Temperature dependence of limiting activity coefficients and Henry’s law constants of cyclic and open-chain ethers in water

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Fluid Phase Equilibria 262 (2007) 121–136

Temperature dependence of limiting activity coefficients and Henry’s law constants of cyclic and open-chain ethers in water Daniel Ondo, Vladim´ır Dohnal ∗ Department of Physical Chemistry, Institute of Chemical Technology, Technick´a 5, 16628 Prague 6, Czech Republic Received 28 June 2007; received in revised form 23 August 2007; accepted 24 August 2007 Available online 30 August 2007

Abstract Limiting activity coefficients (γ1∞ ) of seven selected ethers (diethyl ether, tetrahydrofuran, tetrahydropyran, dimethoxymethane, 1,2dimethoxyethane, 1,3-dioxolane, and 1,4-dioxane) in water were measured at several temperatures in the range from 273 to 373 K. Six experimental techniques were employed for the purpose. A comprehensive review is presented of experimental data on the limiting activity coefficients, infinite dilution partial molar excess enthalpies and heat capacities of these aqueous solutes. For each ether, the compiled data were critically evaluated and together with the data measured in this work correlated with a suitable model equation providing adequate simultaneous description of the equilibrium measurements and the calorimetric information. As a result, a recommended thermodynamically consistent temperature dependence of γ1∞ of superior accuracy was established in the range from the melting point to the normal boiling point of water. Analogous recommendations were derived also for the temperature dependence of their Henry’s law constants (KH ). Variations of the infinite dilution thermodynamic properties of aqueous ethers with temperature and ether molecular structure are subsequently discussed. The performance of three group contribution approaches, namely the modified UNIFAC, and the methods of Cabani et al. and Plyasunov et al., to predict γ1∞ for the aqueous ethers was tested and found quite unsatisfactory. © 2007 Elsevier B.V. All rights reserved. Keywords: Infinite dilution activity coefficient; Henry’s law constant; Ether; Water; Temperature dependence

1. Introduction Ethers are nonassociated amphiphilic oxygenated compounds of moderate polarity. These substances are typically volatile liquids with good chemical stability, low viscosity and strong solvency exhibiting a substantial compatibility with water and a number of organic solvents. Such properties make ethers valuable process media and solvents in many industries. The proton accepting ability of ethers makes them also efficient solvent stabilizers and their favorable combustion characteristics promote their massive and still increasing use as automotive fuel additives. Many ethers are large volume production chemicals which, due to their high vapor pressures and aqueous solubilities, are released in significant amounts to the environment. The environmental contamination by ethers, in particular of surface and ground water, is therefore growing concern. The adverse

∗

Corresponding author. Tel.: +420 220 444 297; fax: +420 220 444 333. E-mail address: [email protected] (V. Dohnal).

0378-3812/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2007.08.013

impact of ether contamination ranges from bad taste characteristics of drinking water to acute toxic and possibly carcinogen effects [1]. To model and predict phase and chemical equilibria, transport effects and other phenomena involved in environmental processes, remediation of contaminated sites, and industrial separations, the thermodynamic properties of highly dilute aqueous solutions of ethers, such as ether limiting activity coefficients (γ1∞ ) or Henry’s law constants in water (KH ), are of essential importance. Accurate knowledge of the thermodynamic quantities of the dissolution and hydration of ethers and their variation with temperature is of extreme interest also for theoretical reasons, in particular for understanding the hydrophobic effect. Recently, we have presented several detailed studies on the temperature dependence of limiting activity coefficients and Henry’s law constants of various oxygen- or nitrogen-containing aqueous solutes of semihydrophobic character [2,3], our systematic effort being especially devoted to alkanols [4–6]. In this work, we extend the scope of this systematic investigation to ethers. Literature data on γ1∞ (T ) or KH (T ) for these aqueous solutes are often fragmentory or insufficiently accurate, which

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calls for experimental reexamination [7]. Seven environmentally important and large volume production ethers were selected for the present study, namely: diethyl ether (DEE), tetrahydrofuran (THF), tetrahydropyran (THP), dimethoxymethane (DMM), 1,2-dimethoxyethane (DME), 1,3-dioxolane (DXL), and 1,4dioxane (DOX). As indicated in Fig. 1, the involvement of these various cyclic and open-chain monoethers and diethers in this selection makes possible various molecular structure effects on thermodynamic behavior of aqueous ethers to be captured. For the selected aqueous ethers we report here results of our accurate measurements of limiting activity coefficients which were performed by several suitable experimental techniques as a function of temperature. The present experimental work is further amended by a comprehensive compilation and critical evaluation of literature experimental data on limiting activity coefficient and related thermal dissolution properties-limiting ¯ E,∞ . ¯ E,∞ , and heat capacity, C partial molar excess enthalpy, H 1 P,1 All the data, measured in this work and taken from literature, are subsequently processed by a simultaneous thermodynamically consistent correlation. The treatment results in a recommended temperature dependence of these infinite dilution properties

which has superior accuracy and is valid in the range from the melting to the normal boiling temperature of water. Analogous recommendations are further generated for the temperature dependence of the Henry’s law constants, hydration enthalpies, and heat capacities. Finally, variation of γ1∞ and KH with temperature and ether molecular structure is discussed and their prediction tested using three group contribution schemes.

2. Experimental 2.1. Materials The solutes studied in this work were analar grade commercial chemicals used without further purification: diethyl ether (Penta, p.a. 99.7%); dimethoxymethane (Aldrich, 99.9%); tetrahydrofuran (Aldrich, 99.9%); tetrahydropyran (Fluka, 99%); 1,2-dimethoxyethane (Fluka, >99.5%); 1,3-dioxolane (Aldrich, 99%); 1,4-dioxane (Aldrich, 99.8%). Water used as the solvent was distilled and subsequently treated by Milli-Q Water Purification System (Millipore, USA).

Fig. 1. Molecular structure of the ethers studied and structural changes linking them: (a) cyclization; (b) methylene addition; (c) replacement of methylene with the second ethereal oxygen.

D. Ondo, V. Dohnal / Fluid Phase Equilibria 262 (2007) 121–136

2.2. Apparatus and procedure To measure the limiting activity coefficients, six experimental techniques were employed in this work: inert gas stripping (IGS) [8–10], headspace analysis in two variants—the classical (HSA) [2] and the relative (RHSA) [6], non-steady-state gas–liquid chromatography (NSGLC) [11], Rayleigh distillation (RDIST) [12] and the method of circulation still (CIRC) [13]. These techniques were applied alternatively, the choice being in accordance with system characteristics and conditions, in order to achieve the best measurement accuracy. Since the methods were used previously and the respective instrumentation and experimental procedures were described in detail, given here is only a brief overview of their principles supplemented with particulars, if any, that are specific to the present application. 2.2.1. Inert gas stripping (IGS) The IGS method involves measuring the rate of elution of a solute as an inert gas is passed through a highly dilute solution where after equilibration the outlet gas is periodically analyzed for the solute content by gas chromatography [8–10]. The IGS method was used for all solutes studied in this work at temperatures ranging from 273.35 to 338.15 K. To optimize the elution rate, the capacity of equilibrium cell and the flow rate of stripping gas were selected depending on the system and conditions. For this purpose, three water thermostated equilibrium glass cells of similar design, but of different capacities 13/17, 30/47 and 160/190 cm3 (liquid loading/total volume) and stripping gas flow rates from 6 to 22 cm3 min−1 were used. Presaturation of the stripping gas with the solvent vapor was used in all experiments. To perform the analysis, a HP 5890 II GC equipped with a 15 m long wide-bore capillary column DB-WAX was employed at oven temperature 40 or 50 ◦ C. 2.2.2. Headspace analysis methods (HSA, RHSA) The classical variant of headspace analysis (HSA) method for determination of γ1∞ consists of gas chromatographic measurement of the equilibrium solute partial pressure above the liquid solution of a known composition, the GC response being calibrated with the equilibrium vapor above the pure solute [2]. For the present systems, the HSA measurements were carried out for DOX in a typical range of mole fractions from 0.0004 to 0.003 and at temperatures 288.15, 298.15 and 308.15 K. For most solutes studied in this work, the headspace analysis method was applied in its relative variant (RHSA) which has been implemented recently in our laboratory [6]. In RHSA the calibration is accomplished using the vapor-phase above the same highly dilute solution as that under study but at some different temperature where γ1∞ has been already accurately determined by another technique. The gas chromatography analysis for both variants was done in the same manner as in the IGS method. 2.2.3. Non-steady-state GLC (NSGLC) In the non-steady-state GLC, the amount of a volatile stationary phase (solvent) in the column decreases as the solvent continuously evaporates into the carrier gas. As a consequence, the retention of gradually injected solute samples decreases.

123

From the slope of the linear dependence of the retention time on the time of injection γ1∞ can be directly calculated [11]. The NSGLC experiments were performed at 328.15 and 338.15 K for all studied solutes whose limiting relative volatility from water is in the applicability range of NSGLC method [11]. The gas chromatograph employed was Agilent 6890 Plus and the solutes were injected in the form of their dilute aqueous solutions by means of a 7683 Series Agilent AutoInjector. 2.2.4. Rayleigh distillation method (RDIST) In RDIST method, a highly dilute binary solution is subjected to the one-stage flask-to-flask distillation at a given temperature and the changes of the solution mass and the solute concentration are measured. The distillation is accomplished by bubbling an inert gas through the thermostated solution and the analysis is carried out by gas chromatography [12]. In this work, the RDIST method was applied at 348.15 K for aqueous solutions of DME, DXL and DOX, the extent of distillation being chosen according to the error analysis [12] to minimize the effect of analytical errors. To analyse RDIST samples, the same GC apparatus, analytical procedure and conditions as for IGS, HSA and RHSA experiments were used. The samples were dosed by a HP 7673 AutoInjector. 2.2.5. Circulation still method (CIRC) In this method, a VLE circulation still operated at constant pressure is employed to provide samples of corresponding vapor and liquid compositions in the region of high dilution, no measurement of temperature being needed as the boiling temperature of the solution is indistinguishable from that of the neat solvent [13]. The CIRC measurements were done for DME, DXL and DOX at a temperature 372.55 K. The samples were analyzed by an Agilent 6890 Plus GC equipped with a 7683 Series Agilent Autoinjector using a 0.5 m Porapak Q packed column at oven temperature 180 ◦ C. 3. Results of measurements Primary measurements by the various techniques were processed to obtain values of limiting activity coefficients. The calculations have been described in detail previously and respective relations can be found in our papers cited above. Due attention was paid to the selection of sufficiently accurate solute vapor pressure data in order to minimize the resulting uncertainty of γ1∞ values. The selected vapor pressure data are given in Table 1 as parameters of the Wagner equation or the Antoine equation. Note that for tetrahydropyran and 1,3-dioxolane these parameters were obtained in this work by fitting simultaneously vapor pressure data from several literature sources. Water vapor pressure was calculated from the reference equation of Wagner and Pruss [14]. The gas-phase non-ideality was accounted for by the truncated virial equation of state, the second virial coefficients being obtained from the Hayden–O’Connell [15] correlation with parameters from Prausnitz et al. [16] and CDATA [17]. L to calculate the Poynting The pure liquid molar volumes Vmi correction were taken from CDATA [17]. For the systems and

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Table 1 Constants of vapor pressure equations used in the calculation of γ1∞ and KH Solute

Equation

Temperature range (K)

a

b

Diethyl ether Tetrahydrofuran Tetrahydropyran Dimethoxymethane 1,2-Dimethoxyethane 1,3-Dioxolane 1,4-Dioxane

Wagnera

250–466 296–373 263–403 283–398 305–392 280–370 285–375

−7.29916 −7.12856 −6.17065 −9.37543 −8.08980 −7.99289 14.74390

1.24828 1.47916 0.60372 5.67097 −2.53555 2.92635 3357.1

a b c d e

Wagnera Wagnera,b Wagnera Wagnerd Wagnera,b Antoinee

c −2.91931 −2.90568 0.11274 −8.25359 −3.48091 −5.55143 −42.888

d

Tc (K)

Pc (kPa)

Source

−3.36740 −1.77712 −6.30849 6.32500 −3.65036 2.43936

466.74 540.1 572.2c 491 537 552c

3646 5190 4770c 5019 3960 5860

[17] [55] [56–58] [59] [60] [61–63] [17]

ln(P s /Pc ) = (aF + bF 1.5 + cF 3 + dF 6 )Tc /T , where F = 1 − T/Tc . Simultaneous fit of vapor pressure data from cited sources. From Ref. [17]. ln(P s /Pc ) = (aF + bF 1.5 + cF 2.5 + dF 5 )Tc /T , where F = 1 − T/Tc . ln(P s /kPa) = a − b/(c + T/K).

conditions under study the vapor-phase nonideality corrections are in most cases quite small, not exceeding 2%. DEE is however an exception here; at 328.15 K, the highest experimental temperature for this solute which is well above its normal boiling point, the vapor-phase nonideality correction reaches 6%. The limiting activity coefficients determined in this work are listed in Table 2, along with their relative standard uncertainties estimated by the error propagation law. As seen from Table 2, most limiting activity coefficients were determined fairly precisely; their uncertainty is typically 2–3 % and never exceeds 5%. 4. Determination of temperature dependence of γ1∞ and KH

S=

nG  [ln γ ∞ (exp) − ln γ ∞ (calc)]2 1

i=1

+

1

(1)

where τ ≡ T/T0 , T0 = 298.15 K was employed to describe the temperature dependence of γ1∞ . If all four adjustable parameters ¯ E,∞ linearly depenA, B, C and D are used, Eq. (1) implies the C P,1 dent on temperature. All four parameters could be considered here for tetrahydrofuran, 1,2-dimethoxyethane and 1,4-dioxane for which the experimental information on temperature depen¯ E,∞ was available. However, for other solutes studied, dence of C P,1 ¯ E,∞ was available only at a single temperature of 298.15 K and C P,1 hence the three-parameter form of Eq. (1)(D = 0), implying a ¯ E,∞ , had to be employed. constant C P,1

i

1

i

¯ E,∞ ) s2 (H 1 i

2 i

nC ¯ E,∞ ¯ E,∞ (calc)]  [CP,1 (exp) − C P,1 i=1

To establish a truly reliable temperature dependence of the limiting activity coefficients, the newly measured values for each aqueous ether studied were supplemented by additional criti¯ E,∞ cally evaluated data on γ1∞ and related thermal properties H 1 E,∞ ¯ and C P,1 compiled from the literature and all these data were then correlated simultaneously by a suitable fitting equation. Comprehensive databases gathered on respective properties and used in this treatment are given in Appendices A–C. Note that only original experimental values were considered here, those estimated by prediction methods or extrapolated from measurements on concentrated solutions were disregarded. The following four-parameter equation:

1

s2 (ln γ1∞ )i

nH ¯ E,∞  ¯ E,∞ (calc)]2 [H (exp) − H i=1

+

4.1. Recommended γ1∞ (T ) data

lnγ1∞ = A + B/τ + Clnτ + Dτ

The adjustable parameters were evaluated by the simultaneous correlation of all available data using the weighted least-squares method. In the least-squares fitting, the objective function minimized was given as

¯ E,∞ ) s2 (C P,1 i

(2)

with data being weighted according to their uncertainties given in Appendices A–C. These uncertainties correspond to standard deviations and comprise contributions from all possible sources of error, both random and systematic. For estimation and assignment of these data uncertainties we followed the policy and procedure which have been described in detail and extensively applied in our previous studies [2,5,6]. In essential, the assigned uncertainties should account for any scatter and/or disparity of existing data, which in other words means that the coherence of all data within their uncertainty bounds should be obtained. As a main criterion of this coherence the residual sum of squares Smin for the simultaneous representation of all data by Eq. (1) was required to range within statistically plausible bounds, i.e. within the respective critical values of χ2 : 2 2 χα/2 (n − p) < Smin < χ1−α/2 (n − p),

(3)

where n = nG + nH + nC is the total number of data points, p the number of adjustable parameters of Eq. (1), and α is the significance level (α = 0.05). The treatment enabled us to discriminate efficiently between existing data and to obtain their thermodynamically consistent description of superior quality. Table 3 gives the calculated values of parameters of Eq. (1), the number of experimental data used for the optimization, the standard deviation of fit and the calculated temperature Tmax at which γ1∞ is maximum. Eq. (1) with parameters from Table 3 is considered to establish the recommended temperature dependence of γ1∞ valid in the range

D. Ondo, V. Dohnal / Fluid Phase Equilibria 262 (2007) 121–136 Table 2 Experimental values of limiting activity coefficients γ1∞ of ethers in water determined in this work along with their relative standard uncertainties srel (γ1∞ ) Solute (1)

T (K)

γ1∞

srel (γ1∞ ) (%)

Techniquea

Diethyl ether

273.35 278.15 283.15 288.15 293.15 298.15 303.15 308.15 318.15 328.15 273.35 278.15 288.15 298.15 308.15 318.15 328.15 328.15 273.35 278.15 288.15 298.15 308.15 318.15 328.15 273.35 283.15 293.15 298.15 303.15 313.15 323.15 333.15 273.35 278.15 288.15 298.15 308.15 318.15 328.15 328.15 338.15 338.15 348.15 372.50 273.35 278.15 288.15 298.15 308.15 318.15 328.15 338.15 338.15 348.15 372.55 273.35 278.15 288.15 288.15 298.15 308.15 308.15

33.9 41.2 49.4 58.5 68.6 77.1 88.8 98.8 121.0 139.5 9.74 11.1 14.3 17.6 20.6 24.0 26.8 28.1 39.6 44.9 58.0 70.5 83.4 96.2 107.5 10.8 13.7 15.6 16.9 18.0 19.9 22.0 23.9 1.59 2.02 3.01 4.22 5.61 7.25 9.12 8.81 11.2 11.1 12.2 14.9 8.13 8.48 9.12 9.71 10.2 10.6 10.9 11.0 11.4 11.3 11.2 3.53 4.00 4.75 5.17 5.88 6.15 6.70

2 2 1 1 1 1 1 2 2 2 2 2 1 1 2 2 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 2 3 3 3 3 2 5 2 1 1 1 1 1 2 3 2 2 3 3 3 3 5 5 3 5

IGS IGS IGS IGS IGS IGS IGS IGS IGS IGS RHSAb RHSAb IGS IGS IGS IGS NSGLC NSGLC RHSAc RHSAc IGS IGS IGS IGS IGS IGS IGS IGS IGS IGS IGS IGS IGS RHSAd RHSAd RHSAd RHSAd RHSAd IGS NSGLC NSGLC NSGLC NSGLC RDIST CIRC RHSAe RHSAe RHSAe RHSAe RHSAe IGS IGS NSGLC NSGLC RDIST CIRC RHSAf,g RHSAf,g RHSAf HSA HSA RHSAf HSA

Tetrahydrofuran

Tetrahydropyran

Dimethoxymethane

1,2-Dimethoxyethane

1,3-Dioxolane

1,4-Dioxane

125

Table 2 (Continued ) Solute (1)

T (K)

γ1∞

srel (γ1∞ ) (%)

Techniquea

318.15 328.15 338.15 338.15 348.15 372.50

6.82 7.76 8.27 8.22 8.43 10.3

3 3 3 2 5 3

RHSAf NSGLC NSGLC IGS RDIST CIRC

a IGS, inert gas stripping; HSA, classical headspace analysis; RHSA, relative HSA; NSGLC, non-steady-state GLC; RDIST, Rayleigh distillation; CIRC, circulation still. b Reference value of γ ∞ (288.15 K) = 14.3 from this work (IGS). 1 c Reference value of γ ∞ (288.15 K) = 58.0 from this work (IGS). 1 d Reference value of γ ∞ (318.15 K) = 7.25 from this work (IGS). 1 e Reference value of γ ∞ (318.15 K) = 10.6 from this work (IGS). 1 f Reference value of γ ∞ (298.15 K) = 5.45 from Park et al. [64]. 1 g Based on the hypothetical subcooled liquid vapor pressure extrapolated from Antoine equation given in Table 1.

from the melting point to the normal boiling point of water. The relative standard uncertainty (68 % confidence level) of the recommended γ1∞ values, as inferred by the error propagation from the parameter variance-covariance matrix, does not exceed 3 %. In Table 4 listed are for a quick reference and illustration the recommended values of the various infinite dilution excess thermodynamic functions at 298.15 K. 4.2. Recommended KH (T ) data To obtain the recommendation for the temperature dependence of the Henry’s law constant and the related hydration properties, the γ1∞ (T ) dependence determined for each solute was combined with existing data on respective pure solute properties using the following relations:  L (P s − P s )  B11 P1s + Vm1 2 1 KH = γ1∞ P1s exp , (4) RT   KH , (5) = RT ln hyd G∞ 1 P0 ¯ E,∞ − vap H10 , hyd H1∞ = H 1

(6)

G,0 ∞ ∞ ¯ P,1 ¯ E,∞ + (CL,• − CG,0 ), =C − CP,1 =C hyd CP,1 P,1 P,1 P,1

(7)

L the pure where Pis is the pure component vapor pressures, Vm1 liquid solute molar volume, B11 the pure solute second virial coefficient, vap H10 the pure solute vaporization enthalpy to the L,• ideal gas standard state (standard vaporization enthalpy), CP,1 G,0 and CP,1 are the pure solute heat capacities at the liquid state ¯ ∞ is the and the ideal gas standard state, respectively, and C P,1 solute partial molar heat capacity at infinite dilution in water. The infinite dilution hydration quantities correspond to a transfer of the solute from the pure ideal gas state at standard P 0 = 100 kPa to a hypothetical infinitely dilute solution of unity solute mole fraction (x1 = 1). The vapor pressures of pure solutes (Table 1), their fugacity coefficients and liquid molar volumes needed for the evaluation of KH according to Eq. (4), are available for the entire

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Table 3 ¯ E,∞ data, number of respective underlying data points nG , nH , nC , the standard deviation ¯ E,∞ and C Parameters of Eq. (1) obtained by simultaneous treatment of γ1∞ , H 1 P,1 of fit s a and temperature Tmax at which γ1∞ is maximum Solute (1)

A

B

C

Diethyl ether Tetrahydrofuran Tetrahydropyran Dimethoxymethane 1,2-Dimethoxyethane 1,3-Dioxolane 1,4-Dioxane

41.4709 29.9281 37.8770 22.3287 33.5261 11.2697 13.5041

−37.1142 −21.7795 −33.6086 −19.5036 −29.6502 −8.9997 −5.3661

−29.2312 −10.4706 −27.7086 −15.2756 −17.9444 −7.1526 4.9951

a

s = [Smin /(n − p)]

1/2

D −5.2733 −2.4337 −6.4313

nG /nH /nC

s

Tmax (K)

45/8/2 15/9/13 9/4/2 9/8/1 13/5/20 12/2/1 21/13/10

1.01 0.99 1.27 0.82 0.85 1.20 1.03

378.6 378.4 361.6 380.7 414.5 375.1 411.7

, where S is given by Eq. (2).

Table 4 Recommended values of excess thermodynamic functions at infinite dilution calculated from Eq. (1) with parameters from Table 3 for ethers in water at 298.15 K Solute(1)

γ1∞

¯ E,∞ (kJ mol−1 ) G 1

¯ E,∞ (kJ mol−1 ) H 1

¯ E,∞ (J mol−1 K−1 ) C P,1

Diethyl ether Tetrahydrofuran Tetrahydropyran Dimethoxymethane 1,2-Dimethoxyethane 1,3-Dioxolane 1,4-Dioxane

78.0 ± 0.3 17.7 ± 0.1 71.4 ± 0.7 16.9 ± 0.1 4.23 ± 0.03 9.68 ± 0.05 5.51 ± 0.05

10.80 ± 0.01 7.13 ± 0.01 10.58 ± 0.03 7.00 ± 0.02 3.58 ± 0.02 5.63 ± 0.01 4.23 ± 0.02

−19.54 ± 0.04 −14.96 ± 0.06 −14.63 ± 0.11 −10.48 ± 0.03 −22.98 ± 0.02 −4.58 ± 0.05 −9.74 ± 0.06

243 ± 4 175 ± 1 230 ± 4 127 ± 3 190 ± 2 60 ± 4 66 ± 2

temperature range of interest (273–373 K) and were obtained in the same way as in Section 3. Thermal properties of pure solutes, however, are mostly available only at a single temperature 298.15 K or in narrow near-ambient temperature ranges, this fact limiting accordingly the temperatures at which the hydration thermal properties can be calculated. The standard enthalpies of vaporization were obtained from reliable standard vaporization internal energy values given by Majer and Svoboda [18] except for DXL and DMM for which they were taken from Fletcher et al. [19] and from Plyasunova et al. [20], respectively. To calculate the hydration heat capacities, we followed two alternative routes depending on the data available. For solutes for which solution heat capacity measurements were available ∞ was obtained at a single temperature only (298.15 K), hyd CP,1 E,∞ L,• G,0 ¯ combining the recommended C P,1 values with CP,1 and CP,1 . For THF, DME, and DOX for which precise experimental values ¯ ∞ are available from Swenson et al. [21] in a broad temperof C P,1 ∞ was obtained by combining directly their ature range, hyd CP,1 ¯ ∞ values with CG,0 data. The reason for taking the original C P,1 P,1 ∞ (T ) latter route was to avoid any possible distortion of hyd CP,1 ¯ E,∞ by Eq. (1). The heat dependence due to the previous fit of C P,1 capacities of pure liquids were taken from the compilation of Z´abransk´y et al. [22,23] and those of ideal gas for most solutes G,0 from CDATA [17]. For DMM we took CP,1 from [20], and for DXL and THP from work of Dorofeeva [24,25], respectively, smoothing the discrete data with the exponential equation of Bureˇs et al. [26]. In order to establish the recommended temperature dependence of hydration properties in a thermodynamically consistent ∞ analytical form, we fitted the data on KH , hyd H1∞ and hyd CP,1

simultaneously to the following equation: ln KH = AH +

BH + CH ln τ + DH τ. τ

(8)

All four parameters could be evaluated only for 1,4-dioxane ∞ temperature dependence is evident; for for which the hyd CP,1 other solutes studied this dependence appears insignificant or the respective data are lacking. The input data for the fits are listed in Appendix D. To provide a simplified way of data weighing, the sum of squares of relative deviations was minimized. The calculated parameters of Eq. (8), along with the corresponding relative standard deviations of the fit srel , are listed in Table 5. In Table 6 given are the values of the recommended infinite dilution hydration functions calculated at 298.15 K from Eq. (8) and their comparison with recent recommendations by PlyaTable 5 Parameters of Eq. (8) obtained by simultaneous treatment of KH , hyd H1∞ and ∞ along with the respective standard deviation of fit s hyd CP,1 rel Solute (1)

AH

BH

CH

Diethyl ether Tetrahydrofuran Tetrahydropyran Dimethoxymethane 1,2-Dimethoxyethane 1,3-Dioxolane 1,4-Dioxane

63.0358 52.0575 60.1345 46.2628 60.3006 33.7721 37.3025

−54.4411 −46.1179 −53.6292 −39.4880 −56.6443 −28.9054 −24.3009

−35.4515 −27.1657 −33.7096 −23.3359 −32.3970 −12.8373 4.8084

nG

srel

−9.7034

0.002 0.010 0.002 0.006 0.012 0.002 0.009

2 1/2 a nsHrel = [Smin∞/(n − p)] ; S∞ = i=1 [K2 H,i (calc)/KH,i (exp) − 1] + [ H (calc)/ H (exp) − 1] + hyd 1,i hyd 1,i i=1 nC 2 ∞ ∞ i=1

[hyd CP,1,i (calc)/hyd CP,1,i (exp) − 1] .

a

DH

D. Ondo, V. Dohnal / Fluid Phase Equilibria 262 (2007) 121–136

127

Table 6 Recommended values of hydration thermodynamic functions calculated from Eq. (8) with parameters from Table 5 for ethers in water at 298.15 K and their comparison with those given by Plyasunov et al. [7] (in parenthesis, if available)a Solute (1)

KH (kPa)

−1 hyd G∞ 1 (kJ mol )

hyd H1∞ (kJ mol−1 )

∞ (J mol−1 K−1 ) hyd CP,1

Diethyl ether Tetrahydrofuran Tetrahydropyran Dimethoxymethane 1,2-Dimethoxyethane 1,3-Dioxolane 1,4-Dioxane

5405 (5270 ± 640) 380 668 876 (800 ± 160) 38.7 (41.9 ± 8.5) 130 27.1

9.89 (9.83 ± 0.3) 3.31 4.71 5.38 (5.16 ± 0.5) −2.35 (−2.15 ± 0.5) 0.65 −3.24

−47.07 (−46.41 ± 1.0) −46.98 −49.38 −40.04 (−41.12 ± 2.0) −60.11 (−59.37 ± 0.5) −39.83 −48.11

295 (304 ± 15) 226 280 194 (180 ± 16) 269 (253 ± 10) 107 121

a

Converted from the molality scale used in Plyasunov et al. [7] to the mole fraction scale used in the present work.

sunov et al. [7] for the open-chain ethers. Mutual agreement of the two recommendations is very good; the present values should be however preferred as they are significantly more accurate. 5. Discussion 5.1. Data and correlation assessment The treatment described in Section 4.1 revealed that most γ1∞ data collected in Appendix A agree within a reasonable scatter. Some data, however, deviate grossly (> 0.2 in ln γ1∞ ) from the fits. According to the evaluation policy we adopted, such data were not strictly rejected, but rather labelled with a larger uncertainty, which reduced appropriately their statistical weight in the treatment. The grossly deviating points, which are indisputably

Fig. 2. Limiting activity coefficient ln γ1∞ of diethyl ether(1) in water(2) as a function of temperature. Experimental values from this work (Table 2): () IGS; data from literature: () Haggard [30]; () Shaffer et al. [31]; () Kety [32]; () Eger et al. [33]; (♦) Signer et al. [27]; () Bachofen et al. [65]; () Villamanan et al. [43]; () Lamarche and Droste [28]; (䊉) Guitart et al. [66]; ( ) Guitart [34]; ( ) Nielsen et al. [67]; () Zhang [68]; ( ) Sherman et al. [29]; ¯ E,∞ , and ( ) Fukuchi et al. [35]; ( ) Atik et al. [36]. Simultaneous fit of γ1∞ , H 1 E,∞ ¯ C P,1 data by Eq. (1): (—) three-parameter equation. The dashed lines indicate the prediction by the Mod. UNIFAC (Dortmund): (– – –) fourth revision [52]; (- - -) fifth revision [53].

subject to large errors, are encountered especially for diethyl ether for which also an exceptionally abundant information from the literature is available (Fig. 2). As seen from Fig. 2, clear outliers are γ1∞ values from Signer et al. [27], Lamarche and Droste [28], and Sherman et al. [29], which appear to be too high, and most older measurements and/or measurements originating from biologically oriented laboratories [30–34], which appear too low. Despite of the numerous outliers, the recommended fit for diethyl ether is pretty stable, being essentially based on accurate measurements of this work and data of Fukuchi et al. [35] and Atik et al. [36] which are all in excellent mutual agreement. Note that the stability of the recommended γ1∞ (T ) was also ¯ E,∞ (T ) data of Dohnal greatly supported by new calorimetric H 1 ˇ and Reh´ak [37] which are fully consistent with the temperature trend of the best γ1∞ data as well as with recent solution heat capacity determinations [38]. A systematic error was further found in the tensimetric measurements of Pividal et al. [39] for THF and the HSA mea¯ E,∞ value surements of Kolb et al. [40] for DOX. The derived H 1 −1 from the former data differs by 4 kJ mol from the calorimetric ¯ E,∞ data reported in literature and the gross positive deviaH 1 tion from the recommended fit for the latter data is analogous to the deviations found recently for measurements of Kolb et al. [40] on 1-butanol [5] and 2-propanol [6]. A few excessive ¯ E,∞ outliers (devideviations were also found for thermal data. H 1 −1 ations 1 kJ mol ) involve an old value tabulated for DMM by Anderson [41], measurements of Nakayama [42] for THF and DOX and a value for DEE extrapolated from heat-of-mixing ¯ E,∞ outliers (deviations measurements of Villamanan [43]. C P,1 > 15 J K−1 mol−1 ) involve data of Cabani et al. [44] for DMM and DME and of Bonner and Cerutti [45] for DOX. As their solution heat capacity measurements were carried out at higher concentrations not sufficiently approximating infinite dilution, one should allow for larger uncertainty of these data. ¯ E,∞ ¯ E,∞ , and C The simultaneous representation of γ1∞ , H 1 P,1 data is illustrated for 1,2-dimethoxyethane in Figs. 3 and 4 and for 1,4-dioxane in Figs. 5 and 6. For these two solutes the four-parameter form of Eq. (1) was used to capture a slight ¯ E,∞ with temperature as experimentally observed. increase of C P,1 In Figs. 3 and 5 one can also note the accord of our γ1∞ measurements carried out by various techniques in a 100 K temperature span.

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D. Ondo, V. Dohnal / Fluid Phase Equilibria 262 (2007) 121–136

Fig. 3. Limiting activity coefficient ln γ1∞ of 1,2-dimethoxyethane(1) in water(2) as a function of temperature. Experimental values from this work (Table 2): ( ) RHSA; () NSGLC; (䊉) IGS; () RDIST; ( ) CIRC; data from ¯ E,∞ data ¯ E,∞ , and C literature: () Cabani et al. [69]. Simultaneous fit of γ1∞ , H P,1 1 by Eq. (1): (—) four-parameter equation. The dashed lines indicate the prediction by the Mod. UNIFAC (Dortmund): (– – –) fourth revision [52]; (- - -) fifth revision [53].

¯ E,∞ (a) and heat capacities Fig. 4. Limiting partial molar excess enthalpies H 1 E,∞ ¯ C (b) of 1,2-dimethoxyethane(1) in water(2) as a function of temperature. P,1

¯ E,∞ values are from Appendix B: ( ) Kusano et al. [70]; () Experimental H 1 ¯ E,∞ values are from Dohnal et al. [71]; () Kustov et al. [72]. Experimental C P,1 Appendix C: () Cabani et al. [44]; () Schr¨oedle et al. [73]; () Swenson et E,∞ ∞ ¯ E,∞ data by ¯ al. [21]; () Slav´ık et al. [38]. Simultaneous fit of γ1 , H1 , and C P,1 Eq. (1): (—) four-parameter equation.

Fig. 5. Limiting activity coefficient ln γ1∞ of 1,4-dioxane(1) in water(2) as a function of temperature. Experimental values from this work (Table 2): ( ) HSA; ( ) RHSA; () NSGLC; (䊉) IGS; () RDIST; ( ) CIRC; and data from literature: () Cabani et al. [61]; () Rohrschneider [74]; () Park et al. [64]; ¯ E,∞ , and (♦) Kolb et al. [40]; () Sherman et al. [29]. Simultaneous fit of γ1∞ , H 1 E,∞ ¯ C P,1 data by Eq. (1): (—) four-parameter equation. The dashed line indicates the prediction by the Mod. UNIFAC (Dortmund) (- - -) (no difference between the parameters of fourth and fifth revision).

¯ E,∞ (a) and heat capacities Fig. 6. Limiting partial molar excess enthalpies H 1 E,∞ ¯ C (b) of 1,4-dioxane(1) in water(2) as a function of temperature. ExperiP,1

¯ E,∞ values are from Appendix B: ( ) Franks et al. [75]; (♦) Belousov mental H 1 and Makarova [76]; () Nakayama [42]; () Cabani et al. [61]; ( ) Arnett et al. [77]; () Dohnal et al. [71]; () Kulikov et al. [78]; ( ) Kustov et al. [72]. ¯ E,∞ values are from Appendix C: () Cabani et al. [79]; () Experimental C P,1 Kiyohara et al. [80]; () Bonner and Cerutti [45]; (♦) Schr¨oedle et al. [73]; () ¯ E,∞ data by Eq. (1): ¯ E,∞ , and C Swenson et al. [21]. Simultaneous fit of γ1∞ , H 1 P,1 (—) four-parameter equation.

D. Ondo, V. Dohnal / Fluid Phase Equilibria 262 (2007) 121–136

Adequate simultaneous fit of infinite dilution equilibrium data and related thermal quantities was obtained also for hydration process properties of all solutes studied (not shown). The treatment ensuring thermodynamic consistent description of the compound information greatly stabilized the fits and allowed sufficiently reliable γ1∞ (T ) and KH (T ) dependences to be obtained from 273 to 373 K even in cases when the underlying equilibrium measurements did not cover fully this entire temperature range. As estimated, the standard uncertainty of the calculated values of γ1∞ and KH at 373.15 K did not exceed 3 and 5%, respectively, for any of the aqueous solutes studied. 5.2. Variation of properties with temperature and molecular structure In Fig. 7, the recommended temperature dependences of γ1∞ and KH for all the ethers studied in this work are plotted for comparison. This figure, along with the data in Tables 3, 4 and 6, discloses essential features of the variations of the properties with temperature and ether molecular structure. Although a detailed analysis of the observed behavior and its molecular interpretation is beyond the scope of this paper, certain inferences in this direction are given below. In the approach adopted here, it is considered that thermodynamic behavior of aqueous ethers, as amphiphilic solutes, is essentially governed by two concurrent effects whose interplay greatly depends on the structure of the ether molecule: (i) the hydrophobic hydration of the nonpolar hydrocarbon moiety promoting water structuring

Fig. 7. Recommended temperature dependences for (a) limiting activity coefficients ln γ1∞ and (b) Henry’s law constants KH of ethers(1) in water(2): (♦) diethyl ether; () tetrahydrofuran; (䊉) tetrahydropyran; ( ) dimethoxymethane; () 1,2-dimethoxyethane; () 1,3-dioxolane; ( ) 1,4dioxane.

129

and (ii) hydrophilic hydration of polar ethereal oxygens forming hydrogen bonds with water. As seen in Fig. 7a, going from 273 K the values of ln γ1∞ rise with temperature, following concave courses which at a higher T display a maximum. The maximum appears around the normal boiling temperature of water, except for DME and DOX for which it is shifted by about 40 K higher (see Table 3). The highest deviations from ideality (γ1∞ ) are observed for DEE and THP. Despite their relatively strong hydrogen bonding with water, which is manifested in a very exothermic dissolution at ambient temperatures, these monoethers exhibit appreciably hydrophobic behavior due to a large aliphatic hydrocarbon backbone of their molecules. Consistent with this picture are the greatly enhanced values of ¯ E,∞ and consequently the enhanced curvature of van’t Hoff C P,1 plots for these solutes, which reflect the coacting decay of both the hydrophobic structuring of the solvent water and the solute-solvent complex formation with the increasing temperature. As expected, diminishing the hydrocarbon portion of the ether or introducing another ethereal oxygen atom into its molecule, γ1∞ decreases. Thus, THF or DOX have lower γ1∞ values than THP. The lowest γ1∞ values of the ethers studied were observed for DME. The hydrogen bond complex formation of this open-chain diether with water is enormous, efficiently reducing its γ1∞ values especially at lower temperatures. When compared at higher temperatures (T > 323 K) where the complex formation declines, DOX, a cyclic analog of DME, exhibits γ1∞ values lower than those of DME. It appears that the stability advantage of DME at lower temperatures is overridden by an entropy gain which for DOX results from less water clustering induced around its more compact cyclic molecule. It can be noted from Fig. 8a that the ring closure increment becomes more negative with the increasing temperature and that for other two open-chain/cyclic structure analog pairs (DMM/DXL and DEE/THF) the cyclic forms exhibit, in comparison with the

Fig. 8. Limiting activity coefficient increment  ln γ1∞ due to the change in ether molecular structure: (a) cyclization () DEE → THF; ( ) DMM → DXL; (䊉) DME → DOX; (b) methylene group (–CH2 –) addition (♦) THF → THP; () DXL → DOX; () DMM → DME. For the corresponding structure changes see Fig. 1.

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D. Ondo, V. Dohnal / Fluid Phase Equilibria 262 (2007) 121–136

open-chain forms, lower γ1∞ values in the entire temperature range investigated. DME and DEE are the two solutes showing the steepest rise of γ1∞ with increasing temperature around the ambient, i.e. the dissolution of these two solutes in water is the most exothermic. This fact appears to indicate that hydrogen bonds responsible for complex formation of these ethers with water are among the ethers studied relatively the strongest. On the other hand, DOX and DXL show the weakest temperature dependence of γ1∞ with ¯ E,∞ values), which together the least curvature, (i.e. the lowest C P,1 ∞ with their quite low γ1 values suggests that these two cyclic diether solutes neither form very strong complexes with water nor appreciably promote water hydrophobic structuring. The comparison between solutes whose molecular structures differ by just one methylene group appears to be especially interesting. The changes  ln γ1∞ due to methylene addition are shown as a function of temperature for three such solute pairs in Fig. 8b. When the CH2 group is introduced into the THF molecule to form THP, the respective  ln γ1∞ is positive and large, and showing a flat maximum at a slightly subambient temperature. This is a result one would normally expect for an enlargement of the hydrophobic moiety of a molecule in a homologous series of alipahtic hydrocarbons and/or their derivatives. However, the introduction of the methylene group into DXL or DMM to form DOX or DME, respectively, leads to a reversed effect, i.e. to negative  ln γ1∞ increments, whose magnitude, especially big at low temperatures, diminishes rapidly with increasing T. This apparently puzzling behavior can be well explained by mutual inductive intramolecular influencing of the two ethereal oxygen atoms due to their proximity in the DXL and DMM molecules. The mutual inductive influencing of the two oxygen atoms separated by only one methylene group weakens their hydrogen bonds with the water molecules, this effect being efficiently cancelled when another methylene group is introduced between these oxygen atoms. Our screening quantum mechanics calculations using Gaussian software package [46] revealed a significant decrease in the electron density on oxygen atoms of DXL and DMM in comparison with those in DOX and DME. The enthalpies of the 1:1 hydrogen bond formation between the ethereal oxygen and water hydrogen calculated by BLYP/6-31G** method are for DXL and DMM smaller (in the absolute magnitude) than those for DOX and DME by 14 and 10 kJ mol−1 , respectively. There is also an experimental support for the above given explanation of the puzzling results of Fig. 8b; a recent NMR study [47] on the oxaalkane-water hydrogen bonding has shown the hydrogen bonds in acetal hydrates to be considerably weaker than those in other diether hydrates. Another interesting behavioral feature observed for the studied set of aqueous ethers is their enthalpy–entropy compensation. The enthalpy–entropy compensation relationship has been widely documented experimentally for a variety of physico-chemical processes in aqueous solutions (nonelectrolyte hydration [48], molecular recognition [49], protein unfolding [50]). This phenomenon appears to be related to the hydrophobic effect, i.e. to the water structure reorganization on

Fig. 9. Enthalpy–entropy compensation for the set of cyclic and open-chain ethers studied in this work at 298.15 K: (♦) diethyl ether; () tetrahydrofuran; (䊉) tetrahydropyran; ( ) dimethoxymethane; () 1,2-dimethoxyethane; () 1,3-dioxolane; ( ) 1,4-dioxane.

the periphery of a nonpolar or amphiphilic molecule when it is introduced as a solute into water solvent. For the ethers studied in this work the phenomenon is illustrated in Fig. 9 by correlations ¯ E,∞ , and G ¯ E,∞ . The linear fits are seen to ¯ E,∞ , C of T S¯ 1E,∞ with H 1 P,1 1 be very good, their respective coefficients of determination (r2 ) being 0.95, 0.95, and 0.81. DME, as the most hydrophilic solute which is extensively complexed to water, could be recognized as a systematic outlier and hence it was excluded from these fits. Note that a relatively lower quality of the Gibbs energy–entropy correlation is inherent, because a small change in Gibbs energy results here typically from the compensation of large but opposite changes in enthalpy and entropy. Contrary to theoretical expectations, analogous correlations between hydration properties (not shown) were found less quantitative, their r 2 values ∞ ), and 0.54 ( ∞ being 0.49 (hyd H1∞ ), 0.92 (hyd CP,1 hyd G1 ). The van’t Hoff plots of Henry’s law constants (Fig. 7b), like those of limiting activity coefficients, exhibit also a concave rise with temperature, but with no maxima. The extrapolation of KH (T ) up to 423 K by Eq. (8) shows no maximum to occur even in this extended range. Trends in KH (T ) and related hydration properties mostly parallel those in γ1∞ (T ) and related excess

D. Ondo, V. Dohnal / Fluid Phase Equilibria 262 (2007) 121–136

131

Table 7 Deviations of the ln γ1∞ values predicted by three group contribution methods from those recommended in this work δ ln γ1∞

Solute(1) Mod. UNIFAC b

Diethyl ether Tetrahydrofuran Tetrahydropyran Dimethoxymethane 1,2-Dimethoxyethane 1,3-Dioxolane 1,4-Dioxane a

a

Cabani et al.c

Plyasunov et al.c

298.15 K

373.15 K

298.15 K

373.15 K

0.99 (0.26) −0.05 −0.53 −0.15 (−2.04) 2.34 (0.46) −0.30 0.21

0.04 (−1.16) −0.82 −1.29 0.66 (−2.41) 2.19 (−0.88) −0.14 0.11

0.42 0.14 −0.12 −1.64 0.81 −6.46 0.00

0.24 0.10 −0.06 −0.69 1.00 −4.45 −0.05

298.15 K 0.00 – – −0.19 0.18 – –

373.15 K −0.22 – – −0.20 0.16 – –

∞ ∞ δ ln γ1∞ = ln γ1,predicted − ln γ1,recommended .

b

Calculated with the latest values of H2 O–CH2 O group interaction parameters from fifth revision [53]; values given in parentheses calculated with parameters from fourth revision [52]. s ∞ ∞ c Group contribution method gives  hyd G1 , P1 needed to calculate γ1 were obtained from equations given in Table 1.

properties, but often they are seen less spectacular. DEE and DOX are the solutes exhibiting the highest volatility and the lowest volatility, respectively, from the aqueous solution at infinite dilution, differing roughly by a factor of 200 throughout the temperature range studied. Compared to this molecular structure variation effect, the effect of temperature appears to be much more dramatic; for DME, the solute with the most exothermic hydration, KH changes in the temperature range studied by a factor of about 850.

A surprisingly accurate estimate for DOX results from the fact that Cabani et al. used just this compound to adjust the respective correction parameter. The predictions by the secondorder group contribution scheme of Plyasunov et al. are very good, but the method, involving a number of corrections, is presently limited to open-chain ethers only. In addition, the presence of higher-order effects identified by Plyasunov et al. for aqueous ethers can cause the predictions for other oxaalkanes than those three explicitly examined in this work much less accurate.

5.3. Prediction by group contribution methods 6. Conclusion As mentioned in the previous section, interactions between water and ethereal oxygen atoms are strongly dependent on the ether molecular structure. This fact however implies that reliable prediction of thermodynamic properties of aqueous ethers by group contribution methods will not be feasible unless various structure specific corrections and/or ad hoc cumulative functional groups capturing the specific structure are introduced and corresponding parameters evaluated from accurate experimental data. To see the state of the art in the prediction of limiting activity coefficients, the performance of three group contribution approaches, namely the Modified UNIFAC (Dortmund) and the methods of Cabani et al. [51] and of Plyasunov et al. [7], was tested against the newly recommended data from this work. The predictions by the Modified UNIFAC were done using the latest parameter values published in the open literature (fourth [52] and fifth revision [53]). Table 7 gives deviations of the predicted ln γ1∞ values from those recommended at two temperatures (298.15 and 373.15 K). As seen, the results are in general quite unsatisfactory. The Modified UNIFAC fails considerably except for the two cyclic diethers. Its poor performance for monoethers, in particular DEE, is especially disappointing. The predicted temperature dependences of γ1∞ are often very discordant as exemplified in Figs. 2, 3 and 5. The method of Cabani et al. fails drastically for all diethers except for DOX, the failure being obviously due to missing correction parameters for the intramolecular interaction of the two oxygen atoms.

Thanks to our new systematic air-water partitioning measurements combined with corresponding calorimetric information from the literature, recommended thermodynamically consistent temperature dependences could be established for γ1∞ and KH of seven environmentally important ethers in water. These recommendations of superior accuracy substantially improve our knowledge of thermodynamic behavior of dilute aqueous ethers and meet respective data needs for environmental and industrial applications. The results, disclosing various effects of molecular structure on the thermodynamic behavior of aqueous ethers, will be also useful for further development of prediction schemes whose current performance for ethers is rather unsatisfactory. Presently, the results contribute to the establishment of a data base on hydration properties of organic nonelectrolytes within an international project [54] conducted under the auspices of IUPAC and IAPWS. List of symbols a, b, c, d parameters of the vapor pressure equation A, B, C, D adjustable parameters of Eq. (1) AH , BH , CH , DH adjustable parameters of Eq. (8) B11 solute second virial coefficient ¯ C, C molar heat capacity, partial molar heat capacity F dimensionless variable (F = 1 − T/Tc ) ¯ G, G molar Gibbs energy, partial molar Gibbs energy

132

¯ H, H KH n p P R s s(X) S T V

D. Ondo, V. Dohnal / Fluid Phase Equilibria 262 (2007) 121–136

molar enthalpy, partial molar enthalpy Henry’s law constant of solute 1 in solvent 2 number of experimental points number of adjustable parameters pressure universal gas constant standard deviation of fit standard uncertainty of quantity X least-squares objective function temperature volume

Greek letters α significance level γ activity coefficient Δ difference τ dimensionless temperature 298.15 K) χ2 χ2 distribution

(τ = T/T0 , T0 =

Subscripts 1 solute 2 solvent c critical hyd hydration m molar min minimal max maximal P constant pressure rel relative vap vaporization Superscripts 0 gas standard state ∞ infinite dilution E excess property L liquid s saturated vapor Acknowledgments We thank to Professor E.M. Woolley from Brigham Young ¯ ∞ values University, Provo, USA for making available to us C P,1 measured in his laboratory before publication. We also thank to our colleagues from ICT Prague, Dr. M. Bureˇs for performing the quantum mechanics computations and Dr. P. Vrbka for helpful advices with the measurements. Financial support to this project from the Ministry of Education of the Czech Republic (grant MSM 604 613 7307) is gratefully acknowledged. Appendix A Values of limiting activity coefficients γ1∞ of ethers(1) in ¯ E,∞ and water(2) used for simultaneous fit with calorimetric H 1 ¯ E,∞ data by Eq. (1) (Table A.1). C P,1

Table A.1 T (K)

γ1∞

Diethyl ether 294.15 71.4d 299.15 73.8d 305.15 79.6d 310.15 81.9d 311.65 80.4d 313.15 80.1d 285.35 58.2d ,e 289.15 56.5d ,e 293.15 58.0d ,e 301.15 65.9d ,e 306.15 75.2d ,e 313.15 83.0d ,e 310.15 82.6 310.15 97.8d 298.15 123 308.15 88.7d 310.15 97.9d 312.15 104.6d 308.15 97.9f 293.05 108.1d 302.9 113.8d 310.15 78.2d 303.15 71.3d 318.15 89.2d 296.15 72.9 298.15 69.7f 298.15 110g 283.15 50.0 293.15 68.0 298.15 77.0 303.15 85.9 303.25 90.1 313.25 113 323.25 135 333.25 158 273.35 33.9 278.15 41.2 283.15 49.4 288.15 58.5 293.15 68.6 298.15 77.1 303.15 88.8 308.15 98.8 318.15 121.0 328.15 139.5 Tetrahydrofuran 298.15 19.5 298.15 18.1 293.15 16.6 308.15 23.5 323.15 32.8 313.15 20.8f 298.15 18.7g ,h 273.35 9.74 278.15 11.1 288.15 14.3 298.15 17.6 308.15 20.6 318.15 24.0 328.15 26.8 328.15 28.1 Tetrahydropyran 298.15 74.1 298.15 78.6g ,h

srel (γ1∞ ) a

Techniqueb

Vaporc

Source

0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.05 0.5 0.1 0.05 0.05 0.02 0.5 0.2 0.2 0.2 0.2 0.03 0.1 0.2 0.03 0.03 0.03 0.05 0.03 0.03 0.03 0.03 0.02 0.02 0.01 0.01 0.01 0.01 0.01 0.02 0.02 0.03

CACP CACP CACP CACP CACP CACP CACP CACP CACP CACP CACP CACP Unknown HSA TENS TENS TENS TENS TENS PRV PRV HSA HSA HSA IGS IGS GLC IGS IGS IGS IGS IGS IGS IGS IGS IGS IGS IGS IGS IGS IGS IGS IGS IGS IGS

IDEAL IDEAL IDEAL IDEAL IDEAL IDEAL IDEAL IDEAL IDEAL IDEAL IDEAL IDEAL Unknown IDEAL IDEAL IDEAL IDEAL IDEAL VIR IDEAL IDEAL IDEAL IDEAL IDEAL IDEAL Unknown VIR IDEAL IDEAL IDEAL IDEAL VIR VIR VIR VIR VIR VIR VIR VIR VIR VIR VIR VIR VIR VIR

[30] [30] [30] [30] [30] [30] [31] [31] [31] [31] [31] [31] [32] [33] [27] [65] [65] [65] [43] [28] [28] [66] [34] [34] [67] [68] [29] [35] [35] [35] [35] [36] [36] [36] [36] This work This work This work This work This work This work This work This work This work This work

0.1 0.03 0.05 0.1 0.2 0.1 0.05 0.02 0.02 0.01 0.01 0.02 0.02 0.03 0.03

TENS TENS TENS TENS TENS IGS HSA RHSA RHSA IGS IGS IGS IGS NSGLC NSGLC

IDEAL IDEAL VIR VIR VIR Unknown Unknown VIR VIR VIR VIR VIR VIR IDEAL IDEAL

[27] [61] [39] [39] [39] [68] [29] This work This work This work This work This work This work This work This work

0.05 0.1

TENS HSA

IDEAL Unknown

[61] [29]

D. Ondo, V. Dohnal / Fluid Phase Equilibria 262 (2007) 121–136 Table A.1 (Continued ) T (K)

γ1∞

273.35 39.6 278.15 44.9 288.15 58.0 298.15 70.5 308.15 83.4 318.15 96.2 328.15 107.5 Dimethoxymethane 297.25 18.5 273.35 10.8 283.15 13.7 293.15 15.6 298.15 16.9 303.15 18.0 313.15 19.9 323.15 22.0 333.15 23.9 1,2-Dimethoxyethane 298.15 4.31 273.35 1.59 278.15 2.02 288.15 3.01 298.15 4.22 308.15 5.61 318.15 7.25 328.15 9.12 328.15 8.81 338.15 11.1 338.15 11.2 348.15 12.2 372.50 14.9 1,3-Dioxolane 298.15 10.1 273.35 8.13 278.15 8.48 288.15 9.12 298.15 9.71 308.15 10.2 318.15 10.6 328.15 10.9 338.15 11.0 338.15 11.4 348.15 11.3 372.55 11.2 1,4-Dioxane 298.15 5.6 298.15 5.19d 298.15 5.45 313.15 8.67d 333.15 9.78d 343.15 10.63d 353.15 10.85d 298.15 5.42g 273.35 3.53 278.15 4.00 288.15 4.75 288.15 5.17 298.15 5.88 308.15 6.15 308.15 6.70 318.15 6.82 328.15 7.76 338.15 8.27 338.15 8.22

133

Table A.1 (Continued ) srel (γ1∞ ) a

Techniqueb

Vaporc

Source

0.03 0.02 0.02 0.02 0.02 0.02 0.02

RHSA RHSA IGS IGS IGS IGS IGS

VIR VIR VIR VIR VIR VIR VIR

This work This work This work This work This work This work This work

0.1 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02

GLC IGS IGS IGS IGS IGS IGS IGS IGS

IDEAL VIR VIR VIR VIR VIR VIR VIR VIR

[81] This work This work This work This work This work This work This work This work

0.05 0.03 0.03 0.03 0.03 0.03 0.01 0.03 0.03 0.03 0.03 0.02 0.05

TENS RHSA RHSA RHSA RHSA RHSA IGS NSGLC NSGLC NSGLC NSGLC RDIST CIRC

IDEAL VIR VIR VIR VIR VIR VIR IDEAL IDEAL IDEAL IDEAL VIR VIR

[69] This work This work This work This work This work This work This work This work This work This work This work This work

0.05 0.02 0.01 0.01 0.01 0.01 0.01 0.02 0.03 0.02 0.02 0.03

TENS RHSA RHSA RHSA RHSA RHSA IGS IGS NSGLC NSGLC RDIST CIRC

IDEAL VIR VIR VIR VIR VIR VIR VIR IDEAL IDEAL VIR VIR

[61] This work This work This work This work This work This work This work This work This work This work This work

0.03 0.05 0.03 0.2 0.2 0.2 0.2 0.05 0.03 0.03 0.03 0.05 0.05 0.03 0.05 0.03 0.03 0.03 0.02

TENS HSA HSA VPC VPC VPC VPC HSA RHSA RHSA RHSA HSA HSA RHSA HSA RHSA NSGLC NSGLC IGS

IDEAL IDEAL IDEAL IDEAL IDEAL IDEAL IDEAL Unknown VIR VIR VIR VIR VIR VIR VIR VIR IDEAL IDEAL VIR

[61] [74] [64] [40] [40] [40] [40] [29] This work This work This work This work This work This work This work This work This work This work This work

T (K) 348.15 372.50

γ1∞ 8.43 10.3

srel (γ1∞ ) a

Techniqueb

Vaporc

Source

0.05 0.03

RDIST CIRC

VIR VIR

This work This work

a Relative standard uncertainty estimate assigned during the process of data evaluation and treatment. b IGS, inert gas stripping; HSA, headspace analysis; RHSA, relative HSA; GLC, measurement of retention time in gas–liquid chromatography; NSGLC, non-steady-state GLC; RDIST, Rayleigh distillation; CIRC, circulation still; VPC, vapor-phase calibration; TENS, tensimetry; CACP, chemical analysis of coexisting phases. c IDEAL, ideal gas; VIR, virial equation of state. d Limiting activity coefficient calculated from liquid/vapor or vapor/liquid distribution coefficient reported in the cited source. e Averaged value of γ ∞ at selected temperatures from cited source. 1 f Limiting activity coefficient calculated from dilute range P– x data reported in the cited source. g Secondary reference citing an original unavailable source (e.g. thesis). h Note that in the paper of Sherman et al. [29] there was an error in the decimal separator of the activity coefficient value.

Appendix B ¯ E,∞ of Values of limiting partial molar excess enthalpies H 1 ∞ ¯ E,∞ ethers(1) in water(2) used for simultaneous fit with γ1 and C P,1 data by Eq. (1) (Table B.1 ). Table B.1 T (K)

¯ E,∞ (kJ mol−1 ) H 1

Diethyl ether 286.15 −24.70 298.15 −19.26 308.15 −15.48c 283.15 −23.10 288.15 −22.00 293.15 −20.83 298.15 −19.66 303.15 −18.24 Tetrahydrofuran 278.15 −18.40 288.15 −16.59 298.15 −14.93 298.15 −16.57 313.15 −12.93 298.15 −14.95 298.15 −15.40 298.15 −15.05 298.15 −14.83 Tetrahydropyran 278.15 −19.70 288.15 −17.14 298.15 −15.10 298.15 −14.58 Dimethoxymethane 298.15 −13.4 288.15 −11.81 293.15 −11.09 298.15 −10.54 303.15 −9.78 308.15 −9.24 313.15 −8.57 318.15 −7.91

¯ E,∞ ) a (kJ mol−1 ) s(H 1

Techniqueb

Source

1.5 0.2 1 0.08 0.08 0.08 0.08 0.08

Unknown BATCH Unknown FLOW FLOW FLOW FLOW FLOW

[41] [77] [43] [37] [37] [37] [37] [37]

0.15 0.14 0.17 1 0.5 0.15 0.3 0.1 0.3

BATCH BATCH BATCH BATCH BATCH BATCH BATCH BATCH FLOW

[75] [75] [75] [42] [42] [61] [77] [82] [71]

0.3 0.3 0.3 0.1

BATCH BATCH BATCH BATCH

[75] [75] [75] [61]

2 0.1 0.1 0.1 0.1 0.09 0.08 0.08

Unknown FLOW FLOW FLOW FLOW FLOW FLOW FLOW

[41] [37] [37] [37] [37] [37] [37] [37]

134

D. Ondo, V. Dohnal / Fluid Phase Equilibria 262 (2007) 121–136

Table B.1 (Continued ) T (K)

¯ E,∞ (kJ mol−1 ) H 1

1,2-Dimethoxyethane 293.15 −23.94 298.15 −22.95 303.15 −22.07 298.15 −22.36 298.15 −22.95 1,3-Dioxolane 298.15 −4.62 298.15 −4.66 1,4-Dioxane 288.15 −10.23 298.15 −9.82 308.15 −9.41 298.15 −9.28 323.15 −7.61 348.15 −6.23 298.15 −10.79 313.15 −9.25 298.15 −9.60 298.15 −8.79 298.15 −9.52d 298.15 −9.49 298.15 −9.82

Table C.1 (Continued ) ¯ E,∞ ) a (kJ mol−1 ) s(H 1

Techniqueb

Source

T (K)

0.05 0.05 0.05 0.4 0.08

BATCH BATCH BATCH FLOW BATCH

[70] [70] [70] [71] [72]

0.05 0.1

BATCH FLOW

[61] [71]

0.15 0.09 0.25 0.3 0.3 0.3 0.5 0.4 0.2 0.5 0.4 0.15 0.15

BATCH BATCH BATCH BATCH BATCH BATCH BATCH BATCH BATCH BATCH FLOW BATCH BATCH

[75] [75] [75] [76] [76] [76] [42] [42] [61] [77] [71] [78] [72]

Dimethoxymethane 298.15 111 1,2-Dimethoxyethane 298.15 171 313.15 165 298.15 188 298.15 189 303.15 190 308.15 192 313.15 193 318.15 194 323.15 195 328.15 196 333.15 197 338.15 197 343.15 198 348.15 198 353.15 198 358.15 197 363.15 197 368.15 196 373.15 195 298.15 185 1,3-Dioxolane 298.15 56.9 1,4-Dioxane 298.15 61.4 298.15 71.5 298.15 85.0 298.15 71.9 288.15 60.0 293.15 62.5 298.15 65.0 303.15 67.4 308.15 69.7 313.15 72.0

a Standard uncertainty estimate assigned during the process of data evaluation and treatment. b BATCH, batch dissolution calorimetry; FLOW, flow mixing calorimetry. c Reevaluated H ¯ E,∞ value from raw H E (x1 ) data published in [43] considering 1 the experimental points from x1 = 0.0029–0.0078. d Reevaluated H ¯ E,∞ value from raw H E (x1 ) data published in [71] where the 1 outlier point at x1 = 0.001096 was omitted.

Appendix C ¯ E,∞ Values of limiting partial molar excess heat capacities C P,1 of ethers(1) in water(2) used for simultaneous fit with γ1∞ and ¯ E,∞ data by Eq. (1) (Table C.1). H 1 Table C.1 T (K)

¯ E,∞ C P,1 (J mol−1 K−1 )

Diethyl ether 298.15 250 298.15 242 Tetrahydrofuran 298.15 171 298.15 172 298.15 172 278.15 169 283.15 171 288.15 174 293.15 175 298.15 177 303.15 178 308.15 179 313.15 180 318.15 181 323.15 182 Tetrahydropyran 298.15 223 298.15 235

¯ E,∞ )a s(C P,1 (J mol−1 K−1 )

Techniqueb

Source

10 5

ADIAB FLOW

[44]c [38]c

2 5 5 3 3 3 3 3 3 3 3 3 3

ADIAB ADIAB FLOW SCAN SCAN SCAN SCAN SCAN SCAN SCAN SCAN SCAN SCAN

[79]c [45] [83] [21]c [21]c [21]c [21]c [21]c [21]c [21]c [21]c [21]c [21]c

8 5

ADIAB FLOW

[79]c [80]c

¯ E,∞ C P,1 (J mol−1 K−1 )

¯ E,∞ )a s(C P,1 (J mol−1 K−1 )

Techniqueb

Source

10

ADIAB

[44]c

10 20 5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 7 5

ADIAB ADIAB FLOW SCAN SCAN SCAN SCAN SCAN SCAN SCAN SCAN SCAN SCAN SCAN SCAN SCAN SCAN SCAN SCAN FLOW

[44]c [44]c [73] [21]c [21]c [21]c [21]c [21]c [21]c [21]c [21]c [21]c [21]c [21]c [21]c [21]c [21]c [21]c [21]c [38]c

3

ADIAB

[79]c

5 5 15 5 3 3 3 3 3 3

ADIAB FLOW ADIAB FLOW SCAN SCAN SCAN SCAN SCAN SCAN

[79]c [80]c [45] [73] [21]c [21]c [21]c [21]c [21]c [21]c

a Standard uncertainty estimate assigned during the process of data evaluation and treatment. b FLOW, flow calorimetry; SCAN, scanning calorimetry; ADIAB, adiabatic calorimetry. c Reference to the source of original C ¯ ∞ value; CL,• values taken from [22,23]. P,1 P,1

Appendix D Valuesa of Henry’s law constants KH , hydration enthalpies ∞ of ethers(1) in hyd H1∞ and hydration heat capacities hyd CP,1 water(2) used in simultaneous fit by Eq. (8) (Table D.1). Table D.1 T (K)

DEE

THF

THP

DMM

DME

DXL

DOX

ln(KH /kPa) 273.16 278.15 283.15 288.15 293.15 298.15 303.15 308.15

6.718 7.142 7.540 7.914 8.265 8.594 8.902 9.191

4.096 4.507 4.896 5.263 5.610 5.938 6.247 6.540

4.551 4.991 5.406 5.796 6.162 6.506 6.829 7.133

5.216 5.562 5.890 6.200 6.494 6.771 7.033 7.281

1.326 1.843 2.333 2.797 3.236 3.652 4.045 4.418

3.346 3.679 3.998 4.301 4.590 4.867 5.130 5.382

1.481 1.878 2.257 2.619 2.965 3.296 3.611 3.913

D. Ondo, V. Dohnal / Fluid Phase Equilibria 262 (2007) 121–136 Table D.1 (Continued ) T (K)

DEE

THF

THP

DMM

313.15 9.461 6.816 7.418 7.515 318.15 9.714 7.077 7.686 7.736 323.15 9.951 7.323 7.938 7.945 328.15 10.172 7.555 8.174 8.143 333.15 10.378 7.774 8.395 8.329 338.15 10.571 7.980 8.603 8.505 343.15 10.750 8.174 8.798 8.671 348.15 10.917 8.357 8.980 8.827 353.15 11.072 8.529 9.150 8.975 358.15 11.216 8.690 9.309 9.114 363.15 11.349 8.842 9.458 9.245 368.15 11.472 8.984 9.597 9.369 373.15 11.585 9.117 9.726 9.486 hyd H1∞ (kJ mol−1 ) 283.15 −51.37 288.15 −49.89 293.15 −48.41 298.15 −46.93 −47.12 −49.29 −39.65 303.15 −45.44 −46.02 308.15 −43.96 −44.91 313.15 −42.48 −43.80 318.15 −42.67 323.15 −41.54 328.15 −40.40 333.15 −39.26 338.15 −38.10 ∞ (J mol−1 K−1 ) hyd CP,1 278.15 218 283.15 220 288.15 222 293.15 223 298.15 296 225 282 196 303.15 226 308.15 227 313.15 228 318.15 228 323.15 229 328.15 229 333.15 229 338.15 229 343.15 228 348.15 228 353.15 227 358.15 226 363.15 225 368.15 224 373.15 222

DME 4.771 5.105 5.421 5.721 6.005 6.273 6.527 6.768 6.995 7.210 7.413 7.605 7.787

DXL 5.622 5.851 6.070 6.280 6.480 6.671 6.854 7.028 7.195 7.355 7.508 7.653 7.793

DOX 4.202 4.478 4.742 4.995 5.237 5.469 5.691 5.903 6.107 6.302 6.488 6.667 6.838

−59.45 −40.08 −48.40

273 273 273 273 273 273 273 273 272 272 271 271 270 269 268 267 265 264 262 261

107

108 112 115 119 122 125 128 131 134 136 139 142 144 147 149 151 153 155 157 159

Calculated at equidistant temperatures with 5 K increment.

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