Calculation of the Density and Activity of Water in ATPSSystems for Separation of Biomolecules

Masood Valavi • Saeed Shirazian • Ali Forouzesh Pour •

Mani Ziary

Received: 22 December 2012 / Accepted: 15 January 2013 / Published online: 3 August 2013� Springer Science+Business Media New York 2013

Abstract In this study, the perturbed hard sphere chain equation of state is utilized to

calculate the activity of water in binary and ternary solutions of polyethylene glycol (PEG),

salt and water. The liquid density of the binary and ternary solutions is also predicted. To

estimate the water activity in PEG–water binary systems, a linear correlation is obtained

for the binary interaction parameter between water and PEG. Then, using this correlation

and without introducing any additional binary parameters, the water activities are predicted

in ternary solutions of water, salt and PEG with different molecular weights (MW). Our

results show that the mean absolute average relative deviation (AARD %) of water activity

for binary PEG–water solutions in 298 K is 0.73 %. In addition, the water activity in

ternary solutions of water and two PEGs with different MW is predicted within 0.31 %

AARD %. Furthermore, the AARD % for prediction of water activities in binary PEG–

water solutions over the temperature range 308–338 K is 0.41 %. Also, the water activities

of aqueous two-phase systems are predicted with AARD % = 0.64 %. In this regard, no

adjustable parameters were correlated between salt and PEG. Finally, liquid densities were

predicted in binary solutions of water–PEG and ternary solutions of water–PEG–salt.

M. Valavi (&)Thermodynamics Research Laboratory, School of Chemical Engineering, Iran University of Scienceand Technology, Tehran 16846-13114, Irane-mail: masudvalavi@yahoo.com

S. ShirazianDepartment of Chemical Engineering, Iran University of Science and Technology, Tehran, Iran

A. F. PourDepartment of Chemical and Oil Engineering, Sharif University of Technology, Tehran, Iran

M. ZiaryDepartment of Chemical Engineering, Amirkabir University of Technology, Tehran, Iran

123

J Solution Chem (2013) 42:1423–1437DOI 10.1007/s10953-013-0040-8

Keywords Thermodynamic modeling � Polymer � Separation � Water activity � Density

List of Symbolskij Binary interaction parameter

R Number of segments

T Temperature (K)

xi Mole fraction of component i

Z Compressibility factor

AbbreviationsEoS Equation of state

OF Objective function

PHSC Perturbed hard sphere chain

SAFT Statistical associating fluid theory

SuperscriptsAssoc Association term

Cal Calculated

Expt Experimental

Ref Reference

Pert Perturbation term

Subscriptsi, j Component index

Greek Letterse Dispersion energy parameter (J)

eAB Energy parameter of the association between sites A and B (J)

q Number density (number of molecules in unit volume), (A-3)

r Temperature independent segment diameter (A)

jAB Volume of interaction between sites A and B

1 Introduction

Aqueous two-phase systems (ATPS) are among the most favorable solutions for separation

of biological molecules and proteins at an appropriate temperature and pH. The most

favorable advantages of these solutions are their low price and low viscosity. ATPS consist

of two main types, polymer–polymer and polymer–salt systems. In ATPS each phase

contains a large amount of water, which, with proper compositions of other components

like buffer and salts, provide a good environment for biological molecules. Due to these

advantages, most biomolecules such as proteins, lipids, nucleic acids and viruses are

separated using ATPS [1].

On the other hand, because of the high concentration of water in each phase, this

component plays crucial role in these solutions. Therefore, prediction of thermodynamic

properties of water has great importance for the calculation of thermodynamic properties of

these solutions. Different approaches, specifically activity coefficient models, have been

applied in PEG–salt–water solutions.

Nini et al. [2] employed the UNIFAC group contribution method for correlation and

prediction of water activities in polymer–water binary mixtures. Herskowitz et al. [3] also

1424 J Solution Chem (2013) 42:1423–1437

123

used the UNIFAC model for calculation of water activity in mixtures of PEG300 and

water. Sadeghi [4] modified the NRTL model for correlation of water activities in poly-

mer–salt–water systems. Haghtalab and Joda [5] proposed a new local composition model

for correlation of water activities in polymer–salt–water binary and ternary solutions. Xu

et al. [6] modified Wilson’s equation for the correlation of vapor–liquid equilibria of

ATPS. All of these studies are based on the application of activity coefficient models for

calculation of vapor–liquid equilibrium of these systems.

On the other hand, among the various physicochemical properties, density has great role

in various industrial process calculations. Equations of state (EoS) are great tools for

calculation of properties such as density and activity. The importance of an EoS is even

more significant in prediction of liquid densities because activity models cannot be used for

this purpose. Song et al. [7] utilized the modified Chiew EoS [8] for hard sphere chains as

the reference term, employed a van der Waals attractive contribution for the perturbation

term, and named the resulting EoS the PHSC EoS; it was then used for numerous systems

including alkanes, alkenes and aromatics. Their great work was the calculation of polymer

properties such as liquid density with high accuracy. Large differences between molecular

size of polymers and other components in polymer containing solutions makes thermo-

dynamic modeling of their solutions a challenging issue. Therefore, an accurate EoS

capable for predicting polymer properties is needed. PHSC is one of the most favorable

EoS for phase behavior in polymer containing systems and it has been used extensively

[9–17].

Valavi and Dehghani [18] modified the PHSC EoS using the mean spherical approxi-

mation model (MSA) and utilized this modified model to predict pressure–temperature

diagrams of hydrates in the presence of salts. Valavi et al. [19] used the PHSC EoS and

MSA model to correlate liquid–liquid equilibria in ATPS systems. In the present study, the

capability of the ePHSC model in the correlation and prediction of water activities and

liquid densities in binary polymer–water and ternary polymer–salt–water solutions is

investigated. In the previous modification [19], the restricted MSA contribution was added

to the original PHSC EoS for treating long-range electrostatic interactions between ions.

Also, in the presence of hydrogen bonding, self and cross association phenomena between

water and PEG molecules must be taken into account. For this purpose, Lee and Kim [20]

added an association contribution, proposed by Wertheim, to the PHSC EoS. With our

modification and the modification described by Lee and Kim [20], we employed the

modified PHSC EoS and applied it to VLE calculations in binary polymer–water solutions

and ternary polymer–salt–water systems.

2 Theory

In the PHSC EoS proposed by Song et al. [7], the molecules are assumed to be chains of

tangent hard spheres. This EoS consists of two main contributions. The first term is the

modified Chiew EoS [8], as the reference contribution, and the second term is the perturbation

contribution which uses the van der Waals theory that accounts for perturbation effects

between segments. The EoS, in terms of the compressibility (Z) factor, is expressed as:

ZPHSC ¼ Zref þ Zpert ð1Þ

where Zref and Zpert are reference and perturbation compressibility factors, respectively.

Since water and PEG show strong hydrogen bonding effects, the thermodynamic model

J Solution Chem (2013) 42:1423–1437 1425

123

must be capable of accounting for association interactions. Lee and Kim [20] added the

association term of Wertheim to the PHSC EoS. Meanwhile, when salt is dissolved in the

solution, it dissociates into cations and anions. These charged particles can affect the

thermodynamic behavior of solutions considerably and thus electrostatic interaction among

ions must be considered. The Debye–Huckel theory was the first important work on

modeling of electrostatic interactions. However, the Debye–Huckel model is precise only

in low concentrations of salt, whereas at higher concentration of ions, and also as the size

of ions increases, the model does not provide an accurate description of solution behavior.

To deal with the electrostatic interactions among ions, considerable developments have

occurred in statistical mechanics methods, such as molecular simulation, perturbation

theory and integral equation theory. The integral equation theory is the most popular one.

In engineering applications, the MSA approach, obtained from the solution of the Orn-

stein–Zernicke integral equation, is commonly preferred.

In this study, the primitive MSA model is used for the long-range interaction energy

between ions and, therefore, an additional contribution is added to Eq. 1 as:

Z ¼ ZPHSC þ Zasc þ ZMSA ð2Þ

In the above equation, Zasc is the compressibility factor contribution for association

interactions introduced by Wertheim [21–26], while ZMSA represents the restricted MSA

model that is used in this work. Formulas of calculation of ZMSA are reported in previous

work [19].

In this study the above model is applied to correlate and predict water activity and liquid

density in PEG–salt–water solutions and the results are introduced in the next section.

3 Results and Discussion

In this section the capability of the modified PHSC EoS for VLE calculation of polymer–

salt–water solutions is examined. In this regard, the water activities and liquid densities of

binary and ternary solutions were investigated. The molecular weight (MW) of PEG

molecules in the considered systems varies from 200 to 20,000 g�mol-1. Salts considered

here are (NH4)2SO4, Na2CO3, K2HPO4, MgSO4, and Na2CO3. The model utilizes only two

adjustable binary interaction parameters (BIP) between water and PEG for all of the nine

binary solutions. Furthermore, using adjustable parameters that were obtained in previous

work, the water activities in ternary PEG–water–salt solutions are predicted. In the last

section, liquid densities of binary mixtures of water–PEG and ternary solutions of water–

PEG–salt are predicted while setting the BIP between components to zero.

3.1 Pure Component Parameters

The PHSC EoS introduced by Song et al. [7] employs three parameters (r, r, e) for non-

hydrogen bonded pure compounds. Since hydrogen bonding affects strongly the thermo-

dynamic properties of substances, self-association phenomena must also be considered in

thermodynamic models for these systems. As mentioned before, following Lee and Kim

[20], the association term proposed by Wertheim [21–26] is added to the PHSC EoS. This

term adds two more parameters (kAB and eAB) to the original PHSC model. To use the

association contribution, an important issue that must be taken into account is the number

of association sites on each associating molecule. For water, authors usually consider 2 or 4

1426 J Solution Chem (2013) 42:1423–1437

123

hydrogen bonding sites. In previous work [18] the phase behavior of pure water was

examined by considering two and four sites. However, it was found that two association

sites models gave better results. Pure component parameters for water are taken from

previous work [18]. Calculation of parameters for the polymer component is more chal-

lenging. Because of the high MW of polymers, they are not present in the vapor phase and

the data available for them is commonly the liquid phase densities. To obtain pure com-

ponent parameters for polymers, three different approaches are common in the literature.

The first approach involves calculation of parameters from the liquid density data. In the

second approach, the parameters are obtained by adjusting the vapor pressure and liquid

densities of low MW compounds and then the obtained parameters are extrapolated to the

real MW of a polymer. The third approach uses the binary mixture data [27]. In this study,

pure parameters obtained in previous work [19] are used for polymers. These parameters

are listed in Table 1.

Furthermore, in the previous work [19] parameters for salts were adjusted with activity

coefficient and liquid density data of aqueous salt solutions.

3.2 Binary Interaction Parameters

To obtain the BIP, we first adjusted the BIP between water and PEG to represent exper-

imental water activity data of nine binary polymer–water systems. The objective function

(OF) is introduced by Eq. 3:

OF ¼XNp

1

ðaexp tw � acal

w Þ2 ð3Þ

where awexpt and aw

cal are experimental and calculated water activities, respectively, and Np is

the number of experimental data points. The OF introduced in Eq. 3 was used to correlate

water activities in 10 PEG aqueous solutions. Molecular weights of PEG ranged from 200

to 20,000 g�mol-1 and 84 data points were correlated together to obtain the BIP. A linear

relationship between the BIP values and MW of PEG molecules was observed for 10

binary aqueous solutions at 298 K that is given in Eq. 4:

kij ¼ �0:078469755� 0:000000413819ðMWÞ ð4Þ

In Eq. 4, kij is BIP between water and PEG and is a function of the MW of the PEGs. It is

worth mentioning that the BIP is related to dispersion energy parameters:

Table 1 Pure component parameters of the modified PHSC EoS [19]

Component r R (A) e/k (J�K-1) eAB/k (J�K-1) kAB

Water 1.6384 2.8921 450.82 1247.38 0.03766

PEG 0.0144(MW) 4.7997 426.05 1928.58 0.00395

(NH4)2SO4 1 3.2171 749.18 – –

MgSO4 1 4.9657 1810.48 – –

Na2SO4 1 3.8921 1837.66 – –

K2HPO4 1 3.6716 749.18 – –

Na2CO3 1 3.9899 1890.22 – –

J Solution Chem (2013) 42:1423–1437 1427

123

eij ¼ffiffiffiffiffiffiffiffieiiejjp ð1� kijÞ ð5Þ

Table 2 lists the results of the above correlation and shows very good agreement

between the experimental data and the calculated results. The absolute average relative

deviation (AARD %) for water activity is defined as:

AARD% ¼ 1

Np

XNp

1

aexpw � acalc

w

aexpw

����

���� ð6Þ

We also compare our results with the poly-NRTL and poly-Wilson models employed by

Sadeghi [4] for correlation of water activities in these systems in Table 2. As is observed,

the deviations of our results are lower than for the NRTL and Wilson models. In Fig. 1 a

comparison is presented between calculated and experimental water activity data in the

presence of PEG 200, 400 and 600 for solutions containing up to 90 % weight fraction of

PEG. Also, the water activities in solutions with PEG1500, 8000 and 20000 are presented

in Fig. 2. With increasing MW of PEG, larger deviations are observed between model and

experimental data.

With the BIP obtained from Eq. 4, the water activities in ternary aqueous solutions with

two PEG components (the MW of PEGs are different) are predicted. For this purpose, no

interaction parameters between two PEG molecules with different MW are employed. The

results are presented in Table 3 and some of them are depicted in Fig. 3.

To examine the capability of correlations introduced by Eq. 4, it is used for the pre-

diction of water activities in the presence of PEG300, 400, 4000 and 6000 at different

temperatures (308–338 K). Although the effect of temperature is not considered in the

derivation of Eq. 4 (derived at 298 K), it is observed in Table 4 that it can produce good

results at other temperatures. The minimum AARD is 0.2 % (for PEG 400 at T = 318 K)

and the maximum error is observed for PEG 6000 at 308 K (AARD = 1.07 %). The mean

average of all 16 systems is 0.44 %. It is observed that with increasing MW of PEGs, the

error of the ePHSC model increases. The correlation for BIP was also used for considering

interactions between water and PEG in ternary systems of water–PEG–salt. Also, the BIP

between salt and PEG were set to zero. (NH4)2SO4, MgSO4, KH2PO4, Na2SO4 and

Na2CO3 are the salts considered in this section. The MW of PEGs are 1000, 4000, 6000,

8000, and 10000 g�mol-1. AARD % values for 14 systems with 267 data points are

Table 2 AARD % for correla-tion of water activities (VLE data)for binary PEG–water solutionsobtained by the modified PHSCEoS developed in this work andthe results obtained by the Wilsonand NRTL models at 298 K; dataare taken from Ref. [2]

a Units: g�mol-1

PEG MWa Np AARD % ofthis work

AARD% ofWilson [4]

AARD% ofNRTL [4]

200 11 1.03 2.36 2.38

400 11 1.37 1.036 1.11

600 11 1.6 1.62 1.65

1,000 7 0.36 0.81 0.83

1,450 9 1.37 0.53 0.55

3,350 7 0.37 0.54 0.55

6,000 7 0.31 0.68 0.69

8,000 7 0.3 0.60 0.61

10,000 7 0.27 0.57 0.59

20,000 7 0.31 0.69 0.7

Total 84 0.73 0.96 0.98

1428 J Solution Chem (2013) 42:1423–1437

123

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.4

0.5

0.6

0.7

0.8

0.9

1

PEG weight fraction

wat

er a

ctiv

ity

Fig. 1 Water activity in the presence of PEG200 (open circle), PEG400 (open square) and PEG600 (opendiamond). Solid lines are predictions using the modified PHSC EOS; experimental data are taken from Ref.[2]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.75

0.8

0.85

0.9

0.95

1

PEG weight fraction

wat

er a

ctiv

ity

Fig. 2 Water activity in the presence of PEG1500 (open circle), PEG8000 (open square) and PEG20000(open diamond). Solid lines are prediction using the modified PHSC EOS; experimental data are taken fromRef. [2]

J Solution Chem (2013) 42:1423–1437 1429

123

presented in Table 5. As is observed in this table, the overall deviation is 0.64 %, which

shows good agreement with the experimental data. In this table we also compare our

results with those from Wilson’s activity coefficient model [4] which also produced

accurate results (AARD = 0.3 %). It is worth noting that although the result for Wilson’s

model is better than the ePHSC EoS. Considering the fact that, in contrast to Wilson’s

model, the ePHSC EoS is a predictive tool, the capability of the ePHSC model in pre-

dicting water activities is acceptable.

The modified PHSC EoS was also utilized for prediction of liquid densities of binary

and ternary aqueous solutions of PEG and salt. We first tried to use the BIP that were

correlated with water activity; however, our result showed that by setting BIPs between all

components to zero, the liquid density was better predicted. Therefore, in this work we set

the BIP to zero for prediction of liquid densities. Due to hydrogen bonding interactions

Table 3 AARD % for predic-tion of water activities in PEG–PEG–water ternary solutions at298 K

The weight fractions of PEGs inthe ternary solution are the same;data are taken from Ref. [2]a Units: g�mol-1

PEG (1), MWa PEG(2), MWa Np AARD %

6,000 2,000 5 0.37

400 20,000 5 0.16

200 1,450 6 0.41

200 600 8 0.63

1,000 10,000 5 0.34

1,000 8,000 5 0.39

Total 6 34 0.38

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350.75

0.8

0.85

0.9

0.95

1

PEG weight fraction

wat

er a

ctiv

ity

Fig. 3 Prediction of water activity in ternary aqueous solutions containing PEG6000, PEG20000 (opencircle), PEG200, PEG1450 (open square), PEG1000 and PEG10000 (open diamond). Weight fractions ofPEGs in ternary solutions are the same. Solid lines are calculation results using the modified PHSC EOS;experimental data are taken from Ref. [2]

1430 J Solution Chem (2013) 42:1423–1437

123

among water molecules, and among PEG molecules, and cross association among water

and PEG molecules, modeling of the liquid phase is very complicated. Also, long range

interactions between charged ions make thermodynamic modeling even more difficult. Our

results for the prediction of liquid densities are satisfactory. The AARD % for prediction of

liquid density is defined as:

Table 4 AARD % for predic-tion of water activities in PEG–water binary solutions at differenttemperatures; data are taken fromRef. [28]

a Units: g�mol-1

PEG MWa Np T (K) AARD %

300 4 308 0.31

400 8 308 0.59

4,000 4 308 0.84

6,000 5 308 1.07

300 4 318 0.15

400 8 318 0.2

4,000 4 318 0.27

6,000 6 318 0.29

300 4 328 0.21

400 8 328 0.45

4,000 4 328 0.74

6,000 6 328 0.91

300 4 338 0.08

400 8 338 0.26

4,000 4 338 0.42

6,000 6 338 0.31

Total 87 0.44

Table 5 AARD % for predic-tion of water activities and BIPbetween PEG and salt in PEG–salt–water ternary solutions

a Units: g�mol-1

PEGMWa

Salt Np AARD % ofthis work(prediction)

AARD %of Wilson[4]

Reference

1,000 (NH4)2SO4 20 0.76 0.25 [29]

4,000 (NH4)2SO4 20 0.46 0.35 [29]

6,000 (NH4)2SO4 28 0.28 0.45 [30]

10,000 (NH4)2SO4 15 0.11 0.27 [31]

1,000 MgSO4 15 1.89 0.24 [29]

4,000 MgSO4 15 1.65 0.33 [29]

8,000 MgSO4 20 1.59 0.29 [32]

10,000 MgSO4 10 0.19 0.34 [31]

4,000 KH2PO4 24 0.13 0.25 [31]

1,000 KH2PO4 24 0.55 0.13 [31]

1,000 Na2SO4 20 0.45 [31]

8,000 Na2SO4 20 0.37 [32]

1,000 Na2CO3 20 0.32 [32]

8,000 Na2CO3 20 0.29 [32]

Total 14 267 0.64 0.3

J Solution Chem (2013) 42:1423–1437 1431

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Table 6 AARD % values of theprediction of liquid densities ofPEG–water binary solutions bythe modified PHSC EoS at dif-ferent temperatures

PEG MWa Np T (K) AARD % Reference

200 9 283 0.62 [33]

200 9 293 0.59 [33]

200 9 303 0.58 [33]

200 9 313 0.62 [33]

200 9 323 0.69 [33]

200 9 333 0.99 [33]

200 9 343 1.23 [33]

200 9 353 1.44 [33]

200 9 363 1.61 [33]

300 5 308 1.38 [34]

300 5 318 0.62 [34]

300 5 328 0.45 [34]

300 5 338 0.42 [34]

400 9 283 0.91 [33]

400 9 293 0.82 [33]

400 9 303 0.72 [33]

400 9 313 0.6 [33]

400 9 323 0.47 [33]

400 9 333 0.22 [33]

400 9 343 0.29 [33]

400 9 353 0.49 [33]

400 9 363 0.76 [33]

600 9 283 1.34 [33]

600 9 293 1.23 [33]

600 9 303 1.21 [33]

600 9 313 0.98 [33]

600 9 323 0.84 [33]

600 9 333 0.45 [33]

600 9 343 0.28 [33]

600 9 353 0.38 [33]

600 9 363 0.58 [33]

1,500 9 283 1.80 [33]

1,500 9 293 1.71 [33]

1,500 9 303 1.54 [33]

1,500 9 313 1.44 [33]

1,500 9 323 1.28 [33]

1,500 9 333 0.87 [33]

1,500 9 343 0.52 [33]

1,500 9 353 0.44 [33]

1,500 9 363 0.52 [33]

3,000 15 283 1.69 [35]

3,000 15 288 1.65 [35]

3,000 15 293 1.61 [35]

3,000 15 298 1.56 [35]

1432 J Solution Chem (2013) 42:1423–1437

123

AARD% ¼ 1

Np

XNp

i

ql;exp ti � ql;cal

i

ql;exp ti

�����

����� ð7Þ

The results of the liquid density prediction for aqueous solutions of PEG with different

MW are summarized in Table 6. The liquid density of 67 aqueous systems of PEG, were

predicted for 735 data points with MW of 200–20,000 g�mol-1 at different temperatures

(T = 283–363 K). Very good agreement between the experimental data and calculated

values is observed with the total AARD % = 1.16 %. In Fig. 4, the liquid densities of

PEG 4000 up to 0.4 weight fraction at different temperatures is depicted. Also, Fig. 5

shows the liquid density of PEG200 up to 0.9 weight fractions at different temperatures. As

one can see, the calculated results show the correct trend with good agreement with

experimental data. We may conclude that considering the non-ideality of these systems, the

modified PHSC EoS produced very good results for prediction of liquid densities of these

67 water and PEG binary systems with MW of PEG ranging from 200 to 20,000 g�mol-1.

Finally, the liquid densities of ternary solutions of water–PEG–salt were calculated and

presented in Table 7. The AARD for all 18 systems including 90 data points is 3.95 %,

which is a satisfactory result. The AARD % for liquid densities of these systems are much

larger than for the binary solutions of PEG and water. Even in the case of aqueous solutions

of PEG6000 and Na2CO3, an error of 8 % is observed. But one should note that these

Table 6 continued

a Units: g�mol-1

PEG MWa Np T (K) AARD % Reference

3,000 15 303 1.52 [35]

3,000 15 308 1.46 [35]

3,000 15 313 1.41 [35]

4,000 8 300 1.36 [36]

4,000 8 310 1.23 [36]

4,000 8 313 1.16 [36]

4,000 8 318 1.11 [36]

4,000 8 323 1.06 [36]

4,000 8 328 0.99 [36]

6,000 16 283 1.76 [35]

6,000 16 288 1.72 [35]

6,000 16 293 1.68 [35]

6,000 16 298 1.61 [35]

6,000 16 303 1.60 [35]

6,000 16 308 1.58 [35]

6,000 16 313 1.49 [35]

20,000 14 283 1.71 [35]

20,000 14 288 1.67 [35]

20,000 14 293 1.63 [35]

20,000 14 298 1.60 [35]

20,000 14 303 1.53 [35]

20,000 14 308 1.50 [35]

20,000 14 313 1.45 [35]

Total 735 1.16

J Solution Chem (2013) 42:1423–1437 1433

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450.98

1

1.02

1.04

1.06

1.08

1.1

weight fraction of PEG

liqui

d de

nsity

(gr

/cm

3)

Fig. 4 Prediction of the liquid density of aqueous solutions of PEG 4000 at different temperatures:T = 300 K (open circle), T = 310 K (open square), T = 313 K (open diamond), T = 318 K (9),T = 323 K (), and T = 328 K (open star). Lines are predictions using the modified PHSC EOS;experimental data are taken from Ref. [36]

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91

1.05

1.1

1.15

weigth fraction of PEG

liqui

d de

nsity

(gr/

cm3)

Fig. 5 Prediction of the liquid density of aqueous solutions of PEG 200 at different temperatures:T = 283 K (open circle), T = 293 (open square), T = 303 K (open diamond), T = 313 K (9), andT = 323 K (open star). Lines are predictions using the modified PHSC EOS; experimental data are takenfrom Ref. [33]

1434 J Solution Chem (2013) 42:1423–1437

123

results are completely predicted; therefore the ePHSC EoS has a good capability for

prediction in the mentioned systems.

4 Conclusions

In this study the electrolyte PHSC EoS that uses the MSA term is utilized for correlation

and prediction of water activities and liquid densities in binary and ternary solutions of

PEG, water and salt. For the calculation of water activities in PEG–water systems, a linear

correlation of the BIP between water and PEG was obtained for 10 solutions. This cor-

relation was used for prediction of water activities and liquid densities in binary and

ternary solutions. The results show that the modified EOS can be employed for calculation

of vapor–liquid equilibrium of ATPS system, especially for the calculation of water

activities. The mean absolute average relative deviation (AARD %) of water activities of

aqueous solutions of PEG at 298 K is 0.73 %. Furthermore, the AARD % for prediction of

water activities in binary PEG–water solutions in the temperature range 308–338 K is

0.41 %. Also, the water activities of ATPS are predicted with AARD % = 0.64 %.

Finally, the liquid densities of binary solution of PEG–water and ternary solution of PEG–

water–salt were modeled with 1.16 and 3.95 % AARDs, respectively.

References

1. Haghtalab, A., Mokhtarani, B.: On extension of UNIQUAC–NRF model to study the phase behavior ofaqueous two phase polymer–salt systems. Fluid Phase Equilib. 180, 139–149 (2001)

Table 7 AARD % values for theprediction of liquid densities ofternary solutions of water–PEG–salt; data are taken from Ref. [37]

a Units: g�mol-1

PEG MWa Salt Np T (K) AARD %

1,000 (NH4)2SO4 5 298 2.69

1,000 (NH4)2SO4 5 308 2.40

1,000 (NH4)2SO4 5 318 2.02

6,000 (NH4)2SO4 5 298 2.47

6,000 (NH4)2SO4 5 308 2.07

6,000 (NH4)2SO4 5 318 1.83

1,000 Na2CO3 5 298 5.06

1,000 Na2CO3 5 308 4.63

1,000 Na2CO3 5 318 4.29

6,000 Na2CO3 5 298 8.22

6,000 Na2CO3 5 308 4.29

6,000 Na2CO3 5 318 4.12

1,000 Na2SO4 5 298 5.17

1,000 Na2SO4 5 308 5.88

1,000 Na2SO4 5 318 3.36

6,000 Na2SO4 5 298 4.39

6,000 Na2SO4 5 308 5.37

6,000 Na2SO4 5 318 2.85

Total 18 90 3.95

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