Thermodynamic studies of solute–solute and solute–solvent interactions in ternary aqueous systems containing {betaine + PEGDME250} and {betaine + K3PO4 or K2HPO4} at 298.15 K

In this work, to evaluate solute–solute, solute–solvent and phase separation in aqueous systems containing {betaine + poly ethylene glycol dimethyl ether with molar mass 250 g mol−1 (PEGDME250)}, {betaine + K3PO4} and {betaine + K2HPO4}, first water activity measurements were made at 298.15 K and atmospheric pressure using the isopiestic technique. The water iso-activity lines of these three systems were obtained which have positive deviations from the semi-ideal solutions. This suggests that betaine-polymer and betaine-K3PO4 or betaine-K2HPO4 interactions are unfavorable; and these mixtures may form aqueous two-phase systems (ATPSs) at certain concentrations. Indeed the formation of ATPSs was observed experimentally. Then, osmotic coefficient values were calculated using the obtained water activity data; and, using the polynomial method the solute activity coefficients were determined. Using these activity coefficients, the transfer Gibbs energy (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta {G}_{tr}^{i}$$\end{document}ΔGtri) values were calculated for the transfer of betaine from aqueous binary to ternary systems consisting polymer (PEGDME250) or salts (K3PO4 and K2HPO4). The obtained positive \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta {G}_{tr}^{i}$$\end{document}ΔGtri values again indicated that there is unfavorable interaction between betaine and these solutes. Finally, the volumetric and ultrasonic studies were made on these systems to examine the evidence for the nature of interactions between betaine and the studied salts or polymer.


Apparatus and procedure
Vapor -liquid equilibrium measurements In this study, for obtaining water activity and vapor pressure at 298.15 K the improved isopiestic apparatus was applied 8,9 .In measuring the water activity for binary aqueous betaine solutions, five leg manifold was used in which for each of pure betaine and standard pure NaCl, two flasks are employed and the central flask was considered for water supply.Also, this procedure with a seven leg manifold was employed in the case of water activity measurements for (betaine + PEGDME 250 + water), (betaine + K 3 PO 4 + water) and (betaine + K 2 HPO 4 + water).In this case, for each of pure betaine, pure salt (K 3 PO 4 /K 2 HPO 4 ) or polymer one flask was used.Two flasks were poured with standard sodium chloride.The remaining two flasks of the manifold are considered for {betaine + K 3 PO 4 / K 2 HPO 4 } or {betaine + PEGDME 250 }.Before doing anything, it was necessary to degas and remove the air of the solutions which was made by evacuating the apparatus slowly and frequently.To reach the equilibrium between solutions, the apparatus was slowly immersed in a bath for about 120 h.The temperature controller with uncertainty of 0.01K was used.Then, the apparatus was removed from the bath and the mass of each flask and therefore solute mass fractions were determined with an analytical balance with a precision of ± 1 × 10 -7 kg.We used the differences between mass fractions of two NaCl solutions as an equilibrium criteria; so that the equilibrium is assumed when this difference is less than 0.1%.In isopiestic equilibrium vapor pressure and water activity in the sample solutions and reference are the same.This enables us to calculate the osmotic coefficient for the reference solution, from which the osmotic coefficient of the each sample in isopiestic equilibrium with the standard NaCl is readily determined.In this work, for the required osmotic coefficient values of the reference solutions the correlation relation suggested by Colin et al. 12 was considered.

Density and speed of sound measurement
For volumetric and ultrasonic studies, {betaine + PEGDME 250 + water}, {betaine + K 3 PO 4 + water} and {betaine + K 2 HPO 4 + water} solutions at different molalities of betaine as a solute and different molality of {polymer or salt + water} as a solvent (i.e.0.1, 0.2, 0.3) mol kg −1 were prepared in glass vessels and the corresponding density (ρ) and speed of sound (u) values were measured with (Anton Paar DSA 5000 densitometer and speed of sound analyzer) at T = 298.15K and atmospheric pressure (≈ 85KPa).
The uncertainty values of 0.15 kg m −3 and 0.5 m s −1 were estimated for density and speed of sound, respectively.Before doing anything, it was necessary to calibrate densitometer device; for this purpose dried air and double distilled water were used.The accurate temperature controller built in the apparatus enable us to carry out these measurements within ± 0.01 K which is a required condition for such measurements 13 .

Water activity results
To study the vapor-liquid equilibrium behavior of binary aqueous solutions of betaine and ternary aqueous betaine solutions containing the salts (K 3 PO 4 /K 2 HPO 4 ) or PEGDME 250 , water activity measurements were carried out at 298.15 K using isopiestic method 8,9 .Considering equal water activity of standard and sample solutions at isopiestic equilibrium, and relation 1, the osmotic coefficient, φ of the solution can be calculated 14 : In Eq. ( 1), m and m R stand for the isopiestic equilibrium molalities (mol kg −1 ) of the sample and reference (NaCl) solutions, respectively;φ R is the osmotic coefficient of the reference solution, ν R and ν denote respectively the sum of stoichiometric numbers of the anion and cation in the reference and the salts (K 3 PO 4 /K 2 HPO 4 ) or PEGDME 250 , solutions.The osmotic coefficient values calculated from Eq. ( 1) can be used along with Eqs. ( 2) and ( 3) to obtain the activity of water and the vapor pressure P in polymer and salt solutions 14 .
Here, M W, V 0 W and P 0 W are used respectively for the molar mass, molar volume and vapor pressure of pure water.B stand for second virial coefficient of water vapor and its value at T = 298.15Kwas obtained from the Rard and Platford equation 15 .R is the gas constant and T is the absolute temperature.The Kell equation 16 and the equation of state proposed by Saul and Wagner 17 were used respectively to determine V 0 W and P 0 W .After the system reaches isopiestic equilibrium, a w and P values were calculated using Eqs.( 2) and (3), respectively.
The for binary (betaine + water) solutions, the values of water activity, osmotic coefficient and vapor pressure are collected in Table 2.In Tables 3, 4, 5, water activity values are given for (betaine + PEGDME 250 + water), (betaine + K 3 PO 4 + water) and (betaine + K 2 HPO 4 + water), and the corresponding iso-activity lines are shown in Figs. 1, 2, 3, respectively.In each of presented lines in these Figures, the four points have the same water activity and therefore the same chemical potential.
The available water activity data for (PEGDME 250 + H 2 O) 9,18 ], (K 3 PO 4 + H 2 O) 19 and (K 2 HPO 4 + H 2 O) 20 make it possible to check the quality of our water activity data for these binary solutions by comparisons of these data, as shown in Figs.S1-S3 presented in supporting material.These Figures show good agreement of our measurements for these binary systems with the literature values.
The vapor pressure data for binary (betaine + water), (PEGDME 250 + water), (K 3 PO 4 + water) and (K 2 HPO 4 + water) solutions may be used for comparison of the extent of solute-solvent interactions in these binary solutions.For this purpose, the vapor pressure depression values p were calculated from the correspond- ing P values reported in Tables 3, 4, 5 and vapor pressure for pure water as follows: The plot of calculated vapor pressure depression values for these binary solutions versus molalities of betaine, polymer and salts have been given in Fig. 4.This Figure and Tables 3, 4, 5 show that the vapor pressure depressions for salt solutions are more that of betaine or polymer solutions, and their values for (K 3 PO 4 + water) are higher than (K 2 HPO 4 + water) at the same solute concentrations.In other words solute-solvent interactions for salts are stronger than that of betaine or polymer.The differences in interactions between these salt systems are related to their anions; so that, a more vapor pressure depression is observed for the salt which has a higher anion charge (K 3 PO 4 ˃ K 2 HPO 4 ) leading to stronger salt-water interactions.In other words, since vapor pressure depression is a colligative property, its value is increased by increasing the number of moles of ions.The higher p value observed for aqueous betaine than PEGDME 250 may be due to the fact that betaine is a hydrogen-bond accepter , from the comparison of lines with equal water activity with the line corresponding to semi-ideal solution, the salting-in or salting-out effect in the ternary (betaine + PEGDME 250 + water), (betaine + K 3 PO 4 + water) and (betaine + K 2 HPO 4 + water) systems may be investigated.The Zdanovskii rule was used for this purpose using (Zdanovskii-Stokes-Robinson) relation at constant water activity 23,24 : In Eq. ( 5), m 1 and m 2 stand for concentration of PEGDME 250 or salts and betaine (in molality basis) in the ternary systems, respectively.Similarly, m 0 1 and m 0 2 show respectively, the concentration of the PEGDME 250 or salts and betaine in the binary solution which has an equal a w .The results of applying Zdanovskii rule 23 are also presented in Figs. 1, 2, 3 which indicate that for all the three studied ternary systems the experimental lines with equal water activity indicate negative deviation in regard with semi-ideal behavior.This means that in these investigated ternary systems the interaction between betaine and polymer or salts are unfavorable (salting-out effect).This salting-out effect may be the main reason for phase separation of these systems at some concentrations.Indeed, formation of aqueous two-phase systems (ATPSs) of betaine with both of the salts (K 3 PO 4 and K 2 HPO 4 ) have been observed by Zeng et al. 6 .Also, it was found that betaine can form an ATPS with polyethylene glycol (PEG) 6 .These ATPSs have found important applications in protein extraction 6 .As expected, we observed that betaine can also form an ATPS with PEGDME 250 which has similar structure with PEG.

Thermodynamic framework
The polynomial method 10 has been frequently used for obtaining relation between activity coefficients of solutes and their molalities in ternary systems 25,26 .According to this method, for each component i of the ternary system, the activity coefficient in molality basis,γ m i , can be expressed in terms of betaine molality, m 1 and salts (K 3 PO 4 / K 2 HPO 4 ) or PEGDME 250 molality, m 2 as follows: (5) b w 2 www.nature.com/scientificreports/Equation 6 can be written as Eq.(7a) when in the power series we neglect all terms higher than fourth: In Eq. (7a) the term γ 0,m i denotes the activity coefficient of component i in aqueous binary solutions which is expressed in terms of its molality, m i , as follows: Within this thermodynamic framework a quantity has been introduced 27 : In above relations, φ 0 R , φ 0 1 , and φ 0 2 are used respectively for the osmotic coefficients of solutions of the reference, betaine, and salts (K 3 PO4/K 2 HPO 4 ) or PEGDME 250 .
Then, using the Gibbs-Duhem relation the Eq. ( 9) can be derived 27 : Using the Eq. 8, the values for m 1 m 2 can be easily calculated from the molality values presented in Tables 3, 4, 5 for the reference, betaine, and salts (K 3 PO 4 /K 2 HPO 4 ) or PEGDME 250 .Then, the parameters A ij in Eq. (9) were determined by the minimization method with the results presented in Tables 6.When these parameters are inserted in Eq. 7, it is easy to calculate concentration dependencies of γ m 1 of betaine in solutions composed of salts (K 3 PO 4 /K 2 HPO 4 ) or PEGDME 250 .
To calculate the required φ 0 1 value for betaine and salts (K 3 PO 4 /K 2 HPO 4 ) or PEGDME 250 at each molality m 1 listed respectively in Tables 2 for betaine and Tables 3, 4, 5 (corresponding to zero betaine concentration) for salts (K 3 PO 4 /K 2 HPO 4 ) or PEGDME 250 , the obtained corresponding osmotic coefficients were correlated with the Eq. ( 10): The symbol E denotes the parameters of Eqs.(10) and (11) which can be determined by minimization method.Using the calculated activity coefficients of betaine in binary and ternary salts (K 3 PO 4 /K 2 HPO 4 ) or PEGDME 250 solutions, the Gibbs energy of transfer, G i tr , of betaine from corresponding binary to the ternary solutions were determined from Eq. 12 28 .The sigh of the G i tr may be used to evaluate solute-solute and solute-solvent interactions in the investigated solutions.
i and γ 0x i are respectively mole fraction based activity coefficient of betaine in the ternary and binary solutions.The required γ x i values were determined from the Eq. ( 13) using the calculated values of activity coefficients in molality basis: where γ ∞ i denotes activity coefficient at infinite dilution.The calculated transfer Gibbs energy values are presented in Table 7.The positive transfer Gibbs energies obtained for all the three mixtures imply unfavorable interaction of betaine with salts (K 3 PO 4 /K 2 HPO 4 ) and PEGDME 250 .It is interesting to note that in Sect.3.1.1we arrived at the same result regarding these unfavorable interactions by observing the negative deviation of iso-activity lines from semi-ideal solution for the investigated solutions.Table 7 also show that the G i tr of betaine becomes more positive by increasing the concentrations of salts or polymer.
Based on McMillan-Mayer theory the obtained G i tr values can also be utilized to get information on the extent of solute-solute interactions in our studied systems 29,30 .According to this theory, at constant temperature and pressure the Gibbs energies of transfer for betaine from water to aqueous salt or polymer solutions are given by Eq. ( 14): In Eq. ( 14), g 12 is called pair interaction parameter; (g 112 and g 122 ) are triplet interaction parameters.These interaction parameters were determined by fitting G i tr data to m 1 and m 2 .From the obtained g 12 values and the following relation the salting coefficient k s can be calculated: The pair and triple interaction parameters obtained from Eq. ( 14) were tabulated in Table 8.Also in this Table salting constant values were included which were calculated from Eq. ( 15).The positive pair interaction parameter, g 12 , found for betaine indicates that pairwise interactions between betaine and salts (K 3 PO 4 /K 2 HPO 4 ) or PEGDME 250 are energetically unfavorable.Also the calculated positive salting constant, k s , tabulated in Table 8 imply that salts (K 3 PO 4 /K 2 HPO 4 ) or PEGDME 250 have the salting-out effect on betaine in these ternary systems.Consequently, the McMillan-Mayer theory also confirms the unfavorable interaction between salts (K 3 PO 4 / K 2 HPO 4 ) or PEGDME 250 and betaine that we concluded in Sect.3.1.1from the negative deviation of constant water activity lines with respect to semi-ideal solutions, and from the positive sign of Gibbs energies of transfer for betaine from water to aqueous salts (K 3 PO 4 /K 2 HPO 4 ) or PEGDME 250 solutions discussed above.

Volumetric properties
The experimental density and speed of sound values for binary aqueous betaine and ternary aqueous betaine solutions containing PEGDME 250 and salts (K 3 PO 4 /K 2 HPO 4 ) are collected respectively in Tables 9, 10, 11, 12.No experimental density or speed of sound data for these systems containing betaine have been reported, previously.In Figures S4-S6, the experimental density values for binary aqueous PEGDME 250 , K 3 PO 4 , and K 2 HPO 4 solutions were compared with the corresponding literature values [31][32][33] , which show fairly good agreement.The density data presented in Tables 10, 11, 12 were used to calculate values of the apparent molar volumes, V and the apparent molar volume at infinite dilution, V 0 , from which some information regarding intermolecular interactions between betaine and polymer or salts may be deduced.To calculate V values for betaine in water and ternary {PEGDME 250 or salt + water} solutions at T = 298.15K the following equation 34 was used: here, M is the molar mass of betaine and m is its molality; ρ 0 and ρ stand for density of the (betaine + H 2 O) and ternary mixtures (betaine + PEGDME 250 + H 2 O) or [(betaine + salts (K 3 PO 4 and K 2 HPO 4 ) + H 2 O], respectively.The obtained V values are collected in Tables S1-S3 and plotted against betaine molality in Figures S10-S12.It was found that for the investigated systems linear dependence between V and betaine molality may be ( 15)     The values of V 0 and empirical slope S V obtained from fitting of V values to betaine molality are presented in Table 13.In Figures S10-S12 the straight lines generated by Eq. ( 17) for the three investigated systems indicate validity of Mason equation 35 .The obtained apparent molar volume at infinite dilution V 0 for different systems were plotted against molality of (polymer or salt + water) in Fig. 5.It is also quite clear from the Table 13 and Fig. 5 that the V 0 values decreased with the increase in molality of (polymer or salt + water) as a solvent and this leads to conclusion that interaction is unfavorable between polymer or salt and betaine.This is consistent with the result of vapor-liquid equilibrium study of the investigated ternary solutions mentioned in section "Result and discussion".
The solute-solute interactions in the studied ternary solutions can be deduced from the behavior of apparent molar volumes as plotted against betaine molality in Figures S10-S12.These Figures show that while the slope for the system containing K 3 PO 4 is negative the corresponding one for the K 2 HPO 4 is positive.This difference may be due to the more pronounced chemical reaction between betaine and K 3 PO 4 than K 2 HPO 4 which leads to decrease of solute-solute interactions in ternary solutions containing betaine and K 3 PO 4 as indicated with the negative S v values reported in Tables 13.The reason for possible reaction between betaine and salts in water is that the used salts hydrolyze in water and form alkaline solution, especially K 3 PO 4 .The betaine is an amphoteric molecule and therefore it can react with the alkaline solution.However, this reaction can be occurred readily with K 3 PO 4 than K 2 HPO 4 .
The standard partial molar volumes of transfer (Δ tr Vφ 0 ) is another important thermodynamic property that can be used for investigating the solute-solvent interactions.Here, the Δ tr Vφ 0 is the difference between V 0 values (17) Table 10.The values of density (ρ) and speed of sound (u) for the betaine in aqueous (PEGDME 250 + water) solutions at T = 298.The V 0 for binary aqueous PEGDME 250 , K 3 PO 4 , and K 2 HPO 4 solutions are available in the literature [31][32][33] .The obtained V 0 value for binary aqueous betaine solution is given Table 13.At infinite dilution solute -solute inter- actions are unaffected on the value of Δ tr Vφ 013,36 .The Δ tr Vφ 0 values for the ternary {betaine + PEGDME 250 + water}, {betaine + K 3 PO 4 + water} and {betaine + K 2 HPO 4 + water} solutions were also reported in Table 13.This Table shows that the Δ tr Vφ 0 values are negative and become more negative at higher concentrations of PEGDME 250 and both salts.This means that the presence of betaine in mentioned systems lead to strong interactions between polymer or salts and water.In other words, the betaine-polymer and betaine-phosphate salts interactions are unfavorable; and therefore, phase separation in these ternary systems may be occurred at certain concentrations of betaine, polymer or salts.Indeed, in these systems we observed the formation of two-phase systems.Therefore, both the VLE and volumetric studies predict phase separation in the investigated solutions.

Acoustic properties
The apparent molar isentropic compressibility κφ and its value at infinite dilution κφ 0 are also important quantities which provide some formation regarding the solvent structure around betaine within the bulk solution.The apparent isentropic molar compression for betaine in aqueous solution of (0.1, 0.2, and 0.3 mol kg −1 ) polymer or salts were calculated and tabulated in Table S1-S3 using the Eq. ( 19) 37 .
(18) � tr V 0 ϕ = V 0 ϕ in aqueous ternary − V 0 ϕ in aqueous binary    Table 13.The obtained values of limiting apparent specific volume ( V 0 ϕ ), experimental slopes ( S V ), standard transfer volume (Δ tr Vφ 0 ) and standard deviation for apparent molar volume σ (Vφ) for different ternary aqueous betaine solutions at T = 298.15K a .a Standard uncertainties u for temperature, pressure, and molality are u(T) = 0.01 K, u(P) = 0.01 kPa, and u(m) = 0.003 mol kg −1 , respectively with confidence level of 0.95.b m is the molality of betaine in water.c σ ( , where n is the number of experimental data.The Laplace -Newton's equation 38 was used for calculation of the required isentropic compressibility κ s of mixture: For speed of sound u and density ρ, the measured values presented in Tables 10, 11, 12 were used.κ s0 is the isentropic compressibility of pure solvent.The measured speed of sound data obtained for (PEGDME 250 + H 2 O), (K 3 PO 4 + H 2 O) and (K 2 HPO 4 + H 2 O) may be compared with literature data as Figures S7-S9 respectively.On the basis of these Figures we observe that there are fairly good agreement between speeds of sound data obtained in this work and the literature [31][32][33] .The calculated κφ values reported in Tables S1-S3 indicate that for all the investigated systems their values decrease with increasing of the betaine concentration.
The apparent molar isentropic compressibility at infinite dilution κφ 0 was also determined from fitting of κφ values to the following relation 38 : here S K is an experimental slope indicative of solute -solvent interactions.The values of κφ 0 and S K together with the standard deviation σ (κφ 0 ) obtained from least square fitting of κφ values to betaine molality m are reported in Table 14.The plots of obtained κφ values against betaine molality shown in Figures S13-S15 indicate that κφ values were satisfactorily correlated with Eq. 21.In Fig. 6, the κφ 0 values were plotted against concentration of polymer or salts for different studied systems.It is seen from Table 14 and Fig. 6 that the κφ 0 values for the studied systems decrease with increasing concentration of betaine.Also these values decrease with increase of polymer or salts concentration.The trend observed in variation of κ 0 ϕ or κφ values again implies that while there is a strong interaction between polymer or salts and water, the betaine-polymer and betaine-salts interactions are unfavorable.These results are consistent with the ones we deduced from volumetric studies regarding solute-solvent and solute-solute interactions.
The transfer molar isotropic compressibility (Δ tr kφ 0 ) of betaine from water to aqueous PEGDME 250 or phosphate salts solutions at infinite dilution were calculated by the help of the following relation: For binary aqueous PEGDME 250 , K 3 PO 4 , and K 2 HPO 4 solutions, the κ 0 ϕ values have been given previously [30][31][32] .The corresponding κ 0 ϕ value for aqueous betaine is given in Table 14.Similar to transfer apparent molar volume values, the obtained negative Δ tr kφ 0 values reported in Table 14 indicate strong polymer or salt water interactions implying unfavorable betaine-polymer and betaine-salt interactions which can be regarded as a main reason for biphasic formation of these systems.Therefore, acoustic studies also give the same results as we obtained from volumetric and isopiestic studies regarding the solute-solute interactions in the investigated systems.

Conclusion
In this work, to investigate a possible interactions between betaine and PEGDME 250 , K 3 PO 4 or K 2 HPO 4 in aqueous media, water activity measurements were made on the aqueous systems composed of {betaine + PEGDME 250 }, {betaine + K 3 PO 4 } and {betaine + K 2 HPO 4 } at T = 298.15K and atmospheric pressure by the isopiestic method.The experimental water iso-activity lines showed that these systems have negative deviation from the semi-ideal www.nature.com/scientificreports/solutions implying unfavorable interactions between betaine and polymer or salts.This means that these ternary solutions have tendency to form two-phase systems, which was confirmed experimentally.From thermodynamic treatment of water activity data, activity coefficients values for betaine, PEGDME 250 , K 3 PO 4 and K 2 HPO 4 in binary and ternary solutions were determined and these values were used to calculate transfer Gibbs energies from binary to ternary solutions.The positive transfer Gibbs energy values obtained for all the studied systems confirm the unfavorable interaction between betaine and PEGDME 250 , K 3 PO 4 or K 2 HPO 4 .We found further evidence for these interactions by studying volumetric and acoustic properties of the systems by measuring density and speed of sound values.It was found that both of the transfer molar volume ( � tr V 0 ϕ ) and transfer partial molar isentropic compressibility ( � tr K 0 ϕ ) for transferring the betaine from water to polymer or salts solutions have negative values; therefore, the same conclusion can be made regarding unfavorable interaction between betaine and polymer or salts and possible phase separation of these ternary solutions as predicted from isopiestic studies.The results obtained in this work serve as a basis for the development of environmentally benign two-phase systems for extraction of drugs and other biomaterials from aqueous media.Table 14.The obtained values of isentropic compressibility at infinite dilution (kφ 0 ), experimental slopes (S k ), standard transfer isentropic compressibility (Δ tr kφ 0 ) and standard deviation for isentropic compressibility σ (kφ) for different ternary aqueous betaine solutions at T = 298.15Ka .a Standard uncertainties u for temperature, pressure, and molality are u(T) = 0.01 K, u(P) = 0.01 kPa, and u(m) = 0.003 mol kg −1 , respectively with confidence level of 0.95.b m is the molality of betaine in water.c σ (kϕ) = water activity (a W ), osmotic coefficient ( φ R ) and vapor pressure (p) for the {betaine (1) + K 3 PO 4 (2) + H 2 O (3)} system at T = 298.15K and P ≈ 85 kPa a .a Standard uncertainties u for temperature, pressure, mass fraction, and molality are u(T) = 0.1 K, u(P) = 0.01 kPa, u(w) = 0.002, and u(m) = 0.003 mol kg−1  , respectively with confidence level of 0.95.The combined standard uncertainty of vapor pressure, water activities, and the osmotic coefficient are u (p) = 0.02 kPa, u (a w ) = 0.002, and u (ϕ R ) = 0.02, respectively with confidence level of 0.68.b w 1 and m 1 are mass fraction and molality of betaine, respectively.c w 2 and m 2 are mass fraction and molality of K 3 PO 4 , respectively.
. a Standard uncertainties u for temperature, pressure, and molality are u(T) = 0.01 K, u(P) = 0.01 kPa, and u(m) = 0.003 mol kg−1  , respectively with confidence level of 0.95.The combined standard uncertainty of density and speed of sound are u c (ρ) = 0.15 kg m −3 and u c (u) = 0.9 m s −1 , respectively with confidence level of 0.68.b m is the betaine molality.

b m/mol kg − 1 ρ. 10 - 3 /kg m − 3 u/m s − 1
15 K and P ≈ 85 kPa a .a Standard uncertainties u for temperature, pressure, and molality are u(T) = 0.01 K, u(P) = 0.01 kPa, and u(m) = 0.003 mol kg −1 , respectively with confidence level of 0.95.The combined standard uncertainty of density and speed of sound are u c (ρ) = 0.15 kg m −3 and u c (u) = 0.9 m s −1 , respectively with confidence level of 0.68.b m b is the betaine molalities dissolved per kg of (PEGDME 250 + water).c m p is the molality of polymer in water.b m b /mol kg −1 ρ.10 -3 /kg m −3 u/m s −1 The values of density (ρ) and speed of sound (u) for the betaine in aqueous (K 3 PO 4 + water) solutions at T = 298.15K and P ≈ 85kPa a .a Standard uncertainties u for temperature, pressure, and molality are u(T) = 0.01 K, u(P) = 0.01 kPa, and u(m) = 0.003 mol kg−1  , respectively with confidence level of 0.95.The combined standard uncertainty of density and speed of sound are u c (ρ) = 0.15 kg m −3 and u c (u) = 0.9 m s −1 , respectively with confidence level of 0.68.b m b is the betaine molalities dissolved per kg of (K 3 PO 4 + water).c m K3PO4 is the molality of K 3 PO 4 in water.b m b /mol kg −1 ρ.10 -3 /kg m −3 u/m s −1 a .a Standard uncertainties u for temperature, pressure, and molality are u(T) = 0.01 K, u(P) = 0.01 kPa, and u(m) = 0.003 mol kg−1  , respectively with confidence level of 0.95.The combined standard uncertainty of density and speed of sound are u c (ρ) = 0.15 kg m −3 and u c (u) = 0.9 m s −1 , respectively with confidence level of 0.68.b m b is the betaine molalities dissolved per kg of (K 2 HPO 4 + water).c m K2HPO4 is the molality of K 2 HPO 4 in water.

Table 1 .
Descriptions of the used chemicals.a As stated by the supplier.b Water contents were determined by Karl-Fischer method.

Table 2 .
Mass b

Table 3 .
Mass b w 1 and m 1 are mass fraction and molality of betaine, respectively.cw 2 and m 2 are mass fraction and molality of PEGDME 250 , respectively.bw

Table 7 .
Gibbs energies of transfer, G i tr , of PEGDME 250 , K 3 PO 4 and K 2 HPO 4 from water to aqueous betaine solutions at 298.15 K a .m 2 is molality of PEGDME 250 , or salts (K 3 PO 4 and K 2 HPO 4 ).

Table 8 .
Values of pair (g 12 ) and triple parameters (g 112 and g 122 ) together with the salting constant (k s ). a m 2 is molality of PEGDME 250 , K 3 PO 4 or K 2 HPO 4 .

Table 12 .
The values of density (ρ) and speed of sound (u) for the betaine in aqueous (K 2 HPO 4 + water) solutions at T = 298.15K and P ≈ 85kPa 3