Solvation-based modeling vapor pressures of (solvent + salt) systems with the application of Cox equation

Fluid Phase Equilibria 361 (2014) 155–170

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Solvation-based modeling vapor pressures of (solvent + salt) systems with the application of Cox equation Aynur Senol ∗ Department of Chemical Engineering, Faculty of Engineering, Istanbul University, 34320 Avcilar, Istanbul, Turkey

a r t i c l e

i n f o

Article history: Received 23 May 2013 Received in revised form 1 October 2013 Accepted 3 October 2013 Available online 12 October 2013 Keywords: Vapor pressure Salt effect Modeling Solvation model Statistical analysis

a b s t r a c t A solvation model framework, based on the LSER (linear solvation energy relationship) principles, is proposed for correlating vapor pressures of (solvent + salt) systems. The LSER method basely uses a linear combination of several solvatochromic indices of solvents to describe their physical properties. The assumption inherent in the solvation-based vapor pressure approach is attributed to an additional effect of several physical quantities, i.e. the vapor pressure of a pure solvent estimated by the Cox equation, the salt concentration, the solvatochromic indicators of the solvent and the physical properties of the ionic salt species. It has been performed independently two structural forms of the generalized solvation model, i.e. the integrated property-basis model using nine physical descriptors USMIP (the unified solvation model with the integrated properties) and the reduced property-basis one. Also, a simplified concentration-dependent vapor pressure model is presented. The observed vapor pressure data of fifteen (solvent + salt) and two (solvent (1) + solvent (2) + salt) systems have been processed to establish the basis for the model reliability analysis using a log-ratio objective function. The proposed vapor pressure approaches reproduce the observed performance relatively accurately, yielding the overall design factors of 1.065 and 1.072 for the solvation-based models with the integrated and reduced properties and 1.017 for the concentration-based model, respectively. Both the integrated property-basis and reduced property-basis solvation models were able to simulate satisfactorily the vapor pressure data of a binary solvent mixture involving a salt, yielding an overall mean error of 5.7%. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Increasing concern about environmental issues has recently directed the attention of the scientific community to greener industrial processes with the interactive salt effect. The effect of the ionic salt species on the phase equilibrium finds an applicable field in many separation processes such as distillation, crystallization, liquid–liquid extraction, salt precipitation from mixed solvents, etc. [1]. In many areas of industry, solvent mixtures accumulate due to recycling difficulties. Typically, biofuel processes produce fermented products such as esters, ethers and alcohols that form azeotropes with water which is abundant in the fermentator. The extractive distillation with the interactive salt effect has emerged as the most common and environmentally beneficial method for separating the azeotropic or close-boiling binary solvent systems. The ionic salt is basely used as an extractive agent (entrainer) to alter the relative volatility of the liquid mixture, so that the components can be more easily separated from the system and reused [1–12]. Regarding the azeotropic mixtures, the salting-in or salting-out

∗ Tel.: +90 212 4737070; fax: +90 212 4737180. E-mail address: [email protected] 0378-3812/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fluid.2013.10.006

effect has been employed by several workers to design a distillation operation for shifting or breaking an azeotrope through a salt as the entrainer. Experimental findings have shown that azeotropes can be entirely eliminated by using a salt or ionic liquid in a mixture of polar solvents [7–13]. The synthesis of a water-free organic solvent is another example for such a separation process. Typically, the dewatering of some organic solvents is carried out by adding soluble electrolytes to the aqueous solutions being capable of affecting dramatically the volatility range of components [5]. However, the experimental results of Apelblat and Korin [6] exhibit that the salting-out effect of ionic salts could modify reasonably the molar enthalpies and vapor pressures of saturated aqueous solutions. The technical details and thermodynamic aspects of a separation process with the interactive salt effect have been excellently reviewed by many researches [1,12–16]. Particularly frustrating aspects of using an inorganic salt as the entrainer are insufficient solubility in nonaqueous solvents and corrosion. Nevertheless, the ionic salt becomes favorably potential alternative to the commonly-used liquid as a separation agent in the azeotropic or extractive distillation [12,13,17]. In the context of many chemical processes it is important to determine and interpret the precise effect of dissolved salts on the phase behavior of solvent mixtures. There are several complexities

156

A. Senol / Fluid Phase Equilibria 361 (2014) 155–170

in describing the salt interaction with the volatile components that generally arise out of the selective interactive effect due to a specific salt on the volatilities of liquid components [13]. It is presumed that the salt would likely induce the formation of different associated complexes or clusters of molecules of the volatile component about its ions. The salt effect is thought to be a complex function of the salt and solvent interaction and self-interaction among all the components of the system [12–17]. The usual approach to quantify the efficiency of a salt-based separation process is to specify the properties of the equilibrium system due to a thermodynamic criterion being obeyed. Typically, the thermodynamic criterion of (vapor + liquid) equilibrium implies that the equality of fugacities of component i in both phases is reached regarding each of (solvent + salt) binary system as a pseudo-component, and is given by Eq. (1) [14–19] yi i Pt = xi i i0 Pi where the prime refers to the salt system; i

(1)

represents the activity coefficient of component i in the salt-containing system; yi and xi denote the mole fractions of component i in the vapor and liquid phases, respectively; Pt is the total pressure of the system involving a salt. The Pi stands for the saturation vapor pressure of component i at the system temperature and in the presence of the salt to which we shall refer hereafter to as P. i and i0 designate the fugacity coefficients of component i in the vapor mixture and the pure solvent at the saturated state, respectively. If the non-ideality of the vapor phase is prevalent, the fugacity coefficients i and i0 of component i can be calculated by the virial state equation truncated after the second term [14,20]. In general, since the total pressures of relevant systems involving the ionic salts are sufficiently low, all fugacity coefficient corrections are similar and can be neglected. Many attempts have been made to describe vapor–liquid equilibria of salt-containing associated mixtures using the groupcontribution theory, the statistical associating fluid theory (SAFT), the associated perturbed anisotropic chain theory (APACT), the lattice quasi-chemical theory, COSMO-SAC method and the concepts of multiscale dipole–dipole association [21–25]. In the design practice, the separation efficiency due to a salt effect is usually interpreted in terms of the properties specified by Eq. (1), where the solubility range of the salt is accounted for explicitly [13–19]. Subsequently, appropriate theoretical models are required to estimate the activity coefficient and vapor pressure of the components. Kumar [13] has broadly classified activity coefficient models into two categories depending upon whether the model is derived from excess Gibbs free energy or not. The first category models are concerned with the concept of excess Gibbs free energy, i.e., Wilson, NRTL, UNIQUAC and UNIFAC models [13], and their modified versions [26,27,23]. In the second type semi-empirical in the nature equations, the presence of salt is accounted for implicitly [18,19]. The starting point is based on either the formation of a binary complex between the ionic salt species and the solvent molecules or the use of the concept of preferential solvation [13,19]. Because the determination of all required experimental vapor pressure data in a wide temperature range is almost impossible, it is a practical way to extrapolate vapor pressure prediction to the overall temperature range based on a generalized correlative model for (solvent + salt) systems. While various models, based on the critical temperature and pressure properties, exist for the vapor pressure of a pure fluid, yet only few works have focused on assessing and generating analytically an efficient equation for vapor pressures of (solvent + salt) systems relative to the solubility range of the salt. In the scope of this study, we came across the need to quantify rigorously vapor pressures of relevant systems through a solvation-based relation. The study was aimed to correlate vapor pressures of (solvent + salt) systems through the principles of linear solvation energy relationship (LSER) [28–30] using the

solvatochromic indicators of the solvent and physical properties of the ionic salt species. Two structural forms of the generalized solvation model framework have been performed independently, i.e. a unified solvation approach based on an integration of the overall properties of the solvent and salt, and a reduced property-basis approach which does not account for the properties of the ionic salt species explicitly. A simplified concentration-dependent vapor pressure model is also presented and checked for consistency. The reliability of existing models has been analyzed statistically on the basis of a log-ratio objective function. Recently, the application of LSER has been extended to correlate the tracer diffusion coefficients, the solubility of compounds and the distribution behavior of associated liquid systems [31–33]. The proposed LSER-based solvation models were able to characterize precisely different properties of associating fluids in VLE and LLE systems [33–35]. This capacity turns the LSER principles applicable to (solvent + salt) systems. In this perspective, attempts have been made to correlate vapor pressures of (solvent + salt) systems by a log-basis approach, namely, a unified solvation model with the integrated properties (USMIP). The model combines the system properties at the limiting conditions attributed to the vapor pressure of a pure solvent (or a solvent mixture) with an expansion term clarifying the complementary effect of the salt concentration, the solubility and solvatochromic parameters of the solvent and the physical properties of the ionic salt species. However, to quantify the magnitude of the salt effect, quantitative knowledge of the vapor pressure of a pure solvent is needed.

1.1. Survey of vapor pressure models for pure solvents: application of Cox equation Numerous correlative and theoretical-based relationships for estimating saturation vapor pressures of pure fluids have been published in the literature, being excellently reviewed by many researchers [36–41]. As a result, for accurate fitting vapor pressure data of a pure solvent, the model must possess at least three parameters [37,39]. Traditionally, comprehensive studies on the vapor pressure have been fulfilled using commonly four equations of practical interest in terms of their abilities to correlate vapor pressure of a pure solvent, i.e. the three-parameters Antoine equation, four- or five-parameters Wagner equation, the SVRC model with two fluid-specific regressed parameters, and the Cox equation including at least three adjustable parameters [36–41]. Several modifications to the Wagner equation including three to six adjustable parameters and relations derived from the corresponding states or kinetic theory have found use to varying degrees [36–41]. Recently, various literature studies which use the structure property information alone to generalize the model parameters have been published [39]. As such, quantitativestructure-property relationship (QSPR) models stand out because they become powerful methods providing reliable vapor pressure estimates based solely on detailed chemical structure information covering an optimum number of structural descriptors [39]. Another approach that has gained significance over the years has been the use of the polynomial-type Cox equation with varying temperature exponent [36,42]. The Cox equation applicable to a wide temperature range has been employed frequently for data fitting, particularly in publications from NIPER (the Bartlesville Laboratory of the Bureau of Mines). Due to its reliable correlative capacity and high flexibility, the Cox equation has been incorporated into the LSER-based solvation model to account for the vapor pressure of a pure solvent. The generalized form of the Cox equation is given by Eq. (2) [36,42,43]. The logarithm of the ratio of P0 (kPa) to an arbitrarily selected vapor pressure, e.g. the critical pressure Pc (kPa)

157

114

[45–49,56,58]

4412.63 563.25

Critical properties of pure fluids due to McGarry [59]. T = coefficients of three parameters Cox equation, Eq. (3); F = coefficients of five parameters Cox equation, Eq. (4). b

c

e¯ = a

0.1008 × 10

−9

100 N

i=1

|(P0,obs − P0,mod )/P0,obs |.

0.7180 × 10−5 −0.5281 × 10−4 −0.6276 × 10−2 0.3056 × 10−1 0.3495 × 101 −0.2041 × 101

N

−0.1138 × 10−4 −0.2548 × 10−3 0.6975 × 10−2 0.1216 × 10−1 0.1165 × 101 0.5620 × 101

−0.3474 × 10

−7

0.6309 × 10−5 −0.5618 × 10−4 −0.5443 × 10−2 0.2132 × 10−1 0.3310 × 101 −0.2322 × 100

0.8263 × 10−6

0.5430 × 10−5 −0.1753 × 10−4 −0.4420 × 10−2 0.1116 × 10−1 0.3059 × 101 0.7086 × 100

0.3602 × 10

−0.7920 × 10−9

508.30 0.1998 × 10

−10 −7

0.2211 × 10−5 0.1179 × 10−4 −0.1897 × 10−2 −0.8388 × 10−3 0.2521 × 101 0.2121 × 101

−0.2765 × 10−7

0.6025 × 10−10

536.78

4742.44

5151.11

78

109

113 6130.87 0.4349 × 10−10

513.92

8085.05 512.64 −0.5520 × 10−10 0.1019 × 10−6 0.1503 × 10−5 −0.6730 × 10−4

−0.4187 × 10−7

[44–46,54]

113

References

397

(kPa) b

Pc

22,122.3 −0.1423 × 10−2 0.1867 × 10−1

where P0 (kPa) is the vapor pressure of a pure solvent at the system temperature T (K). A statistical analysis of the Cox equations with three (TP) and five (FP) adjustable parameters has shown that both approaches coincide with the vapor pressure curves over the entire range from the triple point to the critical temperature, based upon known physical behavior of a pure fluid. Table 1 presents a brief summary of the regressed substance-dependent coefficients qk and the deviation statistics of both three- and five-parameters Cox equations. For the determination of qk coefficients, an extensive database of vapor pressure data given in the current literature has been compiled by means of multivariable regression [44–59]. An overview about the database that has been used to fit the parameters is given in Table 1. In general, the quality of the model development efforts depends on the availability of accurate experimental data. Typically, the three-parameters Cox equation (Eq. (3)) has proved slightly more successful than the five-parameters one (Eq. (4)) due to a reduced number of degrees of freedom. In all the cases, as marked in Table 1, any of the examined solvents was not excluded from the fits, in spite of the fact that they made for excessively large deviations of the fits. The poor prediction very likely comes from an irregular distribution of the experimental dataset points over the entire temperature range, especially being the strongest for 2propanol. Further, the form of Eq. (2) requires the correlated data to pass through the critical point value. However, Godavarthy et al. [39] have discussed the influence of errors in critical point value on model stability and have suggested that critical properties can vary by as much as 3% depending on the experimental method used. A survey of the literature reveals that the vapor pressure prediction of highly polar fluids such as alcohols is generally not as accurate, regarding the average error of 9.6% for the generalized SVRC relation [39]. This places further demands on vapor pressure modeling efforts since reliable low-pressure experimental data are not plentiful. Referring to Table 1 one may conclude that the Cox equation is useful for correlating vapor pressures of pure fluids over a wide temperature range, however, the ability of the three-parameters form to extrapolate toward low pressures has been emphasized in the literature [36,43]. Because of a need for limiting the scope of this work, only the original Cox relationship with three adjustable parameters (Eq. (3)) has been employed in the main structure of the solvation-based model framework.

0.2315 × 101 0.1726 × 100

(4)

(K)

exp(q0 + q1 T + q2 T 2

b

(3)

647.35

+ q3 T 3 + q4 T 4 )



exp(q0 + q1 T + q2 T 2 )

Tc

Tc T



q4



ln (P0 )COX-FP = ln(Pc ) + 1 −

Tc T

q3



q2

k=0

Total runs

(2)

where qk are substance-dependent adjustable parameters. The P0 (kPa) is the vapor pressure of a pure solvent at the system temperature T (K). The Pc (kPa) is the critical pressure with Tc (K) being the critical temperature. In the original relationship the number of adjustable parameters qk is three, but up to nine parameters have been used for describing highly accurate vapor pressures over a wide temperature range [36,42,43]. The most frequently used arbitrarily condition is the critical point or the normal boiling point. Depending on the critical properties of the solvent Pc (kPa) and Tc (K), the Cox equations with three-parameters (TP), Eq. (3), and five-parameters (FP), Eq. (4), are expressed as follows ln (P0 )COX-TP = ln(Pc ) + 1 −

[45–50,53,56,58]

qk T k

[45–47,49,50,56–58]



q1

exp

 m 

q0



Water e¯ = 0.9% Tc e¯ = 2.4% Fc Methanol e¯ = 1.8% T e¯ = 2.1% F Ethanol e¯ = 2.8% T e¯ = 2.3% F 1-Propanol e¯ = 4.6% T e¯ = 3.8% F 2-Propanol e¯ = 7.4% T e¯ = 12.5% F 1-Butanol e¯ = 5.8% T e¯ = 11.9% F

Tc 1− T

Solvent



Table 1 Regressed coefficients (qi ) of three-parameters (Eq. (3)) and five-parameters (Eq. (4)) Cox vapor pressure equations for pure fluids, and mean relative error (¯e)a of model estimates for the relevant systems.

P0 ln = Pc

[45–47,49,51,52,54,56,58]

corresponding to the critical temperature Tc (K), is expressed as an exponential of a polynomial expansion in temperature T (K)

[45–49,51,54–56,58]

A. Senol / Fluid Phase Equilibria 361 (2014) 155–170

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A. Senol / Fluid Phase Equilibria 361 (2014) 155–170

When dealing with a solvent mixture, the corresponding vapor pressure (P0 ) and critical properties (Pc and Tc ) of the solvent mixture can be derived from the properties of the individual components by applying the additional parameter estimation rule, Eq. (5), due to Dalton’s law for the pressure and Kay’s method for the pseudo-critical properties, Pc and Tc , respectively [60] P0 =

 i

pi ; Pc =



yi Pc,i ; Tc =

i



yi Tc,i

(5)

i

2. Basic principles of LSER modeling Marcus and co-workers [28–30] have revealed that the properties such as the distribution of nonelectrolyte solutes between water and an immiscible organic solvent can be well correlated by LSER. The general LSER form, Eq. (6), predicts the property XYZ in terms of five physical interaction parameters XYZ = XYZ0 + mV/100 + s( + dı) + bˇ + a˛

(6)

XYZ0 is an adjustable parameter for the distributed solute. LSER includes a cavity term for the molar volume of the solute (mV/100), a polarity/polarizability term [s( + dı)] measuring the endoergic effects of dipole–dipole and dipole-induced dipole interactions, and hydrogen bond-donation a˛ (HBD) and -acceptance bˇ (HBA) terms. The solvatochromic parameter  is an index of polarity/polarizability and ı is a polarizability correction parameter. The ˇ scale is the HBA (hydrogen-bond acceptor) ability of the solute to accept a proton in a solute-to-solvent hydrogen bond and ˛ is the HBD (hydrogen-bond donor) ability of the solute to donate a proton in a solvent-to-solute hydrogen bond. Marcus [29] has proposed the use of the Hildebrand solubility parameter ıH (MPa0.5 ) in Eq. (6) instead of the cavity term, when dealing with free energies of a solution.

salt species, namely, the charge of the ion, z, the normalized reciprocal of the Pauling crystal ionic radius of the ion (r in nm), r = 0.1/r, the normalized volume (v) occupied by the ion, v = 100v =  = R /10 with (400/3)(r 3 ), the normalized molar refractivity, RD D 3 −1 RD (cm mol ) being the molar refractivity of the ion [30,61], and the softness parameter of the ion ( p ). For large ionic species, the softness parameter should be omitted, because it plays a significant role only for smaller ions with r ≤ 0.25 nm [62]. Normalized properties are approximately in the range from 0 to 1, so that the relative contributions of various variables can be more readily compared. F is a concentration-dependent correction factor accounted for the concentration effect of the salt. For a meaningful statistical analysis of the dependencies of vapor pressure on the properties of the solvent and the ionic salt species, the properties included in the integration term with an exponent value greater than one have been treated as independent variables. The explicit form of the unified solvation model with the integrated properties (USMIP) is given by Eq. (8). The model covers two dependently varying parts, i.e. one part accounted for the vapor pressure of a pure solvent, P0 (log-mean), at the limiting conditions when the salt concentration is zero (Cs = 0%) and an expansion term with respect to the Hildebrand solubility parameter (ıH ) and solvatochromic parameters of the solvent (, ˛, ˇ, and ı), and the overall physical properties characterizing the ionic salt. The related of contributions physical property of the salt is the sum  of the indini ri ; vs =

vidual ions in the salt structure, i.e. rs =





 ni p,i ; RD,s =

i

i

ni vs ; p,s =

i

 , where n stands for the stoichiometric ni RD,i i

i

number of the ion in the salt structure. A concentration-dependent correction factor F = Cs /(100 − Cs ) is incorporated into the expansion term to account for two limiting conditions when either the salt concentration (wt%) is zero (Cs = 0%) or the solvent concentration is zero (Cs = 100%)

2.1. Solvation vapor pressure model for (solvent + salt) systems In this study, to formulate an appropriate expression for the vapor pressures of (solvent + salt) systems being statistically compatible with the vapor pressure data for a large set of solvents and ionic salts, the additional parameter estimation rule has been proceeded depending on the principles of LSER, Eq. (6) [28–30]. The design of a perceptible vapor pressure model for a salt-containing solvent calls for the use of the vapor pressure of a pure solvent, the hydrogen bonding indices of the solvent, the salt concentration and the physical properties of the ionic salt species to establish a unified model structure distinguishing the divergence of the solvent volatility due to a salting-in or salting-out effect. Study of numerous (solvent + salt) systems has revealed that the saturation vapor pressure of a salt-containing solvent P (log-mean) could be well correlated through a solvation-based relation. This property of relevant systems, defined in a log-basis scale, can be expressed through a generalized equation, Eq. (7), where the effect of vapor pressure of a pure salt on the overall magnitude of pressure is neglected. The assumption inherent in this approach is attributed to an additional effect of several physical descriptors. The generalized form of the solvation-based model framework is defined as follows Pr mod = Pr 0 + F(Pr solvent + Pr ion )

(7)

where the Prmod (log-mean) designates the modeled property; the Pr0 (log-mean) stands for the observed property at limiting conditions when the salt concentration (weight %) is zero (Cs = 0%); the Prsolvent represents the overall property of the solvent defined by its solubility and solvatochromic parameters (ıH , , ˇ, ˛ and ı); the Prion denotes the overall property characterizing the ionic

ln(P) = ln (P0 )COX-TP +

 Cs i i [(CH,i (ıH ) + C,i ( − 0.35ı) 100 − Cs i

+ Cˇ,i (ˇ) + C˛,i (˛)i ) + (Cr,i (rs ) + Cv,i (vs ) + C,i (p,s )i i

i

 + CR,i (RD,s ) )] i

i

(8)

where P (kPa) and P0 (kPa) stand for the saturation vapor pressures of the solvent with a salt effect and pure solvent alone, respectively. Ci and Cs (wt%) are the adjustable parameter and the salt concentration, respectively. So, an augmented version of the LSER has been performed to estimate the vapor pressures of relevant systems, which aims to capture the physics of hydrogen bond formation associated with the physical indices of the ionic salt species and solvent and the saturation vapor pressure of the pure solvent. It is presumed that the degree of expansion equal to one is adequate for achieving a reasonable accuracy in representing vapor pressures. It is expected that the integrated property-basis solvation model (USMIP), Eq. (8), should be in accordance with the boundary constraints and the behavior of the physical event. However, the USMIP approach calls for the use of a composition ratio correction factor (F) by reason of reducing the expansion term at the salt composition limit Cs = 0% for which P = P0 , or restricting the upper limit Cs = 100% where an indefinable character of the function appears. When dealing with a solvent mixture, the modified solubility and solvatrochromic parameters of the solvent mixture (ı∗H , *, ˇ*, ˛*, ı*) are derived from the corresponding parameters (ıH,i , i , ˇi , ˛i , ıi ) and liquid phase mol fractions on a salt free-basis (xi ) of the

A. Senol / Fluid Phase Equilibria 361 (2014) 155–170

individual components by applying a mean value estimation rule ı∗H =



xi ıH,i ; (∗ − 0.35ı∗ ) =

i

ˇ∗ =



xi ˇi ; ˛∗ =

i





xi (i − 0.35ıi );

i

xi ˛i

(9)

i

To quantify accurately the salt effect on the vapor pressure by a relation being statistically compatible with the vapor pressure data, it is also suggested the use of a reduced number of variables in the main structure of the unified solvation model framework. This limitation is overly dependent on the fact that the salt effect is accounted for explicitly by its concentration level. Moreover, there is not available database of descriptors for the ionic liquid species, therefore, the generalized model had to be carried out with fewer than all of the independent variables. The general form of the reduced property-basis solvation model is given by Eq. (10) ln(P) = ln (P0 )COX-TP +

 Cs i i [KH,i (ıH ) + K,i ( − 0.35ı) 100 − Cs i

i

i

+ Kˇ,i (ˇ) + K˛,i (˛) ]

(10)

where Ki are the adjustable parameters. Both the integrated property-basis and reduced property-basis solvation models, Eqs. (8) and (10), have been executed using the three-parameters Cox equation, Eq. (3), to account for the vapor pressure of a pure solvent. Further, the reduced property-basis solvation model, Eq. (10), has been performed assuming two ranges of the degree of expansion i, i.e. i = 1 (with four coefficients) and i = 2 (with eight coefficients). For the expansion degree i = 1, the unified solvation model with the integrated properties (USMIP), Eq. (8), is rearranged to eightparameters equation, Eq. (11), involving nine physical descriptors Cs [(CH ıH + C ( − 0.35ı) + Cˇ ˇ ln(P) = ln (P0 )COX-TP + 100 − Cs  + C˛ ˛) + (Cr rs + Cv vs + C p,s + CR RD,s )]

(11)

Similarly, the reduced property-basis solvation model, Eq. (10), is rearranged to four-parameters Eq. (12) and eight-parameters Eq. (13) for the degrees of expansion i = 1 and i = 2, respectively ln(P) = ln (P0 )COX-TP +

Cs (KH,1 ıH + K,1 ( − 0.35ı) 100 − Cs

+ Kˇ,1 ˇ + K˛,1 ˛)

ln(P) = ln (P0 )COX-TP +

(12)

Cs   [(KH,1 ıH + K,1 ( − 0.35ı) 100 − Cs

    + Kˇ,1 ˇ + K˛,1 ˛) + (KH,2 (ıH ) + K,2 ( − 0.35ı) 2

  + Kˇ,2 (ˇ) + K˛,2 (˛)2 )] 2

2

(13)

In Eqs. (11)–(13), P (kPa) and P0 (kPa) stand for the saturation vapor pressures of the solvent with a salt effect and pure solvent alone, respectively, Ci and Ki are the adjustable parameters and Cs (wt%) is the salt concentration. Consequently, both the integrated property-basis and reduced property-basis solvation models, Eqs. (11)–(13), satisfy two limiting conditions in accordance with the theory, i.e. if the salt concentration Cs → 0% the vapor pressure of the (solvent + salt) system P approaches P0 of pure solvent, and if Cs → 100% the vapor pressure P is indefinable in the nature, i.e. the conditions circumscribing the validation limits of prediction. This fact indicates common strategy

159

for designing the unified model structure. To avoid dealing with the indefinable upper solubility range of the salt, it is assumed that the vapor pressure P is continually varying with the salt concentration by neglecting the antagonistic changes about the saturation limit of the salt. So, the solid phase contribution effect of the salt is accounted for implicitly in the solvation-based models. Because the solubility is varying with the temperature and the salt structure, it is highly difficult to achieve a model structure including upper solubility range of the salt, along with distinguishing the divergence of the vapor pressure about this prescribed concentration limit. Indeed, a dependently variation of the vapor pressure and temperature among the entire concentration range has been proceeded by reason of reducing the number of degrees of freedom. For the sake of simplicity, the effect of the physical characteristics of the ionic salt species is neglected or incorporated into the adjustable coefficients Ki of Eq. (10). This consequence is thought to be an acceptable basis for establishing the final reduced model structure. It is expected that both structural forms of the solvation-based model framework are able to simulate satisfactorily the behavior of infinitely or highly soluble ionic liquids and salts for the concentration range applicable to a separation process with the salt effect. In conjunction with the three-parameters Cox equation (Eq. (3)), a simplified approach on the salt concentration-basis can be expressed by assuming a third order polynomial concentration effect to represent well the modeled property given by Eq. (14). The temperature variable T (K) has been incorporated into the polynomial concentration term of Eq. (14), since the solubility of the salt is intimately connected to the temperature 1 lk (Cs )k = ln (P0 )COX-TP T 3

ln(P) = ln (P0 )COX-TP +

k=1

1 + (l1 Cs + l2 Cs2 + l3 Cs3 ) T

(14)

where lk and Cs (wt%) are the adjustable parameter and the salt concentration, respectively. The simplified concentration-dependent model, Eq. (14), allows an alternative computation of vapor pressures of salt-containing solvent systems, being improved especially for the solubility range of the salt. 3. Summary of the evaluated (solvent + salt) systems The reliability analysis of existing vapor pressure models has been processed against fifteen (solvent + salt) systems covering different types of salts with univalent and divalent species capable of dipole–dipole interaction with the solvent, regarding the solubility range of the salt. Table 2 presents a brief summary of the properties of the studied systems, being selected from various classes, i.e. water + lithium chloride (I) [63], ethanol + lithium chloride (II) [64], water + zinc chloride (III) [63], water + LiCl/ZnCl2 (IV) [63], 1-propanol + calcium chloride (V) [65], 2-propanol + calcium chloride (VI) [65], 1-butanol + calcium chloride (VII) [65], methanol + ammonium bromide (VIII) [66], water + ammonium bromide (IX) [66], methanol + lithium nitrate (X) [67], 1-propanol + lithium nitrate (XI) [68], water + lithium nitrate (XII) [68], water + phosphoric ionic liquid (XIII) [69], methanol + phosphoric ionic liquid (XIV) [69], ethanol + phosphoric ionic liquid (XV) [69]. The reliability of existing models has been analyzed statistically using I to XV concise sets of vapor pressure data relative to the considered (solvent + salt) systems. The concentration (Cs ) ranges of the salts (by per cent mass weight), for which the model reliability has been tested, are summarized in Table 2. As evident from Table 2, the study concerns with the experimental vapor pressure data for salt-containing six protic solvents, water (dipole moment  = 6.14 × 10−30 Cm;

160

A. Senol / Fluid Phase Equilibria 361 (2014) 155–170

Table 2 Summary of the properties of the evaluated (solvent + salt) systems.

I. II. III. IV. V. VI. VII. VIII. IX. X. XI. XII. XIII. XIV. XV. a b c d

Systema

Runs

Water + lithium chloride (LiCl) Ethanol + lithium chloride (LiCl) Water + zinc chloride (ZnCl2 ) Water + mixed salt (LiCl/ZnCl2 )c 1-Propanol + calcium chloride (CaCl2 ) 2-Propanol + calcium chloride (CaCl2 ) 1-Butanol + calcium chloride (CaCl2 ) Methanol + ammonium bromide (NH4 Br) Water + ammonium bromide (NH4 Br) Methanol + lithium nitrate (LiNO3 ) 1-Propanol + lithium nitrate (LiNO3 ) Water + lithium nitrate (LiNO3 ) Water + phosphoric ionic liquid [MMIM][DMP]d Methanol + phosphoric ionic liquid [MMIM][DMP]d Ethanol + phosphoric ionic liquid [MMIM][DMP]d

9 30 10 16 24 24 24 30 30 16 17 13 15 15 15

Table 3 Hildebrand solubility parameter (ıH ) and solvatochromic parameters (, ˇ, ˛, ı) of solvents. Solvent

 a,b

ˇ a,b

˛ a,b

ıH

Water Methanol Ethanol 1-Propanol 2-Propanol 1-Butanol

1.09 0.40 0.40 0.40 0.40 0.40

0.47 0.42 0.45 0.45 0.51 0.45

1.17 0.35 0.33 0.33 0.31 0.33

47.9 29.7 26.0 24.3 23.5 23.3

a

c d

Reference

10.0–30.0 1.8–15.9 30.0–50.0 20.0–50.0 5.2–21.6 4.2–17.6 1.5–7.3 6.3–19.3 6.7–38.1 2.0–40.2 1.1–14.5 3.1–36.0 20.0–60.0 20.0–60.0 20.0–60.0

[63] [64] [63] [63] [65] [65] [65] [66] [66] [67] [68] [68] [69] [69] [69]

The (solvent + salt) system categorized as a pseudo-component. The range of the salt concentration (by per cent weight) in which the data are selected. The mass ratio of salts, LiCl/ZnCl2 = 0.75. Ionic liquid [MMIM][DMP], 1-methyl-3-methylimidazolium dimethylphosphate.

dielectric constant ε = 80.4) and five polar alcohols, methanol ( = 5.70 × 10−30 Cm; ε = 32.8), ethanol ( = 5.66 × 10−30 Cm; ε = 24.3), 1-propanol ( = 5.53 × 10−30 Cm; ε = 20.1), 2-propanol ( = 5.54 × 10−30 Cm; ε = 18.3) and 1-butanol ( = 5.54 × 10−30 Cm; ε = 17.1) [40,70]. It is concluded from the physical properties of the solvents that water and methanol are the most polar solvents, thereby promoting more predominantly the salt effect. The small size of the water (or methanol) molecule permits it to approach closer and hence makes its dipole more effective in inducing dipole in the large ions. The induced dipoles are the larger, the larger the polarizabilities of the ions, causing the ions to favor water (or methanol) on this account. The substance-dependent regressed coefficients of the Cox equations, Eqs. (3) and (4), and the critical properties of the solvents are given in Table 1. Table 3 presents the solubility and solvatochromic parameters of the studied solvents. The physical properties of the ions involving in the salt structure are listed in Table 4. The studied systems span the range of various inorganic and organic salts. Characterization of the vapor pressure of a solvent (or a solvent mixture) is dependent strongly on the nature of the salt. It has been found that the reduced (or enhanced) magnitude of vapor pressure is very sensitive to the physical characteristics of the ionic salt species, like the ionic strength, radius and valence of the ions [13,17,30,63–69,71]. Regarding Table 2, lithium salts are selected as the favorable candidates for testing the solvation-based model framework. Because of a small radius and high charge density of the lithium ion, being the first member in the 1A series of alkaline metals, lithium salts exhibit very strong hydration and high solubility in water ranging about 20 mol kg−1 H2 O. This property makes the lithium salt, especially lithium chloride, a serious candidate for

b

Concentration of saltb (wt.%)

Due to Kamlet et al. [28]. Due to Marcus [29]. Due to Riddick et al. [47]. Due to Barton [72].

c,d

(MPa0.5 )

ı a,b 0.0 0.0 0.0 0.0 0.0 0.0

using as a regenerable absorbent in desiccant cooling and drying systems [63]. Vapor pressure data for aqueous solutions of divalent metals and ionic liquids are scarce in the literature, which are mostly limited to the water activity at 298 K. One of the new realms for flourishing applications of ionic liquids is to serve as electrolytes, which possess very low vapor pressures. The novel properties of ionic liquids like their nonvolatile and non-toxic natures in addition of their intrinsic ionic conductivity make them promising alternative for electrolyte application [7–12]. Regarding the ionic liquid as a separation agent, it might be superior to the commonly used entrainers due to its nonvolatility, less causticity and good modifier of the solvent volatilities [17,69,71]. Besides these advantages, the published literature data for the dominating effect of ionic liquids are quite limited and, consecutively, require further analysis [71]. A survey of the literature has revealed that among numerous published models only few works are capable of describing qualitatively the salting effect phenomenon on vapor pressure for the whole concentration range of the salt [13,17–19,26,27,23,63,71]. Attempts have been made to interpret theoretically the interactive salt effect on the vapor pressure and other properties of equilibrium systems by postulating a continual thickening of packet-like aggregated segments between the solvent molecules and the Table 4 The physical properties of the ionic spwucture.a Ion

zb

Cations Li+ NH4 + Ca2+ g Zn2+

1 1 2 2

Anions Cl− Br− NO− 3 SCN−

−1 −1 −1 −1

a

 RD = RD /10 f

r = 0.1/r c

v = 100v d

p

1.45 0.68 0.74 1.33

0.14 1.36 1.05 0.18

−1.02 −0.60 −0.66 0.35

0.01 0.47 0.50 0.12

0.55 0.51 0.53 0.447

2.48 3.15 2.83 4.05

−0.09 0.17 0.03 0.85

0.82 1.16 1.02 1.65

e

Due to Marcus et al. [30]. z denotes the charge of the ion. r (nm) represents the Pauling crystal ionic radius of the ion (coordination number 6). d v (nm3 ) stands for the volume occupied by the ion, v = 100v = (400/3)(r 3 ). e The softness parameter ( p ) played significant role, especially, for smaller ions with r ≤ 0.25 nm [62]. f RD (cm3 mol−1 ) designates the molar refractivity of the ion for the mean sodium D line at infinite dilution in water [61]. g The properties of Ba2+ simulated for Ca2+ . b

c

0.4440 × 10−1 0.1629 × 100 −0.3436 × 100 −0.4687 × 10−1 0.4743 × 10−1 (Solvent + salt) systems as defined in Table 1. a

−0.3438 × 100 0.6543 × 10−2 e¯ = 0.82%

0.1227 × 101

0.1359 × 101 0.1290 × 101 −0.3444 × 100 −0.7220 × 100 −0.3417 × 101 0.3127 × 101 0.3887 × 10−1 e¯ = 0.71%

−0.5305 × 101

−0.1514 × 101 0.1514 × 101 −0.2763 × 100 −0.5137 × 10−1 0.1482 × 102 −0.2366 × 101 0.5853 × 10−1 e¯ = 1.24%

−0.7486 × 101

−0.5189 × 10−1 0.6594 × 100 −0.1169 × 10−1 0.1137 × 100 0.3886 × 100 −0.2675 × 100 −0.1140 × 10−2 e¯ = 0.70%

0.4909 × 10−1

0.5983 × 10−2 0.6150 × 100 −0.1102 × 10−1 0.7630 × 10−1 0.8837 × 10−3 −0.9213 × 100 0.2613 × 10−2 e¯ = 0.27%

0.5849 × 100

0.2320 × 100 −0.3847 × 10−1 −0.4135 × 10−1 −0.4064 × 100 −0.1571 × 101 −0.1531 × 101 −0.4367 × 10−1 e¯ = 1.68%

0.2867 × 101

−0.2182 × 10−1 −0.1727 × 10−1 0.1445 × 10−1 0.1301 × 10−1 −0.1710 × 100 0.2197 × 100 −0.1753 × 10−2 e¯ = 2.91%

−0.1122 × 100

0.2688 × 10−2 −0.1645 × 101 −0.8187 × 10−1 0.7538 × 100 −0.1967 × 101 −0.1615 × 101 −0.4512 × 10−1 e¯ = 0.81%

−0.9141 × 100

−0.5454 × 10−1 0.6171 × 100 0.1877 × 10−1 0.6385 × 10−1 −0.9603 × 100 0.3730 × 100 0.1132 × 10−1 e¯ = 8.38%

−0.9998 × 100

0.5267 × 101 −0.1566 × 102 0.1806 × 101 −0.4083 × 101 −0.3005 × 101 −0.1853 × 101 −0.6459 × 10−1 e¯ = 4.02%

0.2887 × 102

0.8215 × 101 0.2798 × 101 −0.4224 × 101 0.5188 × 101 0.3006 × 102 −0.2827 × 102 −0.5998 × 100 e¯ = 7.32%

0.2397 × 102

0.2419 × 102 0.4308 × 102 −0.2887 × 102 −0.7464 × 102 0.1235 × 102 0.1478 × 101 e¯ = 7.11%

−0.6035 × 102

161

System I  = 3.93; System II  = 0.95; System III  = 2.60; System IV  = 4.37; System V  = 0.83; System VI  = 2.81; System VII  = 1.50; System VIII  = 0.25; System IX  = 0.66; System X  = 1.39; System XI  = 1.02; System XII  = 0.91;

C Cv Cr C˛ Cˇ C CH

Vapor pressure data obtained for fifteen (solvent + salt) systems have been processed to analyze the reliability of both the integrated property-basis and the reduced property-basis solvation relations, Eqs. (11)–(13), and the concentration-based approach, Eq. (14), along with considering P0 property to represent the vapor pressure of a pure solvent given by Eq. (3). The adjustable coefficients Ci , Ki and li pertaining to the unified solvation model USMIP, Eq. (11), the reduced property-basis solvation models, Eqs. (12) and (13), and the concentration-based model, Eq. (14), respectively, have been regressed through the multivariable linpack algorithm [60]. A multivariate convergence has been executed using the substancedependent coefficients of pure solvents from Table 1, and the solubility and solvatochromic indices of the solvent and physical properties of the ionic salt species given in Tables 3 and 4. Depending on the regressed coefficients Ci , Ki and li of Eqs. (11)–(14), the reliability of existing models has been analyzed statistically by comparing the modeled performance with the observed properties in terms of the mean relative error (¯e) and root-mean-square deviation (). The resulting adjustable parameters Ci s, Ki and li of Eqs. (11)–(14) and the deviation statistics of model reliability analysis are summarized in Tables 5–7, respectively. Besides the statistical analysis of existing models, a graphical confidence test has been also applied to confirm the model structure, which in turn, quantifies visually responses of the model fits. In general, the obtained regression results with promising e¯ (P) and (P) values rarely demonstrate a physical disagreement between modeled and observed performance. For the verification, the observed property has been compared with the modeled performance through plotting the mean deviation against the temperature or applying a cross-validated diagonal test, as depicted in Figs. 1–8. Figs. 1–8 and Tables 5–7 present a quantitative assessment of the fits achieved for the solvation-based approaches, Eqs. (11)–(13), and concentration-dependent model Eq. (14) with regard to the mean deviations of P (kPa) depending on T (K). The detailed deviation statistics of the solvation-based models with four- and eight-parameters from Figs. 1–6 and Tables 5 and 6 demonstrate a rigorous validation of the considered model structure and its applicable extension. Inspection of Figs. 1 and 2 and Table 5 reveals that the integrated property-basis solvation model involving nine physical descriptors of the solvent and salt USMIP (Eq. (11)) is precise in reproducing observed vapor pressures of diverse (solvent + salt) systems, generally within their experimental uncertainties (1–3% absolute deviation). The same remarks hold for the reduced property-basis solvation models Eqs. (12) and (13), when either four or eight solvatochromic indicators are used to characterize hydrogen-bond association of each solvent. Figs. 1–6 and Tables 5 and 6 illustrate clearly that both structural forms of the solvation model framework, Eqs. (8) and (10), track well the regular variation of the vapor pressure being overly sensitive

Systema

4.1. Reliability analysis of existing vapor pressure models

Table 5 Coefficients Ci of the unified solvation model with integrated properties (USMIP), Eq. (11), and mean relative errors (¯e) and root-mean-square deviations () evaluated for relevant systems.

4. Results and discussion

CR

salt ions with increasing the salt concentration [13,19,71]. The aggregated structure is presumably deactivated at a rather low composition level of the salt. This tendency could be verified by analyzing the difference between the saturation vapor pressures of a solvent with and without an electrolyte. In this work, it has been endeavored to correlate vapor pressures for the extended matrix of solvents and ions by means of a multivariable regression involving the optimum number of variables. Subsequently, the predictive capability of the solvation models has been examined on the binary solvent mixture involving a salt.

0.1892 × 101

A. Senol / Fluid Phase Equilibria 361 (2014) 155–170

162

Table 6 Coefficients Ki of the reduced property-basis solvation model with four-parameters Eq. (12) and eight-parameters Eq. (13) and mean errors (¯e) and root-mean-square deviations () evaluated for relevant systems. Systema

a b

KH,1

K,1

Kˇ,1

Kˇ,2

K˛,2

−0.2057 × 10−2 0.3537 × 10−1

−0.2300 × 101 0.1497 × 101

0.1993 × 101 0.3777 × 101

0.6161 × 10−1 0.2955 × 101

−0.2521 × 10−2

0.4074 × 100

−0.1359 × 100

−0.3524 × 101

0.1371 × 10−1 −0.1173 × 100

−0.1137 × 102 −0.1271 × 101

0.2259 × 101 0.3391 × 101

−0.4147 × 10−9 0.2655 × 101

0.2283 × 10−2

−0.1642 × 102

−0.8062 × 101

0.6409 × 101

0.1328 × 10−2 0.1198 × 100

−0.4779 × 100 −0.1615 × 101

0.1594 × 101 0.7372 × 101

−0.5854 × 100 −0.9556 × 100

−0.1300 × 10−1

0.9493 × 100

−0.3339 × 101

0.1903 × 101

−0.3333 × 10−2 0.6310 × 10−2

−0.1646 × 100 −0.9516 × 100

0.6708 × 100 0.5834 × 100

−0.6625 × 100 −0.3694 × 100

0.1911 × 10−3

0.2309 × 100

−0.4643 × 100

−0.3772 × 100

−0.3179 × 100 −0.3347 × 10−1

−0.4429 × 101 0.4906 × 101

−0.9358 × 101 0.1392 × 101

0.3996 × 102 0.2309 × 100

−0.5179 × 10−3

−0.3006 × 101

−0.1173 × 102

0.7264 × 101

−0.9804 × 10−3 −0.8154 × 10−3

−0.6963 × 10−1 −0.2506 × 10−1

0.1071 × 100 0.5232 × 10−1

0.3648 × 10−1 0.7716 × 10−1

−0.2782 × 10−4

0.1213 × 100

−0.3785 × 100

−0.2598 × 10−1 −0.1881 × 100

−0.2748 × 101 0.4426 × 101

0.2245 × 101 0.5789 × 100

−0.1912 × 101 0.4334 × 101

0.4138 × 10−2

−0.2744 × 102

−0.9378 × 101

0.3341 × 102

−0.4592 × 10−1 −0.1358 × 10−1

0.7828 × 100 −0.5892 × 100

0.4293 × 100 0.8886 × 100

0.1748 × 101 −0.9212 × 100

−0.6583 × 10−3

0.2894 × 101

0.3357 × 101

−0.1181 × 101

−0.1186 × 10−2 −0.3892 × 10−2

0.2906 × 10−1 −0.4365 × 10−1

−0.5196 × 10−1 0.1649 × 100

−0.8925 × 10−1 −0.1175 × 100

−0.3226 × 10−4

0.2576 × 100

0.1296 × 100

−0.8814 × 10−1

−0.2428 × 10−1 0.5268 × 10−1

0.1646 × 101 −0.4008 × 101

−0.3127 × 101 −0.5107 × 101

0.6580 × 100 −0.4271 × 101

0.6037 × 10−3

0.2771 × 102

−0.1816 × 102

0.6290 × 101

0.2733 × 10−1 0.4501 × 10−1

−0.3072 × 101 0.1258 × 100

−0.1905 × 101 −0.4137 × 101

0.9889 × 100 −0.2164 × 101

0.2565 × 10−2

−0.3026 × 101

0.2510 × 101

−0.1104 × 102

0.7395 × 10−2 −0.2373 × 10−2

−0.7840 × 100 0.2953 × 101

−0.5496 × 100 −0.2806 × 101

0.8622 × 10−1 −0.1636 × 101

0.1035 × 10−2

−0.4049 × 101

0.1115 × 102

−0.4069 × 100

−0.3664 × 10−2 0.3723 × 10−2

0.1682 × 100 −0.3819 × 10−2

−0.5435 × 100 −0.2905 × 100

−0.9095 × 10−2 0.9489 × 10−1

−0.2462 × 10−4

0.1404 × 100

−0.1121 × 101

−0.1971 × 100

−0.5105 × 10−2 −0.5460 × 10−1

−0.1279 × 10−1 0.4805 × 101

−0.1037 × 101 0.8421 × 101

0.5014 × 100 0.7707 × 101

−0.3135 × 10−2

−0.1310 × 102

0.5596 × 101

−0.2512 × 102

−0.1608 × 10−2 0.1017 × 10−1

−0.6755 × 100 −0.1363 × 101

−0.3888 × 100 −0.2109 × 100

−0.3217 × 10−1 0.8685 × 100

−0.4779 × 10−3

−0.2891 × 101

0.2525 × 101

−0.1233 × 101

(Solvent + salt) systems as defined in Table 1. F = coefficients of four-parameters Eq. (12), E = coefficients of eight-parameters Eq. (13).

K˛,1

KH,2

K,2

0.8633 × 10−1

A. Senol / Fluid Phase Equilibria 361 (2014) 155–170

System I  = 3.96; e¯ = 7.07%F b  = 3.96; e¯ = 7.07%E b System II  = 0.95; e¯ = 7.32%F  = 0.95; e¯ = 7.32%E System III  = 2.55; e¯ = 4.06%F  = 2.55; e¯ = 4.06%E System IV  = 4.36; e¯ = 8.38%F  = 4.37; e¯ = 8.38%E System V  = 0.83; e¯ = 0.81%F  = 0.83; e¯ = 0.81%E System VI  = 2.80; e¯ = 2.91%F  = 2.81; e¯ = 2.91%E System VII  = 1.50; e¯ = 1.68%F  = 1.50; e¯ = 1.68%E System VIII  = 0.25; e¯ = 0.27%F  = 0.25; e¯ = 0.27%E System IX  = 0.66; e¯ = 0.70%F  = 0.66; e¯ = 0.70%E System X  = 1.08; e¯ = 0.90%F  = 1.08; e¯ = 0.91%E System XI  = 1.02; e¯ = 0.71%F  = 1.02; e¯ = 0.71%E System XII  = 1.69; e¯ = 1.56%F  = 1.69; e¯ = 1.56%E System XIII  = 1.59; e¯ = 3.49%F  = 1.59; e¯ = 3.49%E System XIV  = 2.08; e¯ = 5.35%F  = 2.08; e¯ = 5.35%E System XV  = 1.85; e¯ = 4.32%F  = 1.85; e¯ = 4.32%E

A. Senol / Fluid Phase Equilibria 361 (2014) 155–170

163

Table 7 Coefficients li of concentration-based vapor pressure equation with a temperature term Eq. (14), and mean errors (¯e) and root-mean-square deviations () evaluated for relevant systems. Systema

Eq. (14)

System I  = 0.82; e¯ = 0.88% System II  = 0.32; e¯ = 1.84% System III  = 0.56; e¯ = 0.65% System IV  = 0.93; e¯ = 1.36% System V  = 0.37; e¯ = 0.35% System VI  = 0.41; e¯ = 0.27% System VII  = 0.54; e¯ = 1.03% System VIII  = 0.29; e¯ = 0.30% System IX  = 0.34; e¯ = 0.23% System X  = 0.45; e¯ = 0.31% System XI  = 0.49; e¯ = 0.36% System XII  = 0.22; e¯ = 0.16% System XIII { = 0.20; e¯ = 0.36%} System XIV  = 0.38; e¯ = 1.02% System XV  = 1.06; e¯ = 2.40% a

l1

l2

l3

−0.2000 × 101

0.1003 × 100

−0.4696 × 10−2

−0.1846 × 101

−0.6761 × 100

−0.3710 × 10−2

0.1257 × 101

−0.7863 × 10−1

−0.1364 × 10−3

−0.2177 × 101

0.8250 × 10−1

−0.3371 × 10−2

−0.4111 × 101

0.2596 × 100

−0.8432 × 10−2

0.7558 × 101

−0.9035 × 100

0.2580 × 10−1

−0.1560 × 102

0.3204 × 101

−0.2389 × 100

−0.8767 × 100

−0.3911 × 10−2

−0.2774 × 10−3

0.1482 × 100

−0.3961 × 10−1

0.2819 × 10−3

−0.3000 × 101

−0.7025 × 10−1

−0.6787 × 10−3

−0.7828 × 101

0.6716 × 100

−0.3231 × 10−1

−0.1245 × 101

−0.6286 × 10−1

−0.5621 × 10−3

−0.6940 × 100

0.2831 × 10−1

−0.9574 × 10−3

−0.8904 × 100

0.3426 × 10−1

−0.1333 × 10−2

−0.4210 × 101

0.1698 × 100

0.2875 × 10−2

(Solvent + salt) systems as defined in Table 1.

30

A

20

100(1 - P. mod )/ P obs

10 0 -10 -20 -30 8

B

4 0 -4 -8 290

320

350

T (K)

380

Fig. 1. Mean deviations in calculated vapor pressures through the integrated property-basis solvation model with eight parameters, Eq. (11). (A) Systems I to VII: I (+), II (*), III (♦), IV (), V (), VI (), VII (). (B) Systems VIII to XII: VIII (), IX (夽), X (), XI (⊕), XII (×).

to the temperature and the composition of the (solvent + salt) system. It is concluded from Figs. 1–6 and Tables 5 and 6 that the solvation-based models correlate the saturation vapor pressure of the associated (solvent + salt) systems reasonably accurately over the entire composition range, yielding the overall mean relative errors (¯e) and root-mean-square deviations () of e¯ (P) = 3.0% ((P) = 1.77) for Eq. (11), e¯ (P) = 3.2% ((P) = 1.81) for Eq. (12), e¯ (P) = 3.3% ((P) = 1.81) for Eq. (13), and e¯ (P) = 0.8% ((P) = 0.49) for Eq. (14), considering all the systems studied. The solvation-based models Eqs. (11)–(13) are quite accurate in reproducing the observed properties with an overall mean deviation of e¯ = 3.2% ( = 1.80) for all the considered systems. The reduced property-basis solvation model (Eqs. (12) and (13)) proved to be slightly less accurate for the system IV with regard to the salt concentrations tested, yielding e¯ = 8.4% ( = 4.37). This moderate success of representation of the data pertaining to system IV could be ascribed to the complementary effect of the binary salt system at extremely high concentration levels of both salts in a narrow temperature range, probably being responsible for an irregular change of the vapor pressure. This leads to an incomplete convergence appearing for both Eqs. (12) and (13) relative to the system IV (Figs. 3 and 5). Except for the system IV, any drastic deviation of estimates has not been found for neither Eq. (11) nor Eqs. (12) and (13) regarding systems I–XV. Referring to Figs. 1 and 2 and Table 5, it turns out that the integrated propertybasis model (USMIP) matches relatively well the observed performance over the entire composition range for all the considered systems. Specifically, Eq. (11) provides quite reliable vapor pressure estimates for systems VIII–XII with a mean deviation of e¯ = 0.7% ( = 0.85) as compared to e¯ = 2.2% ( = 1.28) for Eqs. (12) and (13) regarding systems VIII–XV. The simulating performance of Eq. (11) to reproduce the observed properties satisfactorily manifests the fact that the considered model structure with nine

164

A. Senol / Fluid Phase Equilibria 361 (2014) 155–170

30

A

20

100

100(1 - P. mod )/ P obs

P (kPa), modeled

10

10

0 -10 -20 -30 15

B

10 5 0

1

-5

1

10

P (kPa), observed

100

-10 -15

variables (physical descriptors) is found adequate for modeling vapor pressures of diverse (solvent + salt) systems. As can be seen from Figs. 3–6 and Table 6, excellent model estimates about equally precisely are obtained when either five or ten variables are employed in the structure of the reduced property-basis solvation model. This would call for the assumption that the contribution of the variables on a powered range could not improve the data fit. Inspection of Figs. 7 and 8 and Table 7 reveals that the concentration-based approach Eq. (14) is remarkably successful in representing vapor pressures, that is indicative for the consistency of a third order polynomial concentration effect being in good agreement with the behavior of (solvent + salt) systems. The concentration-based model provides reliable estimates with a mean error ranging from e¯ = 0.2% ( = 0.22) to e¯ = 2.4% ( = 1.06) for systems I–XV. Consequently, Figs. 1–6 and Tables 5 and 6 manifest the fact that the proposed two structural forms of the solvation-based model framework, Eqs. (8) and (10), are able to simulate precisely vapor pressure data attributed to different kinds of (solvent + salt) systems. However, for some circumstances relative to the relevant systems, it is presumed that the vapor pressure is varying continually with the temperature and the salt concentration over the entire working range. This fact indicates a common strategy for designing a unified vapor pressure model framework. The results obtained suggest an underlying physical significance for the model variables and show an excellent potential for generalized predictions, i.e. the concepts of hydrogen-bond association and dipole–dipole interaction combining with the physical properties of the salt and solvent can be contemplated for correlating accurately saturation vapor pressures of solvent systems involving a salt. The graphical confidence tests, i.e. the cross-linked diagonal test and the mean deviation test, indicate that an overall unreliable disagreement does not occur for neither solvation-based models nor concentration-based approach, because the distribution along the nil-error assigned line (for Figs. 1, 3, 5 and 7) or the diagonal line (for Figs. 2, 4, 6 and 8) remained in an acceptable narrow band. Further, the random pattern of comparison points at each side of the nil-error assigned line or the diagonal line implies that the

290

320

350

T (K)

380

410

Fig. 3. Mean deviations in calculated vapor pressures through the reduced propertybasis solvation model with four parameters, Eq. (12). (A) Systems I to VII: I (+), II (*), III (♦), IV (), V (), VI (), VII (). (B) Systems VIII to XV: VIII (), IX (夽), X (), XI (⊕), XII (×), XIII (), XIV (), XV ( ).

existing models are almost free of systematical errors. Referring to these figures one may conclude that the models yielded a fair distribution verifying the goodness-of-fit with a mean error of e¯ = 3.2% for the solvation-based models and e¯ = 0.8% for the concentration-based approach, considering all the systems studied. Typically, the concentration-based approach has proved

100

P (kPa), modeled

Fig. 2. Presentation of cross-validated diagonal test results for the integrated property-basis solvation model, Eq. (11). The observed performance P (kPa) against the modeled values for systems I to XII: I (+), II (*), III (♦), IV (), V (), VI (), VII (), VIII (), IX (夽), X (), XI (⊕), XII (×).

10

1 1

10

P (kPa), observed

100

Fig. 4. Presentation of cross-validated diagonal test results for the reduced property-basis solvation model with four parameters, Eq. (12). The observed performance P (kPa) against the modeled values for systems I to XV: I (+), II (*), III (♦), IV (), V (), VI (), VII (), VIII (), IX (夽), X (), XI (⊕), XII (×), XIII (), XIV (), XV ( ).

A. Senol / Fluid Phase Equilibria 361 (2014) 155–170

12

A

20

8

10

4

0

0

100(1 - P. mod )/ P obs

100(1 - P. mod )/ P obs

30

-10 -20 -30 15

B

10

-8 -12 12

0

0

-5

-4

-10

-8

-15

-12

350

T (K)

380

290

410

Fig. 5. Mean deviations in calculated vapor pressures through the reduced propertybasis solvation model with eight parameters, Eq. (13). (A) Systems I to VII: I (+), II (*), III (♦), IV (), V (), VI (), VII (). (B) Systems VIII to XV: VIII (), IX (夽), X (), XI (⊕), XII (×), XIII (), XIV (), XV ( ).

B

8 4

320

A

-4

5

290

165

320

350

T (K)

380

410

Fig. 7. Mean deviations in calculated vapor pressures through the concentrationdependent vapor pressure model with three parameters, Eq. (14). (A) Systems I to VII: I (+), II (*), III (♦), IV (), V (), VI (), VII (). (B) Systems VIII to XV: VIII (), IX (夽), X (), XI (⊕), XII (×), XIII (), XIV (), XV ( ).

P (kPa), modeled

100

10

1 1

10

P (kPa), observed

100

Fig. 6. Presentation of cross-validated diagonal test results for the reduced property-basis solvation model with eight parameters, Eq. (13). The observed performance P (kPa) against the modeled values for systems I to XV: I (+), II (*), III (♦), IV (), V (), VI (), VII (), VIII (), IX (夽), X (), XI (⊕), XII (×), XIII (), XIV (), XV ( ).

166

A. Senol / Fluid Phase Equilibria 361 (2014) 155–170

P (kPa), modeled

100

10

1 1

10

P (kPa), observed

100

Fig. 8. Presentation of cross-validated diagonal test results for the concentrationdependent vapor pressure model with three parameters, Eq. (14). The observed performance P (kPa) against the modeled values for systems I to XV: I (+), II (*), III (♦), IV (), V (), VI (), VII (), VIII (), IX (夽), X (), XI (⊕), XII (×), XIII (), XIV (), XV ( ).

slightly more successful than the solvation-based ones, due to a reduced number of regressed coefficients. 4.2. Statistical design factors of model reliability Validation of the model structure processes in the most critical assessment for the model stability and its predictive capability. In the design application based on a simulation algorithm involving a vapor pressure model, the average uncertainty of the modeled performance must be quantified. The reliability of existing vapor pressure models has been analyzed by application of a log-ratio objective function (OF) given by Eq. (15) [73]. Unlike e¯ and  statistical factors, the log-ratio OF allows for the degrees of freedom associated with the residual sum of the squares, regarding the critical confidence band. For this reason, the log-ratio objective function is generally a more accurate goodness-of-fit measure than e¯ and  factors X = ln(Yobs /Ymod )

(15)

where X is the objective function of model reliability; Yobs is the observed value of performance (vapor pressure, Pobs ); Ymod is the modeled value. In the design practice of a separation process with a salt, avoidance of failure is assumed to be critical at only one end of the confidence band. Using the overall design factor Fod , ¯ and the safety facthe model normalization factor Fm = exp(X), tor Fs = exp(tS), the following expression for upper (Fod = Fm Fs ) and lower (Fod = Fm /Fs ) critical bounds is obtained relative to the vapor pressure [73] Pdesn = (ex¯ e±tS )Pmod = (Fm Fs±1 Pmod ) = Fod Pmod

(16)

where Pdesn (kPa) is the design value; Pmod (kPa) is the modeled value; X¯ = Xi /N is the mean of the objective function; Xi is OF for the i-th observation; N is the number of observations;



2

0.5

S = [ (X¯ − Xi ) /(N − 1)] is standard deviation of the OF; t is Student’s t for a certain degree of confidence. The overall design factor, Fod , is the overall correction factor that must be applied to the model to achieve a specified probability of success. For a selected

95% probability of success (1 chance in 20 of failures) and an infinite number of degrees of freedom related to a 90% confidence band Student’s t = 1.645 due to Johnson and Leone [74]. The results were generated and assessed for the verification of the solvation-based models with four- and eight-parameters, Eqs. (11)–(13), and the concentration-based approach, Eq. (14), by processing about 300 dataset points. The resulting statistical factors of existing vapor pressure models are presented in Table 8. These factors indicate the need to overdesign as a result of insufficient model reliability, e.g. a safety factor of 1.065 for the unified solvation model with the integrated properties (Eq. (11)) means that the vapor pressure estimated through this model must be, on the average, 6.5% larger than the modeled value to achieve a specified (95%) confidence of success. Study of Table 8 reveals that the standard deviations of solvation-based models are remarkably close to 0.035, Required overall design factors (Fod ) are ranging about 1.07 for upper critical bound and 0.95 for lower critical bound, reflecting the accuracy of correlation of vapor pressures pertaining to the (solvent + salt) systems. Among the performed solvation-based models, the USMIP model, Eq. (11) gives a slightly more reliable overall design factor (Fod ) of 1.065, as compared to 1.072 for Eqs. (12) and (13), respectively, along with considering the upper critical bound. The required Fod factors for the lower critical bound tend toward a limiting value of 0.95 (Table 8). The normalization factor Fm is ranging about 1.01 and the safety factor Fs is ranging about 1.06, being indicative for the ability of the LSER-based solvation models, Eqs. (8) and (10), to accurately represent the vapor pressures of solvent systems with a salt effect. Consequently, the deviation statistics of the solvation model framework refer to a reasonable goodness-offit assessing Fod very close to unity. Actually, the statistical results strengthen the validation of the proposed two structural forms of the solvation model and suggest an underlying physical significance for generalized predictions. Regarding the normalization and safety factors in Table 8, one may conclude that Eq. (8) is the most appropriate approach for a design application yielding Fm = 1.01 and Fs = 1.05. In fact, all the vapor pressure data show good compliance with a model structure involving nine physical descriptors. The concentration-based approach, Eq. (14), reproduces the observed performance with a standard deviation of S = 0.01, being about three times smaller in comparison with those of Eqs. (12) and (13). According to the one-sided upper-bound test the concentration-based relation, Eq. (14), proved to be considerably accurate, yielding Fod = 1.017. This is also confirmed by the ranges of Fm = 1.0002 and Fs = 1.0165, manifesting the fact that this approach matches reliably the observed performance. However, this could be attributable to the reduced number of degrees of freedom for Eq. (14) as compared to that of the solvation-based model. In general, the statistical results would call for the assumption that an appropriate limitation of the exponent degree of expansion term and its variables should presumably improve the precision of data fit. In fact, an important concern is whether the estimates by Eqs. (8) and (10) actually confirm the behavior of the observed P in the whole working range. Inspection of Figs. 1–6 reveals that besides the accuracy of Eqs. (11)–(13), the solvation-based models also track the trend of vapor pressure variations seemly throughout the concentration ranges applicable to a separation process with the salt effect. However, it is essential that this phenomenon should have a significant impact on the implementation of a simulation algorithm incorporating Eqs. (8) and (10) with a rate-based method and equilibrium conditions due to Eq. (1). Consequently, it is expected that Eqs. (8) and (10) could be applicable to any (solvent + salt) system with the characteristic physical properties of the ionic salts and solvents being evaluated through Marcus and co-workers [28–30].

A. Senol / Fluid Phase Equilibria 361 (2014) 155–170

167

Table 8 Statistical design factors of model reliability of the existing vapor pressure models attributed to (solvent + salt) systems. Model

N, runs X S Fm Fs Fod (upper) c Fod (lower) c Student’s t

c

Reduced property-basis solvation modela

Concentration-based modelb

Eq. (11)

Eq. (12)

Eq. (13)

Eq. (14)

243 0.0102 0.0321 1.0103 1.0543 1.0652 0.9582 1.65

288 0.0128 0.0342 1.0129 1.0581 1.0718 0.9573 1.65

288 0.0128 0.0342 1.0129 1.0581 1.0717 0.9572 1.65

288 0.0002 0.0099 1.0002 1.0165 1.0166 0.9840 1.65

Solvation model with four-parameters Eq. (12), and eight-parameters Eq. (13). Concentration-dependent vapor pressure model with a temperature variable Eq. (14). Lower critical bound is Fod = Fm /Fs , upper critical bound is Fod = Fm Fs and 95% confidence.

4.3. Application of solvation-based models to binary solvent systems involving a salt In this section, it is endeavored to describe the vapor pressure properties of the systems (solvent (1) + solvent (2)) and (solvent (1) + solvent (2) + salt) in a predictive mode. To test for consistency of predictions through the Cox equation and the solvation-based models, the equilibrium data for the systems (1-propanol + water) [75] and (1-propanol + water + lithium nitrate) [68] were chosen as the reference ones. Due to the low total pressure of the considered both systems involving a salt or not, the vapor phase for these systems is assumed to be ideal, and the fugacity coefficients and the Poynting correction are neglected. Hence, the system total vapor pressure (Pt ) for the binary solvent mixture [75] at equilibrium can be calculated by Eq. (17). Similarly, by considering the non-volatility of the salt, the total vapor pressure (Pt ) of the system involving a salt [68] is calculated by Eq. (18), where xi and yi are mole fractions of component i in the liquid and vapor phases expressed on salt-free basis. The experimental activity coefficients of components  i and i are calculated through Eqs. (17) and (18), using P0 estimated by the Antoine equation with given Antoine constants [75] for Eq. (17), and P due to the concentration-dependent vapor pressure model (Eq. (14)) for Eq. (18), where P0 and P stand for the saturation vapor pressures of pure solvent alone and the (solvent + salt) system categorized as a pseudo-component, respectively. Then, the three-parameters Cox equation (Eq. (3)) with the evaluated parameters of the pure solvents is applied to predict the total pressure of the binary solvent mixture (1-propanol + water) due to Eq. (17). In the prediction of total vapor pressure of salt-containing binary solvent system (1-propanol + water + lithium nitrate) due to Eq. (18), the vapor pressure P of each (solvent + salt) involved in the system is estimated by Eqs. (11)–(13) using the coefficients given in Tables 1, 5 and 6, and the properties of the solvent and salt listed in Tables 3 and 4 Pt =



xi i P0,i

(17)

xi i P 

(18)

solvent systems involving a salt, the experimental data for the systems (2-propanol + water) from ref. [19,46,76] (total runs 56) and (2-propanol + water + ammonium thiocyanate) from Ref. [19] (total runs 33) were utilized to specify the model parameters. It should be pointed out that the binary solvent system is being regarded as a pseudo (coupled)-solvent. The vapor pressure of a salt-free binary solvent system (P0 ) is calculated by the Cox equation Eq. (3) using the pseudo-critical properties Pc and Tc evaluated due to Eq. (5). The P0 estimates are incorporated into Eqs. (11)–(13) to account for the vapor pressure of salt-free (2-propanol + water) binary system In Eqs. (11)–(13), the modified solubility and solvatrochromic parameters of the solvent mixture, SP* (ı∗H , *, ˇ*, ˛* and ı*), can be calculated by a linear combination of the corresponding parameters, SPi (ıH,i , i , ˇi , ˛i and ıi ), and salt-free mole fractions (xi ) of the individual components due to the mean value estimation rule given by Eq. (9), SP ∗ =



xi SPi . Following this procedure, the

i

vapor pressures (P ) of pseudo (coupled)-solvent with a salt effect (2-propanol + water + ammonium thiocyanate) are calculated by means of multivariable linear algorithm. Inspection of Fig. 10 reveals that the solvation models are relatively successful in

15

10

100(1 - P mod )/ Pobs

a b

Unified solvation model (USMIP)

5

0

-5

i

Pt =



-10

i

Both approaches have provided accurate vapor pressure predictions with mean deviations of e¯ = 1.2% ( = 1.3) for Eq. (17) associated with the Cox equation, and e¯ = 2.1% ( = 3.5) for Eq. (18) depending on the P estimates due to the solvation models. Fig. 9 illustrates the consistency of predictions achieved for binary solvent systems with and without a salt effect. To examine the efficacy of using solvation-based models (Eqs. (11)–(13)) to represent the vapor pressure properties of binary

-15 360

365

T (K)

370

375

Fig. 9. Mean deviations in predicted vapor pressures for (1propanol + water + lithium nitrate) system through solvation-based vapor pressure models: Estimates for binary solvent system without a salt effect due to Eq. (17) (+). Estimates for a salt-containing solvent mixture due to Eq. (18) depending on Eq. (11) (*), Eq. (12) () and Eq. (13) ().

168

A. Senol / Fluid Phase Equilibria 361 (2014) 155–170

30

100(1 - P mod )/ P obs

20

10

0

-10

-20

-30 352

356

360

T (K)

364

368

Fig. 10. Mean deviations in calculated vapor pressures for (2propanol + water + ammonium thiocyanate) system through solvation-based vapor pressure models: Estimates for a pseudo (coupled)-solvent system without a salt effect (+). Estimates for a pseudo (coupled)-solvent system involving a salt due to Eq. (11) (*), Eq. (12) () and Eq. (13) ().

estimating vapor pressures of relevant systems with a mean deviation of e¯ = 9.3% ( = 11.8). Consequently, the solvation-based models Eqs. (8) and (10) offer an attractive alternative since they have the potential to provide accurate model estimates based on the molecular descriptors of the solvents and salt. A survey of the literature reveals that most of common predictive models used for estimating vapor–liquid equilibria involve several adjustable interaction parameters, that need to be regressed from the experimental data for relevant systems (e.g., UNIQUAC, NRTL, SAFT, group contribution method). The same remarks hold for the proposed LSER-based solvation models including adjustable parameters. In the absence of available database of interaction parameters for relevant systems, it is almost impossible to make any calculation in a predictive mode. Currently available predictive (QSPR) models, based on the structure property information involving optimum number of structural descriptors, require a large database of reliable experimental vapor pressure data for generalization of the model parameters [39,77–79]. Since data for the solvent systems with a salt effect are not plentiful, an appropriate generalization of the model parameters for a reliable vapor pressure prediction is a challenging problem and, therefore, requires further efforts. Another fundamental problem that makes calculation in a predictive mode quite difficult is that, in principle, generalization of a large number of model parameters is necessarily required for the solvation models. Although subject to continuous updating and checking for consistency, the proposed solvation models become effective tools for estimating vapor pressure properties of salt included solvent systems being actually encountered in practice. 5. Conclusions The literature does not reveal any insight relating to the validity of a LSER-based vapor pressure model for correlating vapor pressures of associated (solvent + salt) systems. A unified solvation model framework (USMIP) is successful in representing vapor pressure data of relevant systems. All the vapor pressure data show

good compliance with eight-coefficients model structure involving nine physical descriptors, Eq. (8). The suitability of this structural form is verified upon the assumption that the vapor pressure is continually varying with the salt concentration throughout the whole working range. However, particularly frustrating aspect of this approach is that, in principle, determination of a large number of adjustable coefficients in the dependently varying terms of the model is necessarily required. The same remarks hold for the reduced property-basis solvation model Eq. (10), where the additional effect of the salt properties is accounted for implicitly. However, the statistical deviation results of Eqs. (8) and (10) indicate an underlying physical significance for the selected independent variables. As well, Eqs. (8) and (10) were able to predict reliably vapor pressures of binary solvent mixtures involved a salt, yielding a mean error of e¯ = 5.7% ( = 7.65). It is expected that the evaluated two structural forms of the solvation-based model should be in agreement with the boundary constraints and the behavior of the physical event. The salt concentration-basis approach with a third order polynomial concentration term Eq. (14) has also been implemented on the systems studied. This simplified concentration-based model reproduces the observed properties quite precisely, yielding a mean error of e¯ = 0.8% ( = 0.49) and an overall design factor of Fod = 1.017. But this approach requires an extension of the properties characterizing the ionic salt and further analysis of their additional effect. Consequently, it turns out from Figs. 1–10 that the examined two solvation-based model structures, Eqs. (8) and (10), are capable of simulating vapor pressure data of salt-containing solvent systems satisfactorily with an overall mean deviation of e¯ = 4% ( = 3.7). Statistical design factors of model reliability for the solvation-based models refer to a reasonable goodness-of-fit assessing Fod very close to unity. Required overall design factors (Fod ) are ranging about 1.065 (Eq. (11)) and 1.072 (Eqs. (12) and (13)) for upper critical bound and 0.95 for lower critical bound. It is expected that the predicted capabilities of the solvation-based models should be intimately connected to the structural properties of the salt and the polarity and proton-donating ability of the solvent. List of symbols

Ci Cs e¯ F Fm Fod Fs Ki li N ni pi P P0 P0 Pc Pc Pt

coefficient as defined by Eq. (8) salt concentration (wt%) mean relative   error, N e¯ = (100/N) i=1 (Pi,obs − Pi,mod )/Pi,obs  (%) concentration dependent factor model normalization factor overall design factor safety factor coefficient as defined by Eq. (10) coefficient as defined by Eq. (14) number of observations stoichiometric number of the ion in the salt structure partial pressure of a component in the solvent mixture (kPa) vapor pressure of (solvent + salt) system (kPa) vapor pressure of a pure solvent (kPa) vapor pressure of a solvent mixture categorized as a pseudo-component (kPa) critical vapor pressure of a pure solvent (kPa) critical vapor pressure of a solvent mixture categorized as a pseudo-component (kPa) total vapor pressure of a solvent mixture without a salt (kPa)

A. Senol / Fluid Phase Equilibria 361 (2014) 155–170

Pt Pr Pr0 r r  RD ; RD

S T Tc Tc t V v; v X X¯ xi Y yi z

total vapor pressure of a solvent mixture involving a salt (kPa) property as defined by Eq. (7) property as defined by Eq. (7) crystal ionic radius of the ion (nm) the normalized reciprocal of the crystal ionic radius (nm−1 ) molar refractivity of the ion and its normalized value (cm3 mol−1 ) standard deviation temperature (K) critical temperature of a pure solvent (K) critical temperature of a solvent mixture categorized as a pseudo-component (K) Student’s t parameter molar volume of the component (dm3 mol−1 ) volume occupied by the ion and its normalized value (nm3 ) objective function of model reliability mean of objective function mole fraction of the component in the liquid phase independent variable of objective function mole fraction of the component in the vapor phase the charge of the ion

Greek letters solvatochromic parameters ˛; ˛* ˇ; ˇ* solvatochromic parameters solvatochromic parameters ı; ı* ıH ; ı∗H Hildebrand solubility parameter (MPa0.5 ) dielectric constant ε ; 0 fugacity coefficients of the component  activity coefficient of the component  dipole moment (Cm) solvatochromic parameters ; *

N



root-mean-square-deviation,  =

p

softness parameter of the ion

i=1

(Yi,obs −Yi,mod )2 N

Subscript desn design property mod modeled property observed property obs s salt Acknowledgements The author thanks Professor H. W. Xiang and Professor K. ˚ ziˇcka for providing their papers on vapor pressure. This work Ruˇ was supported in part by the Research Fund of Istanbul University; project number BYP33167. References [1] W.F. Furter, Thermodynamic Behavior of Electrolytes in Mixed Solvents: Advances in Chemistry Series, American Chemical Society, Washington, DC, 1976. [2] T. Friese, P. Ulbig, S. Schulz, K. Wagner, J. Chem. Eng. Data 44 (1999) 701–714. [3] B. Mock, L.B. Evans, C.C. Chen, AIChE J. 22 (1986) 1655–1664. [4] F. Gironi, L. Lambert, Fluid Phase Equilib. 105 (1995) 273–286. [5] G. Uhrig, X. Ji, G. Maurer, Fluid Phase Equilib. 228–229 (2005) 5–14. [6] A. Apelblat, E. Korin, J. Chem. Thermodyn. 38 (2006) 152–157. [7] X.-M. Li, C. Shen, C.-X. Li, J. Chem. Thermodyn. 53 (2012) 167–175. [8] L. Zhang, Y. Guo, D. Deng, Y. Ge, J. Chem. Eng. Data 58 (2013) 43−47. [9] J. Dhanalakshmi, P.S.T. Sai, A.R. Balakrishnan, J. Chem. Eng. Data 58 (2013) 560−569. [10] I.-C. Hwang, R.-H. Kwon, S.-J. Park, Fluid Phase Equilib. 344 (2013) 32–37. [11] F. Cai, X. Wu, C. Chen, X. Chen, C. Asumana, M.R. Haque, G. Yu, Fluid Phase Equilib. 352 (2013) 47–53.

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