Viscosities, Densities, and Refractive Indices of Aqueous Propane-1,3

Article pubs.acs.org/jced

Cite This: J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Viscosities, Densities, and Refractive Indices of Aqueous Propane1,3-diol František Mikšík,†,‡ Jan Skolil,*,§ Josef Kotlík,† Josef Č aś lavský,† Takahiko Miyazaki,‡ Marie Kací̌ rkova,́ § and Helena Půcǩ ova†́ †

Faculty of Chemistry, Brno University of Technology, Brno 61200, Czech Republic IGSES, Kyushu University, Kasuga 816-8580, Japan § Antifreeze Development Laboratory, Classic Oil s.r.o., Kladno 27201, Czech Republic Downloaded via DURHAM UNIV on November 17, 2018 at 02:29:31 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

‡

S Supporting Information *

ABSTRACT: A detailed investigation into the viscosity, density, and refractive index of a binary mixture of propane1,3-diol/water was performed for the whole range of mass fractions of propane-1,3-diol. The viscosity and density were measured over a wide range of temperatures from T = 253.15 K to T = 353.15 K where possible, or to the nearest safe point before freezing. The refractive indices were measured for the same dilutions as in the case of the viscosity and density over a reduced temperature range of T = (278.15 to 318.15) K. A mathematical analysis via excess properties was conducted and fitted to the Redlich−Kister equation. Furthermore, a prediction of density data is provided by a polynomial and DIPPR exponential model. The viscosity data are fitted to Grunberg-Nissan, Lederer, McAllister, and Heric models and a new combined model with variable temperature and molar and mass fractions based on these models is presented. Data comparison of the propane-1,3-diol/water binary mixture is performed on propan-1,2-diol/water mixture as the physically closest binary system and application alternative fluid.

1. INTRODUCTION Propane-1,3-diol (PDO) belongs to the group of α,ωalkanediols and like other diols with a short hydrocarbon chain it is fully miscible with water. The potential applications of propane-1,3-diol are similar to the other diols but thanks to its high price it has so far mainly remained as a monomer for polymer production such as polytrimethylene terephthalate (PTT).1,2 However, during the past decade great effort has been made by many researchers and companies to lower the production price3,4 and, therefore, broaden the spectrum of possible applications. In this context, the great industrial potential of propane-1,3-diol is gradually being seen, and new applications are growing in importance. New possible applications of PDO are generally derived from the application of other diols based on their compatibility. Currently, the most utilized diols are propane-1,2-diol (MPG) and ethane-1,2-diol (MEG), which also have physical and chemical properties similar to PDO. Because of the similar properties they can be replaced by PDO in many cases except in the production of polymers, where MPG and MEG are extensively used for their own specific products. Especially great potential for a new PDO application is its usefulness as an antifreeze additive5 or possible aircraft deicer6 for which inexpensive MPG and MEG are used in great quantities. So far, the option of using expensive PDO for low-cost applications, such as antifreeze © XXXX American Chemical Society

additive or deicers, has only been on a theoretical level, but the great demand for new ecological chemical feedstock for polymers has led researchers and companies to a new biological means of PDO production4,7−10 which is gradually lowering the price of the raw chemical. In this context, reliable data on the chemical and physical properties of aqueous PDO for antifreeze additives and deicers are growing in importance. To this date, several researchers have reported some of the physical and chemical data on pure PDO. The data on the density and refractive indices of pure PDO are generally in good agreement in the literature, although they are usually limited by a small temperature range. Likewise, the data on the viscosity of pure PDO are strongly limited in the temperature range and furthermore, there are notable differences between reported values from various sources, as has been shown in Moosavi and Rostami11 and in this work. Unfortunately, the number of available literature sources on the viscosity of pure propane-1,3-diol is very limited, and any further comparison is impossible. The data are compared to the most similar system of common propane-1,2-diol which is also the closest candidate for substitution in freezing protection applications. Received: May 15, 2018 Accepted: November 6, 2018

A

DOI: 10.1021/acs.jced.8b00403 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

Article

Table 1. Specifications of Chemical Samples chemical name

formula

CASRN

source

mass fraction purity

purification method

propane-1,3-diol Ppropane-1,2-diol

C3H8O2 C3H8O2

504-63-2 57-55-6

DuPont Tate&Lylea Sigma-Aldrichb

>0.998a >0.995b

used as supplied used as supplied

a

Determined by gas chromatography. bDeclared by the supplier.

Table 2. Comparison of Experimental Data on Density (ρ)a, Dynamic Viscosity (η)a, and Refractive Indices (nD)b, of Pure Propane-1,3-diol, Propane-1,2-diol, and Water with Available Literature Sources 10−3 ρ/(g·cm−3) T/K

expt

293.15 303.15 313.15 323.15

0.9993 0.9968 0.9932 0.9890 10−3 ρ/(g·cm−3)

η/(mPa·s)

nD

lit.

expt

lit.

expt

lit.

0.9982120 0.9956520 0.9922220 0.9880420

water 1.0407 0.8467 0.6997 0.5967

1.001620 0.7972220 0.6527320 0.5465220

1.3331 1.3320 1.3306

1.3333621 1.3323021 1.3309521

η/(mPa·s)

T/(K)

expt

lit.

AAD (%)

283.15

1.0598

293.15

1.0536

303.15

1.0474

313.15

1.0411

318.15

1.0365

323.15

1.0348

1.0591422 1.0569223 1.0536612 1.056211 1.0528713 1.0536423 1.0470912 1.046716 1.046711 1.0475823 1.0406112 1.040216 1.040411 1.0376012 1.037516 1.0373717 1.037911 1.0345112 1.034116

0.0623 0.2717 0.0057 0.2468 0.0693 0.0038 0.0296 0.0668 0.0668 0.0172 0.0471 0.0864 0.0672 0.1061 0.0965 0.0839 0.1351 0.0280 0.0676

293.15

1.0355

303.15

1.0281

313.15

1.0206

323.15

1.0130

333.15

1.0053

1.0363824 1.0365712 1.0291424 1.0291512 1.0292026 1.0219024 1.0214412 1.0215026 1.0137412 1.0140026 1.0062026

0.0850 0.1033 0.1012 0.1021 0.1070 0.1274 0.0823 0.0882 0.0731 0.0987 0.0945

expt

lit.

nD AAD(%)

expt

lit.

AAD (%)

Propane-1,3-diol

52.007

51.5511

0.8787

1.4395

1.439311

0.0139

32.921

32.7411

0.5498

1.4368

1.436711 1.436816

0.0070 0.0000

1.4340

1.434011 1.434016

0.0000 0.0000

21.791

17.01117

0.6309

1.4325

1.432711 1.432716

0.0140 0.0140

Propane-1,2-diol 57.701 56.62025

1.8728

1.4332

33.28913 33.16425 32.4027 19.92513 19.85425 19.1227 12.64125 12.1027 8.1927

2.8919 2.5055 0.1441 2.0626 1.6989 2.0609 1.0004 3.3222 3.1150

1.4299

1.432613 1.432725 1.429313 1.429325

0.0391 0.0321 0.0420 0.0371

1.425913 1.425925

0.0337 0.0330

17.119

15.000

32.353

19.522

12.516 8.4530

1.4264

a

Standard uncertainties are u(T) = 0.01 K, u(p) = 5 kPa and u(wi) = 0.0002; relative standard uncertainties are ur(ρ) = 0.004 and ur(η) = 0.05. Standard uncertainties are u(T) = 0.05 K, u(p) = 5 kPa, u(wi) = 0.0002 and u(nD) = 0.0001.

b

refractive indices over a temperature range of T = (298.15 to 318.15) K with five different dilutions. Other results have been published by various researches focusing on different aspects of the PDO/water system. Density, speed of sound, relative permittivity, and for the first time viscosity, were reported by George and Sastry17 together with other water/alkanediols systems. Molar enthalpies were measured by Nagamachi and Franczesconi18 and data on heat capacities of some pure α,ωalkanediols, including PDO, were measured by Góralski and

Propane-1,3-diol is reported to have been used in several different binary mixtures with various chemicals. Properties of the mixture with butane-1,4-diol was reported by Li et al.12 Densities, viscosities, and refractive indices of polyethylene glycol with PDO and butyl lactate with PDO were measured by Bajić et al.13,14 and properties for the mixture of pyridine with PDO were provided by Kijevčanin et al.15 The first expanded data on the binary system PDO/water were prepared by Lee et al.16 and consisted of densities, surface tensions, and B


a

C

0.0000 0.0288 0.0635 0.1061 0.1597 0.1920 0.2721 0.3826 0.5449 0.7523 1.0000

0.0000 0.0253 0.0552 0.1049 0.1917 0.3266 0.4933 0.8122 1.0000

0.0000 0.1113 0.2226 0.3340 0.4453 0.5010 0.6123 0.7236 0.8349 0.9277 1.0000

0.0000 0.0988 0.1980 0.3312 0.5005 0.6720 0.8044 0.9481 1.0000

1.0625 1.0708 1.0716 1.0682 1.0658

1.0537 1.0580 1.0682 1.0749 1.0785 1.0796 1.0782

253.15 K

1.0417 1.0598 1.0676 1.0681 1.0647 1.0622

1.0518 1.0559 1.0658 1.0722 1.0755 1.0765 1.0754

258.15 K

1.0400 1.0569 1.0643 1.0645 1.0610 1.0586

1.0352 1.0477 1.0525 1.0621 1.0684 1.0722 1.0732 1.0723

263.15 K

1.02170 1.03815 1.0540 1.0609 1.0611 1.0574 1.0549

1.0221 1.0339 1.0457 1.0503 1.0595 1.0656 1.0693 1.0701 1.0690

268.15 K

1.0008 1.0098 1.0208 1.0363 1.0511 1.0575 1.0576 1.0538 1.0513

1.0008 1.0097 1.0213 1.0325 1.0437 1.0481 1.0569 1.0627 1.0663 1.0671 1.0659

273.15 K

283.15 K Propane-1,3-diol 1.0006 1.0088 1.0192 1.0292 1.0393 1.0433 1.0514 1.0568 1.0602 1.0610 1.0598 Propane-1,2-diol 1.0006 1.0082 1.0178 1.0312 1.0446 1.0505 1.0504 1.0456 1.0429 0.9993 1.0057 1.0145 1.0263 1.0381 1.0432 1.0429 1.0382 1.0355

0.9993 1.0067 1.0163 1.0254 1.0345 1.0383 1.0457 1.0509 1.0540 1.0547 1.0536

293.15 K

0.9968 1.0026 1.0106 1.0210 1.0314 1.0359 1.0354 1.0308 1.0281

0.9968 1.0041 1.0129 1.0211 1.0295 1.0330 1.0399 1.0448 1.0478 1.0485 1.0474

303.15 K

Standard uncertainties are u(T) = 0.01 K, u(p) = 5 kPa and u(wi) = 0.0002; relative standard uncertainty is ur(ρ) = 0.004

xi

wi

0.9932 0.9990 1.0060 1.0153 1.0245 1.0284 1.0278 1.0232 1.0206

0.9932 1.0004 1.0083 1.0163 1.0239 1.0273 1.0338 1.0386 1.0415 1.0422 1.0411

313.15 K

0.9890 0.9944 1.0008 1.0092 1.0174 1.0208 1.0201 1.0156 1.0130

0.9890 0.9962 1.0035 1.0108 1.0181 1.0213 1.0274 1.0321 1.0350 1.0358 1.0348

323.15 K

0.9844 0.9893 0.9954 1.0027 1.0100 1.0130 1.0122 1.0078 1.0053

0.9844 0.9911 0.9983 1.0052 1.0118 1.0148 1.0209 1.0254 1.0283 1.0293 1.0284

333.15 K

0.9786 0.9835 0.9892 0.9960 1.0023 1.0049 1.0041 0.9998 0.9977

0.9786 0.9853 0.9924 0.9993 1.0053 1.0084 1.0141 1.0185 1.0215 1.0226 1.0219

343.15 K

0.9727 0.9772 0.9825 0.9889 0.9946 0.9969 0.9960 0.9919 0.9895

0.9727 0.9786 0.9863 0.9926 0.9985 1.0012 1.0068 1.0117 1.0144 1.0158 1.0153

353.15 K

Table 3. Density ρ/(g·cm−3) Values of Aqueous Propane-1,3-diol for Different Mass Fractions wi and Molar Fractions xi over the Temperature Range T = (253.15 to 353.15) K at Standard Pressure (p = 101.325 kPa)a

Journal of Chemical & Engineering Data Article



Article

Figure 1. Density ρ/(g·cm−3) of aqueous propane-1,3-diol (a, b) and aqueous propane-1,2-diol (c, d). (a, c) Dependency on mass fraction wi for (+) 253.15 K; (×) 258.15 K; (∗) 263.15 K; (□) 268.15 K; (■) 273.15 K; (○) 283.15 K; (●) 293.15 K; (△) 303.15 K; (▲) 313.15 K; (◇) 323.15 K; (◆) 333.15 K; (▽) 343.15 K; (▼) 353.15 K. (b) Dependency on temperature for (+) wi = 0; (×) wi = 0.111; (∗) wi = 0.223; (□) wi = 0.334; (■) wi = 0.445; (○) wi = 0.501; (●) wi = 0.612; (△) wi = 0.724; (▲) wi = 0.835; (◇) wi = 0.928; (◆) wi = 1.0. (d) Dependency on temperature for (+) wi = 0; (×) wi = 0.099; (∗) wi = 0.198; (□) wi = 0.331; (■) wi = 0.501; (○) wi = 0.672; (●) wi = 0.804; (△) wi = 0.948; (▲) wi = 1.0.

Tkaczyk.19 The latest study on the properties of selected α,ωalkanediols/water systems including PDO was prepared by Moosavi and Rostami11 focusing on density, viscosity, refractive index, and excess properties. For most of the current applications in which PDO is used in its pure form, existing data are generally sufficient, but new applications such as the above-mentioned antifreeze additives, deicers, and others, where PDO is used in a mixture with water, require a much wider range of temperatures and mass fractions than has been reported. In the most detailed work on the binary mixture of PDO/water with a full span of measured dilutions from George and Sastry17 values are measured over a limited range of temperatures from 298.15 to 338.15 K and the most important values for use as antifreeze additives and deicers around and below 273.15 K are not measured. Similarly, the work of Moosavi and Rostami11 is limited by temperature range T = (288.15 to 318.15) K and a limited number of points in the lower dilutions of PDO, thanks to molar fractions selected for different purposes. In this work, experimental results on aqueous PDO for the most used engineering data of densities, viscosities, and refractive indices are presented. To compare the measured properties to some common system an experimental measurement of properties under similar conditions of aqueous

propane-1,2-diol is shown. The properties of aqueous propane-1,3-diol were obtained over a whole mass fraction range with nine evenly spread different dilutions over a wide temperature range of T = (253.15 to 353.15) K for viscosity and density and T = (278.15 to 318.15) K for refractive indices. The experimental data on aqueous propane-1,2-diol was obtained in a reduced number of seven dilutions in the same temperature range. The full experimental data below T = 288.15 K and above T = 338.15 K on aqueous propane-1,3diol are presented for the first time. Similarly, the data on aqueous propane-1,2-diol data below T = 288.15 K and above T = 338.15 K are not available and an original set of data is presented.

2. METHODOLOGY 2.1. Chemicals. The propane-1,3-diol used for the purposes of this work was obtained from DuPont Tate&Lyle in high purity of at least 99.8% as a part of DuPont Tate&Lyle BioProducts. Methanol for gas chromatography was supplied by Sigma-Aldrich in GC quality. The purity of the propane-1,3diol was established through gas chromatography using a Shimadzu 2010 GC Plus with Flame Ionization Detector (FID). The preparation of different dilutions was done by gravimetric preparation of pure PDO and volumetric dilution D



Article

Table 4. Density Models’ Constants for Pure Propane-1,3-diol and Pure Propane-1,2-diol A

B

polynomial DIPPR

1.02726 363.87352

0.00227 0.54489

polynomial DIPPR

−0.68798 344.28057

0.02542 0.52417

C Propane-1,3-diol −1.513 × 10−5 697.42222 Propane-1,2-diol −1.32 × 10−4 670.66203

D

E

σ

3.537 × 10−8 0.52552

−3.116 × 10−11

7.45 × 10−5 1.44 × 10−4

2.93 × 10−7 0.60781

−2.43 × 10−10

1.62 × 10−4 2.21 × 10−4

Table 5. Redlich−Kister Equation Coefficients for Modelling Excess Molar Volume VE/(cm3·mol−1) for a Temperature Range of T = (273.15 to 353.15) K of Aqueous Propane-1,3-diol, Aqueous Propane-1,2-diol and the Respective Standard Deviations to Experimental Data T/K

A0

A1

273.15 283.15 293.15 303.15 313.15 323.15 333.15 343.15 353.15

−2.13815 −1.96135 −1.83147 −1.71824 −1.64783 −1.55728 −1.46133 −1.39219 −1.30987

−1.04474 −0.84526 −0.79354 −0.62566 −0.53979 −0.43815 −0.36979 −0.35020 −0.33974

273.15 283.15 293.15 303.15 313.15 323.15 333.15 343.15 353.15

−2.72823 −2.68183 −2.48697 −2.33637 −2.21786 −2.10453 −2.00803 −1.88166 −1.83766

−2.25756 −2.06884 −1.84505 −1.63801 −1.48424 −1.35422 −1.21405 −1.19662 −1.05768

A2 Propane-1,3-diol −1.00299 −0.85436 −0.45801 −0.49776 −0.35839 −0.27570 −0.21233 −0.15933 −0.03945 Propane-1,2-diol −2.07347 −1.43295 −1.30531 −1.19507 −0.95139 −0.88406 −0.68639 −0.42670 −0.57097

A3

A4

σ

0.66726 0.47921 0.53301 0.30613 0.26038 0.11339 0.12763 0.08362 0.19667

1.73058 1.59933 1.04601 1.05976 0.80534 0.65961 0.46425 0.17932 −0.01046

0.008 0.006 0.006 0.004 0.002 0.001 0.003 0.003 0.007

1.06123 1.16643 1.07814 0.95274 0.74377 0.65616 0.54686 0.32478 0.30513

1.97296 1.59356 1.49898 1.34506 0.96229 0.81600 0.54308 0.16785 0.29532

0.015 0.014 0.013 0.011 0.009 0.007 0.007 0.006 0.006

density was found to be 0.01% upon calibration carried out by the manufacturer by several different standards (APS3, APN7.5, APN26, APN415). The uncertainty for viscosity was acquired in the same way as for density and was defined as a set of expanded uncertainties for different viscosity ranges. The uncertainty for the range 0.3 mm2·s−1 to 10 mm2·s−1 is less than 0.1%, for the range 10 mm2·s−1 to 1000 mm2·s−1 is less than 0.2% and for the range 1000 mm2·s−1 to 82 500 mm2· s−1 is equal or less than 0.23%. For the measurement of refractive indices, an Anton Paar Abbemat 350 Automatic Refractometer with a measured wavelength of 589 nm was used. The temperature stability for the device is stated as 0.002 K with uncertainty 0.05 K. The measurement uncertainty of refractive indices for the measured samples is claimed by the manufacturer to be 0.0001 point of the refractive index scale. The values were verified by measuring the standards of well-known refractive indices over the whole intended temperature range.

by deionized water prepared by reverse osmosis. The mass fractions of the prepared solutions were then verified by gas chromatography with nitrogen as a carrier gas. For the measurement using the GC, the samples were diluted in a ratio of 1:25 with methanol. Propane-1,2-diol was acquired from Sigma-Aldrich in high purity of ≥99.5%. All the chemicals were used as received without further modification or purification. The specification of the samples is provided in Table 1. The densities, viscosities, and refractive indices of pure propane-1,3-diol were measured first and compared with available literature sources. The measured data on propane-1,2diol were measured later for practical comparison of a similar system. The comparison with available literature sources is listed in Table 2. Data on the density and refractive indices of pure PDO are abundant; however, proper comparison is possible only for the temperature range of T = (293.15 to 338.15) K for most of the sources. The data on viscosity of pure PDO are also limited in amount and within even more reduced temperature ranges. 2.2. Apparatus and Procedures. The dynamic viscosity and density of aqueous PDO were measured together using an Anton Paar Stabinger SVM 3000 automatic viscometer. The temperature was maintained by a Julabo thermostat/cryostat. The uncertainty of the set temperature measured by a highprecision platinum resistance thermometer at the point of measurement was 0.01 K. The expanded uncertainty for

3. RESULTS AND DISCUSSION 3.1. Density. The measured data on density for the whole range of dilutions and temperatures are listed in Table 3. The dependence of density on mass fraction and on temperature of all measured points is then presented in Figure 1. The acquired data of pure PDO are in good agreement with available literature sources and the standard deviation does not exceed σ = 0.002 for the compared range of T = (293.15 to 338.15) K, E



Article

Figure 3. Excess molar volume VE/(cm3·mol−1) as a function of molar fraction xi of proapane-1,3-diol and propane-1,2-diol in water with their respective calculated Redlich−Kister curves (full lines correspond to propane-1,3-diol (this work); dashed lines correspond to propane-1,2-diol (this work)) in comparison to selected literature data. The respective symbols correspond to (+) propane-1,3-diol (this work); (×) propane-1,2-diol (this work); (■) propane-1,3-diol (Zemánková et al.22); (●) propane-1,3-diol (Moosavi and Rostami11); (⧫) propane-1,2-diol (Zemánková et al.22).

ρ=

A 1 + (1 − TC )

D

B

The respective constants and deviations for both models are listed in Table 4. The root-mean-square deviations σ (RMSD) were calculated as ÄÅ n É ÅÅ ∑ (y − y )2 ÑÑÑ1/2 ÅÅ i = 0 expt cal Ñ ÑÑ σ = ÅÅÅ ÑÑ ÅÅ ÑÑ n ÅÅÇ ÑÑÖ (3)

Figure 2. Experimental values of excess molar volume VE/(cm3· mol−1) for aqueous propane-1,3-diol (a), aqueous propane-1,2-diol (b) and the respective Redlich−Kister curves. The progress of the Redlich−Kister eq 5 is shown as a continuous line belonging to the respective symbols for experimental data at (+) 273.15 K; (×) 283.15 K; (∗) 293.15 K; (□) 303.15 K; (■) 313.15 K; (○) 323.15 K; (●) 333.15 K; (△) 343.15 K; (▲) 353.15 K.

where yexpt represents the experimental values, ycal is the calculated values, and n is the dimension of the correlated set. The respective models are sufficiently precise within the measured range (deviation is less than σ = 0.0002), although the polynomial model shows better agreement with the measured data as the variability of the model is better. To describe the influence of the water dilution on the density of PDO we used the excess molar volume method. Because the basic data were acquired in the form of mass fractions, for the purpose of calculation and subsequent analysis of excess molar volume and other excess units, the mass fraction values were recalculated also to molar fractions. The excess molar volume for VE/(cm3·mol−1) was calculated by the following equation: ÄÅ É n Åi y i yÑÑÑ Å j z 1 1 Å Ñ j z V E = ∑ xiMiÅÅÅÅjjj zzz − jjjj zzzzÑÑÑÑ ÅÅk ρ { k ρi {ÑÑ (4) i=1 ÅÇ ÑÖ

comparable results show the data of propane-1,2-diol. For a comparison of propane-1,3-diol data a broader range T = (283.15 to 363.15) K of experimental data is only offered in the work of Zoreb̨ ski et al.,28 and according to our best knowledge, a comparison below T = 283.1 K is not possible because of a lack of any valid data source. The densities of aqueous propane-1,3-diol are comparable to aqueous propane1,2-diol in magnitude and shape as well. For the purpose of future works and simplifying the data output of this work, the collected data of pure propane-1,3-diol and propane-1,2-diol were used to fit the standard density models. The Tait equation and its derivatives can be used for the pressure dependence; however, for the dependence of density on temperature at constant pressure the empirical polynomial model as a function of temperature is usually applied. The basic density dependence can then be written as ρ = A + BT + CT 2 + DT 3 + ET 4

(2)

where xi is the molar fraction, Mi is the molar weight and ρi is the density of the pure respective substances. The density ρ without the index i then states for the measured density at different molar fractions. The excess molar volumes were fitted to the Redlich−Kister30,31 equation (expansion) described as

(1)

k

where ρ is the density, T is the temperature, and A, B, C, D and E are constants based on experimental data. The exponential model is recommended by the DIPPR29 and describes the dependence of density on temperature by four constants A, B, C, and D as the equation DIPPR-105:

Y E = xixj ∑ Ak (xi − xj)k n=0

(5)

E

where the Y states for the specific excess property, xi and xj represent the respective molar fractions of the components of F


a

G

0.0000 0.0288 0.0635 0.1061 0.1597 0.1920 0.2721 0.3826 0.5449 0.7523 1.0000

0.0000 0.0253 0.0552 0.1049 0.1917 0.3266 0.4933 0.8122 1.0000

0.0000 0.1113 0.2226 0.3340 0.4453 0.5010 0.6123 0.7236 0.8349 0.9277 1.0000

0.0000 0.0988 0.1980 0.3312 0.5005 0.6720 0.8044 0.9481 1.0000

84.936 204.36 415.71 1066.95 1772.35

40.115 49.086 85.871 158.67 267.81 418.56 574.39

253.15 K

21.949 55.335 128.54 256.20 638.89 1038.30

29.024 35.240 63.149 110.64 184.18 293.30 409.88

258.15 K

15.587 37.517 84.222 164.62 396.61 631.08

12.021 19.914 24.970 44.198 76.181 132.29 206.53 289.50

263.15 K

5.7797 11.445 26.373 57.218 109.37 254.76 396.99

5.9079 9.2516 15.119 18.810 32.816 55.718 95.750 149.24 207.88

268.15 K

1.8253 2.8771 4.5523 8.6548 19.118 40.161 75.048 169.13 258.00

1.8253 2.8940 4.6953 7.2567 11.739 14.517 24.879 41.578 70.714 109.69 152.75

273.15 K

283.15 K Propane-1,3-diol 1.3453 2.0548 3.1650 4.7484 7.4508 9.0999 15.098 24.571 40.764 62.320 86.483 Propane-1,2-diol 1.3453 1.9913 3.0171 5.3058 10.826 21.240 38.416 86.694 116.68 1.0407 1.4543 2.1715 3.5359 6.7409 12.577 21.746 45.665 59.749

1.0407 1.5287 2.2507 3.3182 5.0346 6.0744 9.7704 15.480 24.995 37.604 52.007

293.15 K

0.8467 1.1483 1.5988 2.5027 4.5104 8.0096 13.276 26.095 33.263

0.8467 1.1893 1.6894 2.4215 3.5854 4.3059 6.7242 10.323 16.257 24.030 32.921

303.15 K

Standard uncertainties are u(T) = 0.01 K, u(p) = 5 kPa and u(wi) = 0.0002; relative standard uncertainty is ur(η) = 0.05.

xi

wi

0.6997 0.9194 1.2408 1.8584 3.1659 5.3981 8.6379 15.998 19.925

0.6997 0.9825 1.3321 1.8575 2.6594 3.1485 4.7253 7.1844 11.049 16.044 21.791

313.15 K

0.5967 0.7541 0.9933 1.4361 2.3446 3.8524 5.9378 10.396 12.679

0.5967 0.7887 1.0758 1.4647 2.0513 2.4010 3.5225 5.2193 7.8010 11.149 15.000

323.15 K

0.5091 0.6304 0.8167 1.1453 1.8017 2.8519 4.2473 7.0936 8.4977

0.5091 0.6769 0.8845 1.1940 1.6408 1.8845 2.6983 3.8996 5.7131 8.0231 10.720

333.15 K

0.4355 0.5375 0.6869 0.9408 1.4252 2.1875 3.1609 5.0531 5.9497

0.4355 0.5651 0.7245 0.9821 1.3015 1.5117 2.1229 3.0098 4.3029 5.9334 7.8549

343.15 K

0.3911 0.4666 0.5856 0.7888 1.1617 1.7328 2.4329 3.7341 4.3322

0.3911 0.4871 0.6668 0.8548 1.0901 1.2104 1.7310 2.4228 3.3174 4.5146 5.9158

353.15 K

Table 6. Dynamic Viscosity η/(mPa·s) Values of Aqueous Propane-1,3-diol for Different Mass Fractions wi and Molar Fractions xi over the Temperature Range T = (253.15 to 353.15) K at Standard Pressure (p = 101.325 kPa)a




Article

Figure 4. Dynamic viscosity η/(mPa·s) of aqueous propane-1,3-diol (a,b) and aqueous propane-1,2-diol (c,d). (a,c) Dependency on mass fraction wi for (+) T = 253.15 K; (×) 258.15 K; (∗) 263.15 K; (□) 268.15 K; (■) 273.15 K; (○) 283.15 K; (●) 293.15 K; (△) 303.15 K; (▲) 313.15 K; (◇) 323.15 K; (◆) 333.15 K; (▽) 343.15 K; (▼) 353.15 K. (b) Dependency on temperature for (+) wi = 0; (×) wi = 0.111; (∗) wi = 0.223; (□) wi = 0.334; (■) wi = 0.445; (○) wi = 0.501; (●) wi = 0.612; (△) wi = 0.724; (▲) wi = 0.835; (◇) wi = 0.928; (◆) wi = 1.0. (d) Dependency on temperature for (+) wi = 0; (×) wi = 0.099; (∗) wi = 0.198; (□) wi = 0.331; (■) wi = 0.501; (○) wi = 0.672; (●) wi = 0.804; (△) wi = 0.948; (▲) wi = 1.0.

the binary fluid and Ak is used for expansion fitted parameters. The number of fitted parameters was chosen as k = 4, which leads to a total number of 5 fitted parameters. The number of parameters was estimated by the F-test method. The resulting parameters A0 to A4 and the resulting deviations according to equation 3 are listed in Table 5. Fitting of the experimental data with the Redlich−Kister equation curve is shown in Figure 2. The aqueous PDO naturally shows the highest excess molar volumes at lower temperatures with the minimum around xi = 0.4 (wi = 0.7−0.8) following the pattern on propane-1,3-diol as the main driving force for the activity of this binary fluid. The shift in the excess molar volume is influenced by the fact that the practical density is actually higher in the region from pure PDO until the amount of water reaches roughly equimolar quantity, as can be seen in Figure 1. Similar results are shown also by the aqueous propane-1,2-diol. However, the presence of the free methyl group and the distance of the hydroxyl groups have stronger influence on the excess properties, and the aqueous propane-1,2-diol reaches the highest excess molar volume sooner than the propane-1,3-diol, and the values are higher as well. This behavior is caused by the hydroxyl−water interaction and is quite common to other dialcohols, as has

been demonstrated by the works of Nagamachi and Francesconi18 and Takahashi and Nishi.32 The positions of the two hydroxyl groups give the binary dialcohol mixtures specific properties when hydrophilic hydration prevails over hydrophobic hydration. When a small amount of water is introduced to PDO, the water molecules primarily interact with the hydroxyl groups, subsequently influencing the interaction between the PDO molecules itself. This leads to a temporary rise in the mixture’s density in the concentration region of xi = (0.5 to 1.0) (wi = 0.8−1.0). The main difference of the methyl group existence in the case of propane-1,2-diol then shifts the peak position more to a lower concentration area where hydrophobic hydration takes over the hydrophilic hydration. The comparison with the literature data is provided in Figure 3 for temperatures T = (293.15; 303.15; 313.15) K by comparing the excess molar volume values. The data on propane-1,2-diol shows very good fit with the available data of Zemánková et al.;22 however, the data on propane-1,3-diol shows some certain negative shifts from the data of Zemánková et al.22 and more in case of Moosavi and Rostami.11 The actual deviation of density values from this work for both propaneH



Article

Table 7. Redlich-Kister Equation Coefficients for Dynamic Viscosity Deviation Δη/(mPa·s) for a Temperature Range of T = (273.15 to 353.15) K. of Aqueous Propane-1,3-diol, Aqueous Propane-1,2-diol and the Respective Standard Deviations to Experimental Data T/K

A0

A1

273.15 283.15 293.15 303.15 313.15 323.15 333.15 343.15 353.15

−59.5404 −30.9695 −16.8351 −9.1614 −5.0749 −2.8345 −1.5987 −0.7713 −0.2484

−67.0263 −34.6122 −18.4173 −10.4895 −6.5922 −3.8867 −2.3796 −1.3829 −0.7505

273.15 283.15 293.15 303.15 313.15 323.15 333.15 343.15 353.15

−213.0586 −79.1466 −32.9136 −14.1690 −6.1264 −2.4393 −0.7874 0.0349 0.4010

29.8891 −13.5494 −9.5030 −6.5334 −4.5094 −2.8738 −1.8120 −1.0798 −0.7693

A2

A4

σ

10.9521 6.1082 3.4410 1.5100 1.6265 0.7721 0.6638 0.2328 0.2269

17.4700 7.5684 3.0276 0.3515 1.4938 0.0739 0.1173 −0.0317 0.6008

0.135 0.079 0.051 0.032 0.018 0.016 0.012 0.009 0.027

48.8258 −21.4633 −8.5416 −2.8316 −0.5014 −0.0713 0.1286 −0.0193 0.2636

−60.5436 −19.6973 −11.4576 −8.9208 −5.8837 −4.2281 −3.1661 −1.7400 −1.9865

0.065 0.033 0.019 0.010 0.005 0.003 0.002 0.002 0.002

A3

Propane-1,3-diol −18.4234 −11.3212 −7.4799 −4.4198 −4.0212 −2.4569 −1.8282 −1.5010 −1.6743 Propane-1,2-diol −36.9031 35.6356 16.9040 9.4321 5.1496 3.0307 2.0494 1.1845 1.0664

Figure 6. Comparison of experimental dynamic viscosity deviations Δη/(mPa·s) as a function of molar fraction xi of proapane-1,3-diol and propane-1,2-diol in water with their respective calculated Redlich−Kister curves (full lines correspond to propane-1,3-diol (this work); dashed lines correspond to propane-1,2-diol (this work)) with selected literature data. The respective symbols correspond to (+) propane-1,3-diol (this work); (×) propane-1,2-diol (this work); (■) propane-1,3-diol (Moosavi and Rostami11); (●) propane-1,2diol (Yang et al.34); (◆) propane-1,2-diol (Kapadi et al.35).

1,3-diol and propane-1,2-diol is then within σ = 0.004 of the literature values. 3.2. Viscosity. The experimental data on the viscosity of propane-1,3-diol and propane-1,2-diol are summarized in Table 6 with the full graphical representation in Figure 4. To easily compare the viscosity deviation of an actual mixture against an ideal mixture and establish the dependence of the viscosity, the viscosity deviation is defined as follows:

Figure 5. Dynamic viscosity deviation Δη/(mPa·s) for aqueous propane-1,3-diol (a), aqueous propane-1,2-diol (b), and the respective Redlich−Kister eq 5 curves. The progress of the Redlich−Kister equation is shown as a continuous line belonging to the respective symbols for experimental data at (+) 273.15 K; (×) 283.15 K; (∗) 293.15 K; (□) 303.15 K; (■) 313.15 K; (○) 323.15 K; (●) 333.15 K; (△) 343.15 K; (▲) 353.15 K.

Δη = η12 − (x1η1 + x 2η2) I

(6) DOI: 10.1021/acs.jced.8b00403 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

J

y0 y1 y2 R2

273.15 283.15 293.15 303.15 313.15 323.15 333.15 343.15 353.15 σ̅ (%)

1.84 1.54 1.17 0.76 1.39 0.80 1.42 1.20 2.88 1.44

−0.4251 −0.5646 −0.6700 −0.7567 −0.7954 −0.8316 −0.8054 −0.7840 −0.7431

13.72209 −0.08895 1.360 × 10−04 0.9961

σ (%)

g12

33.72556 −0.15677 1.990 × 10−04 0.9997 Grunberg - Nissan

y0 y1 y2 R2

24.56 22.08 19.93 17.67 16.24 14.62 14.12 13.31 13.02 17.28

σ (%)

5.7412 5.2956 4.8769 4.4836 4.1433 3.8227 3.5780 3.3869 3.1631

g12

273.15 283.15 293.15 303.15 313.15 323.15 333.15 343.15 353.15 σ̅ (%)

T/K

Grunberg - Nissan

12.58199 −0.08256 1.270 × 10−04 0.9898

−0.4881 −0.6183 −0.7060 −0.7732 −0.8343 −0.8378 −0.8138 −0.7777 −0.7521

g0

29.91495 −0.13801 1.733 × 10−04 0.9996

5.1429 4.7327 4.3534 4.0147 3.6781 3.4061 3.1541 2.9983 2.7738

g0 2.7489 2.3554 2.0112 1.6174 1.5568 1.2723 1.3227 1.1207 1.1324

g2

−2.11475 0.01568 −2.930 × 10−05 0.7923

−0.0104 −0.0326 −0.0477 0.0011 −0.1307 −0.0748 −0.1829 −0.1313 −0.2493

g1

9.35669 −0.05301 7.534 × 10−05 0.8322

0.4900 0.4154 0.2767 0.1259 0.3007 0.0477 0.0699 −0.0451 0.0854

g2

−27.09493 35.37534 0.14021 −0.19657 −1.936 × 10−04 2.822 × 10−04 0.9989 0.9880 Grunberg - Nissan 2

−3.2391 −2.9191 −2.6547 −2.3947 −2.1592 −1.9865 −1.8852 −1.8137 −1.7161

g1

Grunberg - Nissan 2 xi s

wi s

1.22 1.1007 0.99 1.1453 0.86 1.1870 0.73 1.2295 0.78 1.2611 0.70 1.2948 0.81 1.3045 0.90 1.3129 2.65 1.3178 1.07 Regression Constants −2.94424 0.02408 −3.400 × 10−05 0.9972

σ (%)

3.47 0.2606 2.92 0.2711 2.31 0.2810 1.46 0.2911 2.13 0.2986 1.24 0.3065 1.58 0.3088 1.20 0.3108 3.10 0.3120 2.16 Regression Constants −0.69704 0.00570 −8.048 × 10−06 0.9972 Lederer

σ (%)

Lederer

1.85 1.55 1.15 0.94 1.18 0.69 0.91 0.82 2.57 1.30

σ (%)

1.85 1.55 1.15 0.94 1.18 0.69 0.91 0.82 2.57 1.30

σ (%)

νf1

32.95214 −0.16568 2.093 × 10−04 0.9996

27.9917 16.5022 10.4225 7.1236 4.8378 3.6158 2.6887 2.1391 1.6794

νf1

28.52967 −0.13222 1.549 × 10−04 0.9998

54.1000 33.1676 21.2928 14.5843 10.1600 7.3901 5.3512 4.0742 3.1406

28.54827 −0.15165 2.009 × 10−04 0.9991

8.4396 5.4981 3.8154 2.7429 2.1772 1.6901 1.4293 1.1569 1.0399

νf 2

46.27118 −0.23177 3.001 × 10−04 0.9998 McAllister

214.9199 109.4002 60.6928 35.6943 22.6181 15.0541 10.9197 8.0701 6.2021

νf 2

McAllister

2.58 2.27 1.83 1.36 1.88 1.09 1.26 1.08 2.63 1.77

σ (%)

8.22 7.14 6.15 4.96 5.04 4.06 4.29 3.67 4.57 5.35

σ (%)

α0

14.34432 −0.08722 1.337 × 10−04 0.9959

0.5027 0.3693 0.2692 0.1861 0.1497 0.1166 0.1465 0.1688 0.2136

α0

34.29499 −0.15721 2.010 × 10−04 0.9998

6.3471 5.9024 5.4909 5.1098 4.7737 4.4692 4.2289 4.0489 3.8312

α1

−2.51413 0.01521 −2.846 × 10−05 0.8011

−0.4779 −0.4986 −0.5162 −0.4682 −0.5925 −0.5431 −0.6482 −0.5970 −0.7121

α1

−35.73222 0.18557 −2.588 × 10−04 0.9988 Heric

−4.3455 −3.9374 −3.5948 −3.2424 −2.9905 −2.7511 −2.6659 −2.5441 −2.4518

Heric

2.58 2.27 1.83 1.36 1.88 1.09 1.26 1.08 2.63 1.77

σ (%)

8.22 7.14 6.15 4.96 5.04 4.06 4.29 3.67 4.57 5.35

σ (%)

Table 8. Fitting Parameters and Standardized Model Percentage Deviations for Grunberg−Nissan (g12, gi), Lederer (s), McAllister (νf1, νf 2), and Heric’s (αi) Equation for Temperature Range T = (273.15 to 353.15) K at Standard Pressure (p = 101.325 kPa) for both Molar xi and Mass wi Fraction of Propane-1,3-diol




Article

of excess viscosity for aqueous propane-1,2-diol reach below −50 mPa·s, whereas the minimum values of aqueous propane1,3-diol do not exceed −20 mPa·s at T = 273.15 K. The immediate values of dynamic viscosity Δη at low temperatures are then significantly lower in the case of aqueous propane-1,3diol which is advantageous specially in low-temperature applications such as freezing depression in coolant mixtures. The viscosity deviation and excess molar volume determination against the ideal mixture and subsequent Redlich− Kister approximation is possible only for the temperatures where the pure substances stay in their liquid form. Therefore, regardless of the experimental data of the mixtures are expanding to T = 253.15 K, the calculation is limited by the freezing point of water at T = 273.15 K. If we compare the viscosity deviations with literature data some significant deviations for propane-1,3-diol can be observed in the region of 0.2 < xi < 0.6 where the deviation from the very scarce literature data reaches σ = 0.05. Data on viscosity of aqueous propane-1,2-diol are in better agreement with literature. That, however, is also given by better availability of the data and wider comparability. For illustration, the literature data comparison of the equivalent available viscosity deviations Δη/(mPa·s) is provided in Figure 6. High deviations in the region 0.2 < xi < 0.6 can be then observed between the literature sources for both propane-1,3-diol and propane-1,2diol in general. 3.3. Viscosity Modeling of Aqueous Propane-1,3-diol. The natural dependency of aqueous propane-1,3-diol is nonlinear on temperature or molar and mass fraction as a typical property of aqueous mixtures with alcohols and diols.33 To describe these dependencies, many models have been suggested by various authors based on the theoretical background or empirical findings. Many of these models are based on the ideal mixture calculation originating from the Arrhenius equation:

Figure 7. Comparison of experimental data on dynamic viscosity η/ (mPa·s) of pure propane-1,3-diol with literature and models. Model 1 plotted according to eq 14, Model 2 according to eq 15. References: [1] Moosavi and Rostami;11 [2] George and Sastry;17 [3] Maximino.33

In eq 6 the viscosity deviation Δη is defined as the difference between the experimental value of dynamic viscosity η12 and the linear behavior of the ideal mixture, where η1 and η2 are the viscosities of the pure substances and x1 and x2 are their respective molar fractions. The viscosity deviation can then be fitted, similarly to the density, by the Redlich−Kister eq 5. The coefficients are shown in Table 7 for the temperature region of T = (273.15 to 353.15) K and the comparison against the experimental data is shown in Figure 5. The viscosity deviation minimum is shifted even more toward the lower concentrations of propane-1,3-diol than in the case of the density and equally determines the leading factor for the viscosity dependence of propane-1,3-diol. On the contrary to the propane-1,3-diol results, the viscosity deviations of aqueous propane-1,2-diol show much stronger influence of propane1,2-diol on the viscosity in the water mixture. The actual values

ln η12 = x1 ln η1 + x 2 ln η2

(7)

where η12 is the dynamic viscosity of the mixture and x1, x2 and η1, η2 are the molar fractions and viscosities of the pure

Table 9. Absolute Average Deviation (AAD(%)) of the Combined Viscosity Models and Experimental Data of Aqueous Propane-1,3-diol. Deviations Are Averaged for the Whole Fraction Range for Each Model and Temperature

T/K 253.15 258.15 263.15 268.15 273.15 283.15 293.15 303.15 313.15 323.15 333.15 343.15 353.15 AADAverage AADMax

Grunberg - Nissan 2

Lederer

McAllister

Heric

AAD(%)

AAD(%)

AAD(%)

AAD(%)

xi

wi

xi

wi

xi

wi

xi

wi

7.31 5.87 3.17 3.13 2.49 1.89 1.59 0.98 1.42 1.29 1.42 0.91 1.85

7.04 6.03 2.58 1.86 0.99 0.77 0.97 0.68 0.66 0.77 0.96 0.93 1.80

3.84 3.90 1.93 1.98 1.48 1.13 0.82 0.64 0.75 0.73 0.95 0.80 1.86

3.84 3.90 0.16 1.98 1.48 1.13 0.82 0.64 0.75 0.73 0.95 0.80 1.86

12.05 9.52 6.89 6.87 6.00 5.32 4.66 3.73 3.79 3.36 3.39 2.48 3.44

12.73 9.95 5.08 3.54 2.15 1.67 1.44 1.03 1.24 1.31 1.53 0.95 1.78

7.08 6.69 6.48 7.03 7.45 5.24 4.51 3.64 3.78 3.37 3.41 2.53 3.32

5.80 5.11 2.42 2.33 1.83 1.61 1.27 1.07 1.24 1.08 1.20 0.98 1.75

1.54 15.48

0.95 16.02

1.02 9.34

1.02 9.34

4.02 15.48

1.45 20.58

4.14 17.61

1.34 10.43

K



Article

Figure 8. A comparison of dynamic viscosity η/(mPa·s) and kinematic viscosity ν/(mm2·s−1) of the combined models with experimental data (□) of aqueous propane-1,3-diol.

specified binary mixture is proportional to the interchange energy and therefore to the temperature. The resulting equation is then written as ln η12 = x1 ln η1 + x 2 ln η2 + g12x1x 2

(8)

Equation 8 can also be written in a more empirical form (GN2) further enhancing the precision: n

ln η12 = x1 ln η1 + x 2η2 +

∑ x1x2gi(x1 − x2)i i=0

(9)

where interaction parameter g12 from eq 8 (G-N1) becomes a part of the geometric sequence. By introducing more fitting parameters to the equation, the precision is increased, however, the calculation method becomes more complex and the dependence on temperature is reduced; consequently the modeling of viscosity for different temperatures will become unreliable. For that purpose, the maximum number of fitting parameters was set to n = 2, where the dependence on temperature is still sustainable for all the parameters. Another form of eq 7 is presented by Lederer’s38,39 modification:

Figure 9. Lederer’s combined model of dynamic viscosity η/(mPa·s) of aqueous propane-1,3-diol on mass fraction wi in comparison with available literature sources. References: [1] Moosavi and Rostami;11 [2] George and Sastry; [3] Maximino.33

ji x1 zyz ji x 2s zyz ln η12 = jjj ln η1 + jjj ln η2 j x + x s zzz j x + x s zzz 2 { 2 { k 1 k 1

substances, respectively. To achieve good agreement with our experimental data, we tested several different modifications which had been previously tested on different alcohols and alkanediols. A basic and simple modification with a significant improvement in precision is provided by the Grunberg−Nissan36,37 equation, which introduces the fitting interaction parameter g12 to the original eq 7. The interaction fitting parameter g12 of the

(10)

Lederer’s eq 10 emphasizes the significance of mixing two components with a substantial difference between their viscosities in the pure state, where the molar fraction x2 in eq 10 represents the more viscous propane-1,3-diol. The L



Article

Table 10. Refractive Index (nD) Values of Aqueous Propane-1,3-diol and Aqueous Propane-1,2-diol for Different Mass Fractions wi or Molar Fractions xi over a Temperature Range from 278.15 to 318.15 K at Standard Pressure (p = 101.325 kPa)a wi

xi

278.15 K

283.15 K

288.15 K

0.0000 0.1113 0.2226 0.3340 0.4453 0.5010 0.6123 0.7236 0.8349 0.9277 1.0000

0.0000 0.0288 0.0635 0.1061 0.1597 0.1920 0.2721 0.3826 0.5449 0.7523 1.0000

1.3341 1.3466 1.3599 1.3740 1.3866 1.3931 1.4066 1.4181 1.4289 1.4371 1.4436

1.3338 1.3462 1.3592 1.3729 1.3856 1.3921 1.4055 1.4169 1.4276 1.4358 1.4423

1.3335 1.3457 1.3586 1.3719 1.3845 1.3910 1.4041 1.4157 1.4263 1.4345 1.4409

0.0000 0.0988 0.1980 0.3312 0.5005 0.6720 0.8044 0.9481 1.0000

0.0000 0.0253 0.0552 0.1049 0.1917 0.3266 0.4933 0.8122 1.0000

1.3341 1.3458 1.3578 1.3740 1.3946 1.4112 1.4231 1.4343 1.4380

1.3338 1.3454 1.3571 1.3730 1.3932 1.4098 1.4216 1.4328 1.4364

1.3335 1.3449 1.3564 1.3720 1.3918 1.4084 1.4200 1.4312 1.4348

293.15 K Propane-1,3-diol 1.3331 1.3452 1.3579 1.3710 1.3835 1.3899 1.4029 1.4145 1.4250 1.4332 1.4395 Propane-1,2-diol 1.3331 1.3444 1.3557 1.3710 1.3903 1.4069 1.4184 1.4296 1.4332

298.15 K

303.15 K

308.15 K

313.15 K

318.15 K

1.3326 1.3446 1.3572 1.3702 1.3825 1.3888 1.4015 1.4133 1.4237 1.4319 1.4381

1.3320 1.3440 1.3564 1.3693 1.3815 1.3876 1.4002 1.4118 1.4224 1.4304 1.4368

1.3313 1.3434 1.3558 1.3684 1.3804 1.3865 1.3989 1.4103 1.4211 1.4289 1.4354

1.3306 1.3426 1.3550 1.3676 1.3793 1.3854 1.3977 1.4089 1.4197 1.4275 1.4340

1.3298 1.3418 1.3541 1.3668 1.3782 1.3845 1.3965 1.4075 1.4182 1.4261 1.4325

1.3326 1.3432 1.3550 1.3699 1.3889 1.4054 1.4168 1.4280 1.4315

1.3320 1.3432 1.3542 1.3689 1.3874 1.4039 1.4150 1.4264 1.4299

1.3313 1.3424 1.3534 1.3680 1.3862 1.4024 1.4135 1.4248 1.4281

1.3306 1.3416 1.3534 1.3671 1.3851 1.4008 1.4119 1.4231 1.4264

1.3298 1.3408 1.3516 1.3660 1.3840 1.3994 1.4103 1.4214 1.4247

a

Standard uncertainties are u(T) = 0.05 K, u(p) = 5 kPa, u(wi) = 0.0002, and u(nD) = 0.0001.

interaction between the components is then adjusted by a single empirical constant s dependent on temperature. A similar approach to the Grunberg−Nissan was presented in the work of Heric and Brewer40 which expands the original Arrhenius equation following the Eyring theory of ideal mixture with zero excess of activation Gibbs free energy ΔG*E = 0 and adds the molar weight constants of the mixed liquids M1 and M2 with the resulting equation of kinematic viscosity ν12 as ij n yz + jjjj∑ αix1x 2(x1 − x 2)i zzzz j i=0 z k {

All the parameters and variables of McAllister’s eq 12 are identical to Heric’s eq 11 except for the fitting parameters νf1 and νf 2 which can be obtained by calculating the activation entropy and enthalpy energies of the mixing liquids, as was proposed in McAllister’s original paper,41 or as in this case as an empirical fitting parameter. The respective fitting parameters of the presented models are summarized in Table 8. The percentage models’ deviations were calculated according to eq 13, where ηexp is the experimentally measured viscosity (dynamic or kinetic based on model) and ηcal is the viscosity calculated by the model. To compensate for the number of fitting parameters by each model, the number of points in series n is deduced by the number of fitting parameters k in the model.

ln ν12 = x1 ln(ν1M1) + x 2 ln(ν2M 2) − ln(x1M1 + x 2M 2)

(11)

where xi, νi, and Mi represent the molar fraction, kinematic viscosity, and molar weight of the respective components. The number of power series parameter αi can vary for different mixtures, nevertheless, the most common that appears in literature is n = 1, and this form was kept through this work as well. Likewise, McAllister’s41 model is also based on Eyring’s theory of absolute rates and describes the binary mixtures by three-body interaction involving two different molecules. The equation describing McAllister’s model is designed for kinematic viscosity ν12 and is written in its condensed form as

σ(%) =

1 n−k

ij 100(η − η ) yz exp cal z zz ∑ jjjjj zz ηexp j z k {

2

(13)

For fitting the models, the nonlinear optimization method in the form of a generalized reduced gradient algorithm was used. To ensure correct convergence and to find the minimal optimum, several different calculations were made with different starting values. Because all the considered models are based on a calculation from the viscosity of the pure components, the final temperature range was limited similarly to excess properties to T = (273.15 to 353.15). Although, the models were devised for use with molar fractions, it is generally possible to use them with mass fractions as well, based on their empirical validity. When fitting the parameters of aqueous PDO according to mass fraction dependency, a better fit is usually achieved through the whole data set. The Grunberg-Nissan equation could serve as a good example. The cumulative deviation σ̅ of viscosity dependency on molar fraction for G-N1 is σ̅ = 17.28%

ln ν12 = x13 ln ν1 + 3x12x 2 ln νf 1 + 3x1x 22 ln νf 2 + x 23 ln ν2 − ln(x1 + x 2M 2 /M1) + 3x12x 2 ln[(2 + M 2 /M1)/3] + 3x1x 22 ln[(1 + 2M 2 /M1)/3] + x12 ln(M 2 /M1), (12) M



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Figure 10. Refractive indices (nD) of aqueous propane-1,3-diol (a,b,c) and aqueous propane-1,2-diol (d,e,f). (a) Dependency on temperature for (+) wi = 0.0; (×) wi = 0.111; (∗) wi = 0.223; (□) wi = 0.334; (■) wi = 0.445; (○) wi = 0.501; (●) wi = 0.612; (△) wi = 0.724; (▲) wi = 0.835; (◇) wi = 0.928; (◆) wi = 1.0. (d) Dependency on temperature for (+) wi = 0.0; (×) wi = 0.099; (∗) wi = 0.198; (□) wi = 0.331; (■) wi = 0.501; (○) wi = 0.672; (●) wi = 0.804; (△) wi = 0.948; (▲) wi = 1.0. Dependency on (b,e) mass and (c,f) molar fractions for (+) T = 278.15 K; (×) T = 283.15 K; (∗) T = 288.15 K; (□) T = 293.15 K; (■) T = 298.15 K; (○) T = 303.15 K; (●) T = 308.15 K; (△) T = 313.15 K; (▲) T = 318.15 K.

while for the mass fraction it is σ̅ = 1.44%, which is one order higher than the deviation in the molar fraction. Similar practice can be observed with other models, too. The only exception is Lederer’s model, in which the fitting parameter itself serves as an adjusting parameter between mass and molar fraction, and the deviations for molar fraction and mass fraction are therefore identical. An interesting comparison is also offered by McAllister’s and Heric’s models. As has been proven elsewhere42 these two models are mathematically equivalent for binary mixtures, therefore, McAllister’s and Heric’s models should produce identical deviations after fitting, which has been achieved in this work. A common disadvantage of these models is the need for a great amount of experimental data because the pure mathematical concepts for polar fluids usually lack the necessary accuracy and are limited by the measurable temperature range of the mixed liquids in their pure state. However, because of the direct dependency of the fitting parameters on temperature, by acquiring a sufficiently big data sample some of these models can be extended over the whole molar or mass fraction region. Such an attempt was done in this work for all stated models with varying success. All of the model fitting parameters’ dependencies on temperature were expressed by a standard second order polynomial equation in the form of y = y0 + y1T + y2T2. The polynomial regression was

applied on the basis of least-squares, and the resulting parameters and coefficients of determination are stated under each of the data set columns of the model fitting parameters in Table 8. Here, contrary to the previous statement, it is shown that the molar fraction dependent parameters produce a better regression fitting overall than those of the mass fraction, again with Lederer’s model as a known exception. To further expand the application of the acquired measurements, we attempted to fit the data of the pure PDO and water to the viscosity model dependent on the temperature. As the basic model, the Arrhenius equation was used in the form: η = η0 e Ea /(RT )

(14)

where η is viscosity, η0 is the preexponential (entropic) factor and Ea, R, and T are the Arrhenius activation energy, universal gas constant and absolute temperature, respectively. Unfortunately, this equation does not offer the necessary accuracy for polar liquids43 and therefore an empirical substitution was used instead: ij yz A zz η = η0 expjjj 2 j t + t T + t T zz 1 2 ko {

N

(15) DOI: 10.1021/acs.jced.8b00403 J. Chem. Eng. Data XXXX, XXX, XXX−XXX


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Figure 11. Molar refraction Rm/(mol−1) (a,c) and polarizability α/(cm3·mol−1) (b,d) of aqueous propane-1,3-diol (a,b) and aqueous propane-1,2diol (c,d) on temperature T/(K) and molar fraction xi.

Table 11. Linear Dependence of Molar Refraction Rm/ (mol−1) of Aqueous Propane-1,3-diol and Aqueous Propane-1,2-diol on Molar Fraction xi at Temperatures T = 283.15 K, T = 293.15 K, T = 303.15 K, and T = 313.15 K 283.15 K a b

15.2789 3.7217

a b

15.3688 3.7130

293.15 K

303.15 K

Propane-1,3-diol 15.2927 15.3040 3.7186 3.7161 Propane-1,2-diol 15.3845 15.3957 7.108 3.7097

propane-1,3-diol. A comparison of the pure propane-1,3-diol model with experimental data, and very scarce literature data, is shown in Figure 7. Equation 15 is theoretically not dependent on the fusion temperature and can project the viscosity far beyond its practical use. Although it is not possible or difficult to experimentally prove the validity below freezing point, it is still possible to use eq 16 in combination with one of the standard viscosity models and test the validity below the temperature of fusion indirectly. For that purpose, we chose Grunberg−Nissan’s second model with three fitting parameters, Lederer’s model and McAllister’s and Heric’s models by creating a combined form of eq 16 with the respective models. As an example, the combined equation of eqs 9 and 16 will be written as

313.15 K 15.3087 3.7171 15.3990 3.7140

ij yz A1 zz ln η12 = x1jjjln η10 + 2z j z t + t T + t T 10 11 12 k { ij yz A2 zz + x 2jjjln η20 + z 2 j t 20 + t 21T + t 22T z{ k

where η is viscosity, η0 and A are fitting parameters based on the Arrhenius equation, and ti are fitted parameters of the temperature geometric sequence. Equation 15 in logarithmic arrangement will be written as A ln η = ln η0 + t0 + t1T + t 2T 2 (16) In this form, eqs 15 and 16 preserve the original shape of the Arrhenius eq 14 while sufficiently compensating for the excess properties of the mixture at the cost of adding three more parameters to the function. The resulting cumulative percentage deviation according to eq 13 is less than σ (%) < 1% for propane-1,3-diol and σ (%) < 1.5% for water. The Arrhenius activation energy can be easily deduced as A = Ea/R while the preexponential factor is identical to the original eq 14. The Arrhenius activation energy and preexponential factor was found to be Ea = 15.252 kJ·mol−1 and ln(η0) = −13.091 for water and Ea = 33.865 kJ·mol−1 and ln(η0) = −16.777 for

+ x1x 2((g00 + g01T + g02T 2) + (g10 + g11T + g12T 2)(x1 − x 2) + (g20 + g21T + g22T 2)(x1 − x 2)2 )

(17)

where the first low indexes state the relevance to the substance in the case of eq 16 and each parameter’s polynomial representation is as shown in Table 8. In the case of McAllister’s and Heric’s models, which are used for kinematic viscosity, eq 9 was applied only as an empirical model. The O



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describes the relative permittivity εr (dielectric constant) of a material or a mixture as

resulting absolute average deviations of the experimental data and the combined models are listed in Table 9 as calculated according to eq 18. A visual comparison of the combined models with the experimental data is provided in Figure 8. AAD(%) = 100

εr − 1 Nα′ = 3ε0 εr + 2

|ηexp − ηcal | ηexp

(19)

where N is the number of particles per volume, ε0 is vacuum permittivity, and α′ is polarizability volume. Polarizability volume α′ can be expressed generally in SI units giving simple polarizability α = 4πε0α′, expressed by its relation to the permittivity of a vacuum. Furthermore, in the case of a mixture, the right side of the equation 19 is expressed as the sum of the calculated individual bodies. The polarizability in this expresses the ease of polarization of the substance by light or electric field. It is often used as an optical property in a secondary analytical method for defining the mixture ratio as the refractive index. A direct link between the relative permittivity and refractive index nD is then brought by the Lorenz−Lorenz equation by the Maxwell relation:

(18)

On average, the combined models provide a very good fit with the experimental data and do not exceed 5% of the absolute average deviation, AAD(%). Especially Grunberg− Nissan’s second model and Lederer’s equation show an exceptionally good fit around AAD(%) = 1%. However, the problematic of this fitting does not lie in the overall fit but rather in the local maxima. Table 9 also shows the maxima AAD(%) for the all data sets, which usually reach 10% or more. Such a model cannot be used in practical applications as the results could exceed the safety margins. The AAD(%) generally increases with decreasing temperature and is at the highest around the fusion temperatures of the mixtures. Near the fusion temperatures the properties of the mixture change rapidly even with a small change in temperature and can strongly influence the measurements. For comparison with the literature data, we chose our Lederer’s combined model. A comparison with literature sources is provided in Figure 9. Because the experimental data of this work in the compared region are below AAD(%) = 1% of the selected model, we attempted to compare the results directly. The work of Moosavi and Rostami11 reported data only for three different close temperatures T = 293.15, 298.15, and 303.15 K and the resulting average AAD(%) values with our combined model were calculated as AAD(%) = 7.76%, 7.30%, and 5.55%, respectively. More profound data provided by George and Sastry17 show a greater divergence from our combined models, and therefore a greater divergence from our experimental findings as well, reaching an average of AAD(%) = 7.76% at T = 298.15 K, AAD(%) = 5.16% at T = 308.15 K, AAD(%) = 6.59% at T = 318.15 K, AAD(%) = 13.89% at T = 328.15 K, and AAD(%) = 5.16% at T = 338.15 K for Lederer’s combined model. In the work of Maximino33 viscosity η/ (mPa·s) was established only for one temperature T = 298.15 K with AAD(%) = 4.44% compared to our combined Lederer’s model. A comparison of data from these three works with our other combined models from this work showed a similar variance as our experimental data toward these models with relative differences approximated from the original values. As was stated earlier, the literature offers only a limited number of relevant sources dealing with aqueous propane-1,3-diol, and hence the comparison is lacking especially because of the disparity of the data of the biggest data set from the work of George and Sastry.17 The difference was most likely caused by the selection of a different method, when in the work of George and Sastry 17 a standard capillary Ubbelohde viscometer was used. The other two works of Moosavi and Rostami11 and Maximino33 are in better agreement; however, they offer only a fractional comparable data set. 3.4. Refractive Indices. The refractive indices as a function of temperature and mass and molar fraction for the whole set of dilutions are summarized in Table 10 and in Figure 10. A basic link between the optical and volumetric properties is offered by the Clausius−Mossotti equation, which

nD2 − 1 nD2 + 2

=

4π Nα 3

(20)

where nD is the refractive index expressed as = εr, N is again the number of particles in the volume, and α represents the polarizability. The molar refraction Rm/(mol−1) is then expressed as n2D

Rm =

nD2 − 1 M 4π NAα = 2 3 nD + 2 ρ

(21)

where M is the average molar mass, ρ is density. and NA is the Avogadro constant. The molar refraction Rm of water, propane-1,3-diol, and propane-1,2-diol calculated from eq 21 at T = 293.15 K was Rm = 3.7092 mol−1, Rm = 19.0154 mol−1, and Rm = 19.1046 mol−1, respectively. The polarizability of water at T = 293.15 K was then calculated as α = 1.47 × 10−24 cm3·mol−1, which is in good agreement with the literature.44 The polarizability of propane-1,3-diol at T = 293.15 K is higher than that of water and was calculated as α = 7.54 × 10−24 cm3·mol−1 and together with the molar refraction value corresponds to the values of Moosavi and Rostami.11 The value of polarizability of propane1,2-diol is very similar and equals to α = 7.57 × 10−24 cm3· mol−1. The practical meaning of this difference is in higher dipole moment of propane-1,2-diol and lower permittivity. The main cause for the higher values of molar refraction compared to that of water are the alkyl chains in the dialcohols and its molecular size expressed as molar volume M/ρ. The relation of chemical bonds to the refractive index and therefore to polarizability is well-known and regularly used to predict the optical behavior of various chemicals.45 Because the values of molar refraction and polarizability are higher for propane-1,3diol and propane-1,2-diol, the factors rise by adding the diols to the mixture, as can be seen in Figure 11. The actual dependency of the molar refraction Rm on molar fraction xi of propane-1,3-diol and propane-1,2-diol is strictly linear and can be expressed as Rm = axi + b. The individual factors a and b are listed in Table 11. The optical properties of propane-1,3-diol and propane-1,2-diol are typical for alkanediols and do not deviate from other alkanediols.11,17 Though small in scale, the effect of temperature on the change in the trend deserves a note. The molar refraction Rm and polarizability α for water drop with rising temperature; P



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however, the values for propane-1,3-diol and propane-1,2-diol rise in the measured temperature range. This difference is caused by the different temperature dependency on the intramolecular forces such as hydrogen bonds, when the rapid change in the density in propane-1,3-diol causes positive swing over the observed temperature range. For example, the difference in polarizability between T = 283.15 K and T = 313.15 K is Δα = 0.1 × 10−24 cm3·mol−1 for propane-1,3-diol and Δα = −0.02 × 10−24 cm3·mol−1 for water, which also shows much greater dependency of activity on the temperature of propane-1,3-diol. Both dependences follow a distinguishable trend.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jced.8b00403.

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REFERENCES

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4. CONCLUSION The properties of aqueous propane-1,3-diol were measured for the whole dilution range for density, viscosity and refractive index. The temperatures were selected over a wide range of T = (253.15 to 353.15) K for density and viscosity and T = (278.15 to 318.15) K for refractive index. The density values of aqueous propane-1,3-diol show a distinguishable maximum in the last part of the concentration row, which is a typical property of higher α,ω-alkanediols. Because this attribute is not as strong for propane-1,3-diol as for higher diols, it is of limited practical use. The viscosity of propane-1,3-diol increases with the temperature decrease with increasing speed. Several viscosity models were tried to describe the dependency of viscosity on temperature and dilution based on mass fraction or molar fraction. The molar fraction-based modeling offers a “cleaner” solution and better dependency correlation of the models’ parameters on temperature, as they were optimized for that. However, the mass fraction dependency offers overall better conformity with the experimental data. One exception is Lederer’s model, which shows the same properties for both the mass fraction and molar fraction-based calculation. By combining the viscosity models, we achieved a single formula for calculating viscosity based solely on dilution and temperature variables with an average deviation as low as 1.02% with local maxima less than 10%, which were found only over a limited range. Polarizability and molar refraction were calculated from the refractive indices of aqueous propane-1,3-diol and show greater activity of propane-1,3-diol based on temperature than water. Fitting the experimental refractive index data as a function of mass or molar fraction can serve as a secondary verification of the fraction amount, which is common practice.

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Information on the purity of the propane-1,3-diol used (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

František Mikšík: 0000-0001-6337-2328 Notes

The authors declare no competing financial interest. Q



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