P4

ARD

SD

1.126 −7.182 00

40.439 2.304 × 103 01 x1 0.242

−61.247 408.299 02 x21 −0.878

636.811 −28.854 10 T −6.506 × 10−5

3.55 × 10−4 8.67 × 10−3

4.52 × 10−4 9.74 × 10−3

3.87 × 10−5

7.34 × 10−5

0.305

11 Tx1 −6.683 × 10−4

12 Tx21 2.967 × 10−3

Standard uncertainties u are u(P1) = u(P2) = u(P3) = u(P4) = 0.001; u(cij) = 0.001. E

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Table 6. Standard Partial Molar Volume V0ϕ, the Limit Rate of Apparent Molar Volume Av, and Empirical Constant Bv for Sodium Lactobionate Aqueous Solutions at T = 288.15− 323.15 Ka T (K)

V0ϕ

Av

Bν

288.15 293.15 298.15 303.15 308.15 313.15 318.15 323.15

61.20 126.73 163.68 167.59 188.17 150.23 102.21 64.20

626.03 320.03 119.69 92.99 −32.74 156.28 392.07 572.85

−753.70 −380.05 −112.65 −72.28 107.78 −130.14 −424.50 −643.36

rate is positive; however it is negative when the temperature is higher than 303.15 K. This phenomenon reveals that the molecular interactions between sodium lactobionate and solvent molecules first decrease and then increase with increasing temperature. Equation 12 is employed to calculate the excess molar volume for sodium lactobionate aqueous solutions.23−28 ⎛1 ⎛1 1⎞ 1⎞ V E = x1M1⎜⎜ − ⎟⎟ + x 2M 2⎜⎜ − ⎟⎟ ρ1 ⎠ ρ2 ⎠ ⎝ρ ⎝ρ

in which M1 and M2 are the molecular weights of pure components 1 and 2, respectively, and ρ1 and ρ2 are the corresponding densities. The density of sodium lactobionate in solid state has not been reported in the previous publications. Here it is determined by employing methanol as the sink/float medium. The method was described in detail in ref 29. The measured density of sodium lactobionate in solid state at 298 K is 1.7703 g·ml−1. The calculated excess molar volumes are listed in Table 2 and plotted in Figure 5. The values of the excess

Standard uncertainty: u(V0ϕ) = 1 × 10−2 cm3·mol−1; u(AV) = u(BV) = 1 × 10−2.

a

system at 308.15 K, which shows stronger solute interactions. This case results from more violent thermal motion of the molecules at the studied temperature, so the interaction force of the solute−solute is reduced. The variation of V0ϕ, Av, and Bv with temperature is described by eqs 8−10.3,21 V ϕ0 = a0 + a1T + a 2T 2

A v = b0 + b1T + b2T

(12)

(8)

2

(9)

Bv = c0 + c1T + c 2T 2

(10)

Differentiating of eq 8 with regard to T at constant pressure, the expansion rate of standard partial molar volume (∂V0ϕ/∂T)P is obtained.3 ⎛ ∂V 0 ⎞ ⎜⎜ ϕ ⎟⎟ = a1 + 2a 2T ⎝ ∂T ⎠P

(11)

Figure 5. Excess molar volume VE of aqueous solutions of sodium lactobionate as a function of temperature at different concentrations (m, molality; mol·kg−1): ■, 0.0502; ●, 0.0753; ▲, 0.100; ▼, 0.126; ◁, 0.151; ▷, 0.176; and ⧫, 0.201.

The empirical parameters’ values of a0, a1, and a2 in eq 8, b0, b1, and b2 in eq 9, and c0, c1 and c2 in eq 10 are presented in Table 7, and the values of the expansion rate of standard partial molar volume for sodium lactobionate aqueous solutions at different temperatures are shown in Table 8.

molar volumes for the studied system are all negative; they increase with increasing in concentration of sodium lactobionate at a fixed temperature. However, with a certain concentration of sodium lactobionate, the values of the excess molar volumes increase first and then decrease with increasing temperature. The relationship between the excess molar volume (VE) and composition are also described by means of a Redlich−Kister equation:24−27

Table 7. Values of Empirical Parameters in Equations 8−10 a0 × 104

a1

a2

V0ϕ

−3.563 ± 0.001 b0 × 105

234.593 ± 0.002 b1 × 103

−0.384 ± 0.001 b2

Av

1.774 ± 0.001 c0 × 105

−1.161 ± 0.001 c1 × 103

1.899 ± 0.001 c2

Bv

−2.237 ± 0.001

1.462 ± 0.001

−2.389 ± 0.001

N

V E = x1(1 − x1) ∑ Ai (1 − 2x1)i

It can be observed that the values of expansion rate of standard partial molar volume are diminished with increasing in temperature over the studied temperature range. When the temperature is equal to or lower than 303.15 K, the expansion

(13)

i=0

where Ai represents the coefficients of Redlich−Kister polynomial, which can be acquired by a least-squares regression

Table 8. Expansion Rate of Standard Partial Molar Volume at Different Temperatures for Sodium Lactobionate Aqueous Solutions at T = 288.15−323.15 Ka

a

T (K)

288.15

293.15

298.15

303.15

308.15

313.15

318.15

323.15

expansion rate

13.159

9.317

5.474

1.632

−2.210

−6.053

−9.895

−13.737

Standard uncertainties u are u((∂V0ϕ/∂T)P) = 0.001. F

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Table 9. Coefficients Ai of Fitting Equation 13 and Viscosity B Coefficient of Equation 18a

a

T (K)

A0

A1

A2

σ

T (K)

A

B

σ

288.15 293.15 298.15 303.15 308.15 313.15 318.15 323.15

−1359285 −700050 −387797 −437246 −299169 −589537 −943022 −1358443

2735244 1408784 780847 880329 602930 1187262 1898685 2734551

−1376044 −708791 −393099 −443134 −303810 −597786 −955739 −1376199

0.0027 0.0025 0.0027 0.0029 0.0036 0.0040 0.0020 0.0016

288.15 293.15 298.15 303.15 308.15 313.15 318.15 323.15

−8.08 −8.30 −8.66 −8.94 −9.32 −9.42 −9.55 −9.83

1796.70 1799.88 1822.77 1839.08 1856.71 1858.44 1861.87 1879.04

0.0069 0.0068 0.0065 0.0065 0.0065 0.0064 0.0062 0.0060

Standard uncertainties u are u(A0) = u(A1) = u(A2) = 1; u(A) = u(B) = 0.01.

Table 10. Partial Molar Volumes, Partial Molar Volume at Infinite Dilution, and Excess Partial Molar Volumes at Infinite Dilution for the System of Sodium Lactobionate + Watera m 0.0502 0.0753 0.100 0.126 0.151 0.176 0.201 T = 293.15 K 0.0502 0.0753 0.100 0.126 0.151 0.176 0.201 T = 298.15 K 0.0502 0.0753 0.100 0.126 0.151 0.176 0.201 T = 303.15 K 0.0502 0.0753 0.100 0.126 0.151 0.176 0.201 a

V̅ 1

V̅ 2

187.49 201.07 207.92 208.10 201.63 188.55 168.90

T = 288.15 K 18.03 18.02 18.01 18.01 18.02 18.06 18.13

191.85 199.08 202.84 203.16 200.06 193.55 183.66

18.04 18.03 18.03 18.03 18.04 18.06 18.09

191.56 196.43 199.37 200.41 199.54 196.78 192.14

18.07 18.06 18.06 18.06 18.06 18.07 18.08

191.36 196.68 199.83 200.83 199.68 196.40 191.01

18.09 18.09 18.08 18.08 18.08 18.09 18.11

V̅ ∞ 1

140.05

166.95

176.02

174.19

V̅ E∞ 1

−84.98

−58.08

−49.01

−50.84

m 0.0502 0.0753 0.100 0.126 0.151 0.176 0.201 T = 313.15 K 0.0502 0.0753 0.100 0.126 0.151 0.176 0.201 T = 318.15 K 0.0502 0.0753 0.100 0.126 0.151 0.176 0.201 T = 323.15 K 0.0502 0.0753 0.100 0.126 0.151 0.176 0.201

V̅ ∞ 1

V̅ E∞ 1

176.03

−49.00

18.15 18.14 18.14 18.13 18.14 18.15 18.17

164.44

−60.59

186.48 197.89 204.62 206.71 204.17 197.04 185.34

18.18 18.17 18.16 18.16 18.16 18.18 18.22

149.56

−75.47

184.94 200.31 208.97 210.93 206.25 194.95 177.07

18.22 18.20 18.19 18.18 18.20 18.23 18.29

133.87

−91.16

V̅ 1

V̅ 2

190.06 194.83 198.11 199.91 200.24 199.10 196.51

T = 308.15 K 18.12 18.12 18.11 18.11 18.11 18.11 18.12

188.67 196.37 201.14 203.01 201.99 198.09 191.33

E∞ −3 Standard uncertainty: u(m) = 1 × 10−3 mol·kg−1; u(V̅ ) (includes V̅ 1, V̅ 1, V̅ ∞ cm3·mol−1. 1 , and V̅ 1 ) = 1 × 10

n

method. The Ai values together with the standard deviations are given in Table 9 for the studied mixtures. Within the composition range for the studied system, the partial molar volumes (V̅ 1) of the mixtures can also be evaluated by using eqs 14 and 15.24

V2 = V2 + x12 ∑ Ai (1 − 2x1)i + 2x12(1 − x1) i=0 n

∑ Aii(1 − 2x1)i− 1

The excess partial molar volume and partial molar volume at

n

V1 = V1 + (1 − x1)2

∑ Ai(1 − 2x1)i − 2x1(1 − x1)2

infinite dilution are evaluated via eq 16.24

i=0 n

∑ Aii(1 − 2x1)i− 1 i=0

(15)

i=0

n

V1∞ = V1 +

(14)

i=0

G

n

∑ Ai and V1E ∞ = ∑ Ai i=0

(16)

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sodium lactobionate.32,33 The viscosity B coefficient rises with an increase in temperature, so the dB/dT values are positive. In general, the dB/dT was considered as a good criterion for judging the structure-making/breaking nature of solute. The positive values of dB/dT in the studied system illustrate that sodium lactobionate appears to be structure maker, which is in accord with the conclusion obtained from volumetric properties for the system of sodium lactobionate + water.

The calculated values of partial molar volumes, partial molar volume at infinite dilution, and excess partial molar volumes at infinite dilution for the system of sodium lactobionate aqueous solution are reported in Table 10 at different temperatures. It can be found that the V̅ ∞ 1 values are slightly smaller than the corresponding molar volume of the pure sodium lactobionate (V1). All of the V̅ E∞ values are negative, which shows the 1 existence of significant interactions of the solvent−solute among molecules of different natures. Relative Changes in Viscosity and Viscosity B Coefficient. The relative changes in viscosity of sodium lactobionate in water (Δη) can be expressed as3,21 η − η0 η Δη = = −1 η0 η0 (17)

■

CONCLUSIONS In the present work, density (ρ), viscosity (η), and refractive index (n) were determined experimentally for the system of sodium lactobionate + water. These thermodynamic property data were acquired with the composition range of 0.025−0.21 mol·kg−1 at the temperature range of 288.15−323.15 K in intervals of 5 K under atmospheric pressure. The three thermodynamic properties decrease with increasing temperature for constant composition of sodium lactobionate and increase with increasing sodium lactobionate concentration at a given temperature. The dependency of density and viscosity on molality (m) of sodium lactobionate and temperature (T) were correlated by the Vogel−Tamman−Fulcher equation, and the refractive index was fitted by the liner equation. The parameter values for the two models were obtained by least-squares regression method; the calculated values agreed well with experimental data for density, viscosity, and refractive index. On the basis of the thermodynamic property data determined experimentally, the apparent molar volume, excess molar volume, partial molar volume, standard partial molar volume, the limit rate of apparent molar volume, expansion rate of standard partial molar volume, the relative viscosity, and viscosity B coefficient were calculated for the system of sodium lactobionate + water. The apparent molar volume and excess molar volume increase with an increase in concentration of sodium lactobionate at a fixed temperature; nevertheless, they increase at first and then decrease with increasing temperature. The interactions between sodium lactobionate and water molecules are weak; however for solute, the interactions are strong. In the sodium lactobionate aqueous solution, sodium lactobionate behaves as structure maker.

where η0 represents the viscosity of pure water. The evaluated values of Δη by eq 17 based on the experimental data are presented in Table 3 and plotted in Figure 6. We can find that

Figure 6. Relative viscosity Δη of aqueous solutions of sodium lactobionate as a function of concentration and temperature.

Δη values increase with an increase in concentration of sodium lactobionate and decrease with an increase in temperature. This case illustrates that the interaction of solute−solvent strengthens with an increase in concentration of sodium lactobionate and weakens with an increase in temperature. The thermal motion of molecules of the sodium lactobionate aqueous solution intensifies as the temperature increases and weakens the interaction of solute−solvent. In the range of studied compositions and temperatures, Δη values are all positive. The results indicate that the internal force of molecules rises as the solution is formed and also show that the solution exhibits negative deviation with Raoult’s law. On the basis of the Jones−Dole equation,30−32 the dependency of relative viscosity (ηr = η/η0) on the solute concentration, m, can be expressed by η = 1 + A m + Bm ηr = η0 (18)

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AUTHOR INFORMATION

Corresponding Author

*Tel.: + 86 514 87975568. Fax: + 86 514 87975244. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We thank the National Natural Science Foundation of China for their support (project number: 21406192). Furthermore, the Yangzhou City Science and Technology Bureau, China (project number: 2012038-3 and YZ2011139), are also appreciated. In addition, we also give our thanks to the Practice Innovation Project of Jiangsu Province for Post Graduate Students (Project number: SJZZ15_0180).

where m represents the molarity of the sodium lactobionate in water (mol·kg−1). The viscosity B coefficient, B, which is a main contributor to the relative viscosity (ηr), stands for size and shape effect of solute, Coulombic interactions, and effect of Einstein, structural, and solvation caused by interaction of solute−solvent.3 Table 9 shows the regressed values of B coefficient by means of least-squares analysis. It can be found that the values of the B coefficient for sodium lactobionate are all positive. This shows the tendency of structure-making for

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