Vapor pressures of aqueous NaCl and CaCl2 solutions at elevated temperatures

Fluid Phase Equilibria 213 (2003) 39–51

Vapor pressures of aqueous NaCl and CaCl2 solutions at elevated temperatures Yasuhiro Shibue∗ Department of Natural Sciences, Hyogo University of Teacher Education, Yashiro-cho, Kato-gun, Hyogo 673-1494, Japan Received 23 October 2001; accepted 19 March 2003

Abstract The Wagner–Pruss equation for the saturation pressure of pure water is modified for the calculations of the vapor pressures of aqueous NaCl and CaCl2 solutions. The modified equation is applicable to the temperatures from 273.15 to 643 K and to the concentrations up to 30 mass% for aqueous NaCl solution and 18.9 mass% for aqueous CaCl2 solution. The modified equation involves concentration-dependent terms and multiplies the reduced temperature by Tc /647.096, where Tc is the critical temperature of the aqueous solution, in the WagnerPruss equation. © 2003 Elsevier B.V. All rights reserved. Keywords: Vapor pressure; Mixture; Aqueous NaCl solution; Aqueous CaCl2 solution

1. Introduction Vapor pressures of aqueous electrolyte solutions at elevated temperatures have been correlated by the extended Debye–Hückel equation [1], equations of state [2–5], or the corresponding state equations [6,7]. The extended Debye–Hückel equation such as the Pitzer equation has been used for many aqueous electrolyte solutions mainly below 573.15 K [1]. The association phenomena of the dissolved electrolytes often prevent the applicability of the Debye–Hückel approach near the critical point of pure water. For example, Simonson et al. [8] showed that the enthalpy of dilution of aqueous CaCl2 solution is different from that deduced by the Debye–Hückel equation at the infinitely dilute region above 573.15 K. Thereby, it seems to be difficult to formulate the vapor pressures of aqueous CaCl2 solution above that temperature with the extended Debye–Hückel equation. At high temperatures, equation-of-state approaches are often useful for the calculations of vapor pressures. However, many of those approaches are limited for its applicability at low temperatures. Very few equations place the applicable temperatures as low as the triple point of pure water. Therefore, it is necessary to develop the equation for predicting or correlating the vapor pressures of aqueous electrolyte solutions from low (below 373.15 K) to ∗

Fax: +81-795-44-2189. E-mail address: [email protected] (Y. Shibue). 0378-3812/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0378-3812(03)00284-X

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Y. Shibue / Fluid Phase Equilibria 213 (2003) 39–51

elevated temperatures. This study employs an empirical approach to calculate the vapor pressures of aqueous NaCl and CaCl2 solutions partly based on the corresponding state approach. Critical temperatures and pressures of those aqueous solutions have been determined up to high salt concentrations (see later section). Thus, aqueous NaCl and CaCl2 solutions are considered suitable for the present purpose. For the salt-free system, many correlation equations for saturation pressures have been proposed based on the critical pressure and temperature (see summary by Reid et al. [9]). Recently, Wagner and Pruss [10] derived the accurate equation for the vapor pressure of water up to the critical point. Their equation was authorized by the International Association for the Properties of Water and Steam in 1992. The present study employs their equation, which can be written as follows.
 P 1 ln = [−7.85951783(1 − TR ) + 1.84408259(1 − TR )1.5 − 11.7866497(1 − TR )3 ] Pc TR 1 + [22.6807411(1 − TR )3.5 − 15.9618719(1 − TR )4 + 1.80122502(1 − TR )7.5 ] (1) TR where Pc and Tc stand for the critical pressure and critical temperature, respectively, and TR is the reduced temperature (T/Tc ). The critical pressure and temperature are 22.064 MPa and 647.096 K, respectively. The right-hand side of Eq. (1) is designated as f(TR ) hereafter for the sake of brevity. If the aqueous solutions obey the corresponding state principle, then the reduced pressure (PR ) of the aqueous solution can be expressed as a function of the reduced temperature (TR ). The deviation from the corresponding state principle is corrected by a function of the mole fraction of the salt (X) here. Furthermore, the present study multiplies the reduced temperature by an empirical factor (Θ). The equation of the present concern is, thus, as follows. ln PR = f(ΘTR ) + h(X)

(2)

The function h(X) in Eq. (2) is the empirical correction term for the effect of the dissolved salt. 2. Critical properties and vapor pressures of aqueous NaCl and CaCl2 solutions 2.1. Critical properties of aqueous NaCl and CaCl2 solutions Critical properties of aqueous NaCl solutions have been studied extensively [11–15]. Shibue [15] recalculated the experimental results on critical pressures of Knight and Bodnar [16] at the concentrations from 3.2 to 30 mass%. The recalculation was carried out with use of the equation of Anderko and Pitzer [2]. Shibue [15] showed that critical pressures differ among the previous works at 8–20 mass%. In that interval, this study uses the results of Urusova [17] and Rosenbauer and Bischoff [18]. Thus, these three works [15,17,18] are considered for the calculations of the critical pressures. The critical temperature versus composition relation shown by Knight and Bodnar [16] agrees well with the other works [15] in their experimental concentrations (above 3.2 mass%). However, Povodyrev et al. [14] pointed out that the Knight–Bodnar equation [16] gives inaccurate temperatures below 3.2 mass%. To improve the fits at the dilute region, this study incorporates two data points at 0.14 and 0.3m concentrations from Marshall [12]. Thus, the results of Knight and Bodnar [16] and Marshall [12] are used here for the calculation of the critical temperatures.

Y. Shibue / Fluid Phase Equilibria 213 (2003) 39–51

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Table 1 Critical properties of H2 O, H2 O + NaCl, and H2 O + CaCl2

Tc (K) Pc (MPa) q1 q2 q3 q4 q5 q6 q7 q8 q9 q10

H2 O

H2 O + NaCl (X ≤ 0.117 (≤30 mass%))

H2 O + CaCl2 (X ≤ 0.0378 (≤19.5 mass%))

647.096 22.064

647.096 + q1 X0.5 + q2 X + q3 X2 + q4 X4 22.064 + q5 X + q6 X2 + q7 X3 + q8 X4 + q9 X5 + q10 X6 8.78054 × 10 2.42541 × 103 −6.07779 × 103 1.17033 × 106 9.00404 × 102 −2.92542 × 104 1.39806 × 106 −2.80756 × 107 2.41637 × 108 −7.18726 × 108

647.096 + (q2 X)/(X + q12 ) + q3 X2 + q4 X4 22.064 + (q6 X)/(X + q52 ) + q7 X2 + q8 X4 1.47872 × 10−1 7.44241 × 10 −2.19504 × 104 5.42132 × 107 −6.47588 × 10−3 8.58985 × 100 5.97149 × 103 −2.22618 × 107

Critical temperatures and critical pressures of aqueous NaCl solutions are regressed with the following equations. Tc = 647.096 + q1 X0.5 + q2 X + q3 X2 + q4 X4

(3)

Pc = 22.064 + q5 X + q6 X2 + q7 X3 + q8 X4 + q9 X5 + q10 X6

(4)

where q1 to q10 are the regression coefficients and X stands for the mole fraction of NaCl. Those regression coefficients are calculated by linear least-square regressions with the minimization of the following function (S). 2 
Y calc S= (5) −1 Y ref where Y designates the critical temperature or critical pressure of the solution. The calculated results are listed in Table 1. For aqueous CaCl2 solutions, this study considers the experimental results of Oakes et al. [19], Shmulovich et al. [20], and Bischoff et al. [21]. The results of Marshall and Jones [11] do not agree with the later works and are not considered here. Shmulovich et al. [20] did not show the critical properties. Thus, the present study computes the critical temperatures and pressures from their results up to 19.5 mass% by the method and procedure of Bischoff et al. [21]. Calculations of the critical properties from their results should be based on the vapor–liquid equilibrium compositions under the isothermal and isobaric conditions. However, the pressures measured on vapor phase compositions differed from those on liquid phase by several bars in their study. Thus, this study takes the midpoints of the measured pressures of the two phases under the isothermal conditions. Above 19.5 mass%, only the experimental results of Shmulovich et al. [20] are available for the deduction of the critical properties. Considering the uncertainties of the experimental results and the method of the calculation, this study does not calculate the critical properties above 19.5 mass%. The critical temperature versus composition relation for aqueous CaCl2 solution shows that the critical temperature rises steeply at the very dilute region (Fig. 1). Bischoff et al. [21] showed the critical

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Y. Shibue / Fluid Phase Equilibria 213 (2003) 39–51 800

Critical temperature (K)

Oakes et al. [19] Shmulovich et al. [20] 750

Bischoff et al. [21] calc.

700

650

600

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Mole fraction of CaCl2 Fig. 1. Critical temperatures of aqueous CaCl2 solutions. The solid line is obtained by the relevant equation and coefficients listed in Table 1.

temperature at X = 0.0009 to be 653.15 K. The elevation of the critical temperature is about twice the value for aqueous NaCl solution at the same mole fraction [12]. On the other hand, the elevation of the critical temperature at X = 0.016 for aqueous CaCl2 solution [21] is about half the value for aqueous NaCl solution [16]. The relation between the critical pressure and composition for aqueous CaCl2 solution also shows the initial rise at the very dilute region (Fig. 2). Bischoff et al. [21] noted that it was 90 Shmulovich et al. [20]

Critical pressure (MPa)

80 Bischoff et al. [21]

70

calc.

60 50 40 30 20 0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Mole fraction of CaCl2 Fig. 2. Critical pressures of aqueous CaCl2 solutions. The solid line is obtained by the relevant equation and coefficients listed in Table 1.

Y. Shibue / Fluid Phase Equilibria 213 (2003) 39–51

43

difficult to express the critical temperature versus composition relation with simple polynomials. Then, this study employs the following rational function for the regressions of the critical temperatures and pressures. Y = q0 +

q2 X + q3 X 2 + q4 X 2 X + q12

(6)

where q0 is 647.096 for the critical temperature and q1 to q4 are the regression coefficients. Critical pressures are also regressed with Eq. (6), where q0 is then 22.064. Those regression coefficients are calculated by the nonlinear least-square regression technique [22] with the objective function of Eq. (5). The regression coefficients on the critical properties of aqueous CaCl2 solutions are listed in Table 1. Further improvements of the regression equations may be possible by the addition of the higher-degree terms to the regression equations. However, the uncertainties of the measurements and the discrepancy among the experimental results (Figs. 1 and 2) do not warrant the accurate representations of the critical behavior of aqueous CaCl2 solution. 2.2. Vapor pressure data Vapor pressures of aqueous NaCl solutions are taken from the literature data [23–33] listed in Table 2. The maximum concentration corresponds to 30 mass%. Pitzer et al. [29] and Clarke and Glew [31] compiled the thermodynamic properties of aqueous NaCl solutions. The present study calculates vapor pressures from their tabulated values to four significant digits. The other compilations on the thermodynamics of aqueous NaCl solutions, such as Archer [34], are not considered here. Experimental results on the vapor pressures have also been reported by Gehrig et al. [35], Sourirajan and Kennedy [36], and Hubert et al. [37]. The first study was based on the indirect measurements, and the second work often disagrees with the later works. The experimental temperatures and concentrations in Hubert et al. [37] were covered by Clarke and Glew [31], and the experimental results are consistent with Clarke and Glew [31]. Therefore, the incorporation of the results of Hubert Table 2 Vapor pressure data for aqueous NaCl solution Temperature (K)

Mole fraction of NaCl

Data points

Reference

583.15–643.15 623.15 398.15–548.15 623.15 623.15 593.15–633.15 393.15–573.15 598.15–623.15 273.15–383.15 582.65–643.35 622.99–623.12

0.00310–0.0932 0.0331–0.117 0.106–0.115 0.0208–0.0909 0.0331–0.116 0.0220–0.116 0.00893–0.0975 0.00893–0.102 0.00893–0.0975 0.0101 0.00448–0.0513

63 6 7 5 3 19 133 24 84 4 13

[23] [24] [25] [26] [27] [28] [29]a [30] [31]a [32] [33]

a

Computed values after the compilation.

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Y. Shibue / Fluid Phase Equilibria 213 (2003) 39–51

et al. [37] simply weighs the low-temperature data. Busey et al. [38] calculated the osmotic coefficients up to 623.15 K from their measurements on the enthalpies of dilution. Then they employed the Pitzer ion-interaction approach for the data treatment, which results in the vapor pressures coinciding with the measurements of Wood et al. [30]. The association phenomenon of the dissolved electrolyte was not taken into account while recent measurements of Ho et al. [39] indicate the significant degree of the ion-pairing at 623.15 K. Owing to its uncertainty of the data treatment, this study does not include Busey et al. [38] in the end. Previous compilations showed the large arrays of vapor pressures [31] or osmotic coefficients [29] against the temperature and concentration. Among the tabulated values in Clarke and Glew [31], this study takes the data at 273.15–383.15 K in step of 10 K. Concentrations of those data are 0.5 and from 1 to 6m in step of 1m at each temperature. The osmotic coefficients in Pitzer et al. [29] are taken from 393.15 to 573.15 K in step of 10 K. The NaCl concentrations are 0.5 and 1–6m in step of 1m. The data of less than 0.5m concentrations in both studies are not included to avoid weighting the dilute region. The osmotic coefficients (ϕ) in Pitzer et al. [29] are converted into the vapor pressures (P) by using the following relation [40].     Po
Vg 1000 Po 1 Vl o ϕ= ln − − (P − P) (7) dP − 2m 18.0152 P P RT RT P where m, R, and T stand for the molality of the electrolyte, universal gas constant, and absolute temperature, respectively. In Eq. (7), Po , Vg , and Vl stand for the vapor pressure of pure water, molar volume of the gas phase, and molar volume of the liquid phase, respectively. Those water properties are computed with the equation of Haar et al. [41]. Vapor pressures of aqueous CaCl2 solutions are taken from the literature data [30,33,42–45] listed in Table 3. The maximum concentration corresponds to 18.9 mass%. Ananthaswamy and Atkinson [43] summarized the thermodynamic properties of aqueous CaCl2 solutions up to 373.15 K and showed the large arrays of the osmotic coefficients against the temperature and concentration. From their tabulated values, the present study calculates the vapor pressures to four significant digits. This study takes the data at 273.15–373.15 K in step of 10 K. Concentrations of those solutions are 0.2–2.0m in step of 0.2m at each temperature. The conversion from the osmotic coefficient into the vapor pressure is carried out by using Eq. (7) after replacing 2m by 3m in the right-hand side of that equation. The osmotic coefficients at 0 ◦ C are converted into the vapor pressures at the temperature of the triple point (273.16 K). Above 373.15 K, Pitzer and Oakes [46] derived the osmotic coefficients of aqueous CaCl2 solutions from the equation of Table 3 Vapor pressure data for aqueous CaCl2 solution Temperature (K)

Mole fraction of CaCl2

Data points

Reference

423–623 548.15–623.15 273.15–373.15 303.15–343.15 622.98–642.88 423.15–523.15

0.00441–0.0208 0.00893–0.0365 0.00359–0.0348 0.0177 0.00404–0.0173 0.00180–0.0348

6 16 110 5 15 15

[42] [30] [43]a [44] [33] [45]a

a

Computed values after the compilation.

Y. Shibue / Fluid Phase Equilibria 213 (2003) 39–51

45

Holmes et al. [45]. The present study uses their tabulated values at 423.15, 473.15, and 523.15 K. The concentrations are 0.1, 0.2, 0.5, 1.0, and 2.0m at the three temperatures.

3. Results and discussion 3.1. Deviation from corresponding state equation This study calculates the ln PR values from the vapor pressure data (Tables 2 and 3) with use of the critical pressure equations (Table 1). Then, those ln PR values are compared with the values of f(TR ), which are the values of the right-hand side of Eq. (1). The reduced temperatures are calculated from the critical temperature equations (Table 1). Then the ln PR − f(TR ) is plotted against the mole fraction of NaCl (Fig. 3) or CaCl2 (Fig. 4). Figs. 3 and 4 show that the ln PR − f(TR ) values are dependent on both temperature and salt concentration for both aqueous solutions. 3.2. Modification of corresponding state equation This study tried to reduce the complexities of ln PR − f(TR ) by using an empirical term Θ in Eq. (2). After several trials, this study finds that the substitution of T/647.096 for TR in Eq. (1) results in nearly temperature-independent plots of ln PR − f (T/647.096) against the mole fraction of the salt. The value of 9 8

273.15 K

573.15 K

7

373.15 K

623.15 K

InPR - F (TR)

6

473.15 K

5 4 3 2 1 0 0

0.02

0.04

0.06

0.08

0.1

0.12

Mole fraction of NaCl Fig. 3. Plots of ln PR − f(TR ) against mole fraction of NaCl. f(TR ) is the right-hand side of Eq. (1). Data sources are listed in Table 2. Experimental data of Bischoff and Rosenbauer [32] at 623.25 K and those of Crovetto et al. [33] at 622.99–623.12 K are included in 623.15 K plots.

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Y. Shibue / Fluid Phase Equilibria 213 (2003) 39–51 2 273.15 K

573.15 K

373.15 K

623.15 K

InPR - f (TR)

1.5 1

473.15 K

0.5 0 -0.5 0

0.01

0.02

0.03

0.04

Mole fraction of CaCl2

Fig. 4. Plots of ln PR − f(TR ) against mole fraction of CaCl2 . f(TR ) is the right-hand side of Eq. (1). Data sources are listed in Table 3. Experimental data of Zarembo et al. [42] are included in 573.15 and 623.15 K plots. Those data of Crovetto et al. [33] at 622.98–623.02 K are included in 623.15 K plots.

647.096 corresponds to the critical temperature of pure water. Therefore, this study substitutes Tc /647.096 for Θ in Eq. (2). The resultant equation, which is designated as g(T), is as follows.  
 
 1.5

647.096 T T g(T ) = −7.85951783 1 − + 1.84408259 1 − T 647.096 647.096  
 3  3.5


647.096 T T + −11.7866497 1 − + 22.6807411 1 − T 647.096 647.096  
 4 
 7.5

647.096 T T + −15.9618719 1 − + 1.80122502 1 − T 647.096 647.096 (8) The plots of ln PR − g(T ) against the mole fraction of the salt are shown in Fig. 5 for aqueous NaCl solution and in Fig. 6 for aqueous CaCl2 solution. For aqueous NaCl solutions, ln PR − g(T ) values at X = 0.00893 (2.84 mass%) range from −0.276 (273.15 K) to −0.290 (623.04 K) and those values at X = 0.0975 (26.0 mass%) range from −1.798 (273.15 K) to −1.761 (573.15 K). For aqueous CaCl2 solutions, ln PR − g(T ) values at X = 0.00359 (2.17 mass%) are −0.178 from 273.15 to 473.15 K and those values at X = 0.0348 (18.2 mass%) range from −1.298 (273.15 K) to −1.218 (573.15 K). The values of ln PR − g(T ) are almost independent of the temperature for both solutions. Thus, this study expresses ln PR − g(T ) as a function of the salt concentration. That function is designated as h(X). It should be noted that ln PR − g(T ) values depend slightly on temperature for both aqueous solutions at high concentrations. The dependency is obvious for the data on aqueous CaCl2 solutions at 623.15 K, where plots at the mole fractions of larger than 0.015 are displaced from low-temperature data. Even so, this study does not consider the addition of the temperature-dependent term to h(X).

Y. Shibue / Fluid Phase Equilibria 213 (2003) 39–51

47

0 273.15 K, 373.15 K, 473.15 K, 573.15K 623.15 K

-0.5

InPR- g(T)

calc.

-1 -1.5 -2 -2.5 0

0.02

0.04

0.06

0.08

0.1

0.12

Mole fraction of NaCl Fig. 5. Plots of ln PR − g(T ) against mole fraction of NaCl. g(T) is taken from Eq. (8). Data sources are the same with those in Fig. 3. The solid line is obtained by Eqs. (10) and (13).

The relations between ln PR − g(T ) and mole fraction of the salt (Figs. 5 and 6) seem to be expressed by simple polynomial functions. After ln PR − g(T ) values were regressed by h(X) of a certain degree polynomial, vapor pressures were calculated under isothermal conditions by the following relation. ln P calc = ln Pc + g(T ) + h(X)

(9)

0 -0.2

InPR- g(T)

-0.4 -0.6 -0.8 -1 273.15 K, 373.15 K, 473.15 K, 573.15 K

-1.2

623.15 K

-1.4

calc.

-1.6 0

0.01

0.02

0.03

0.04

Mole fraction of CaCl2 Fig. 6. Plots of ln PR − g(T ) against mole fraction of CaCl2 . g(T) is taken from Eq. (8). Data sources are the same with those in Fig. 4. The solid line is obtained by Eqs. (10) and (13).

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Y. Shibue / Fluid Phase Equilibria 213 (2003) 39–51

The calculated pressures often increased with the increase in the salt concentration, which is the incorrect behavior. Thus, this study divides the fitted region into two parts, dilute and concentrated regions. Then the regression equations are connected at a composition (u) with the same slope of dh(X)/dX. For both aqueous solutions, h(X) decreases sharply near X = 0 and then decreases less steeply. With the further increase in X, h(X) decreases sharply. This study expresses h(X) at the dilute region as follows. h(X) =

a2 X + a3 X 2 2 X + a1

(10)

where a1 , a2 , and a3 are the regression coefficients. At the concentrated region, the function h(X) is first considered as follows. a2 u h(X) = b0 (X − u) + b1 (X − u)X + b2 (X − u)X2 + + a3 u2 (11) u + a12 where b0 , b1 , and b2 are the regression coefficients. The values of h(u) from Eqs. (10) and (11) should be the same. Furthermore, this study equates dh(X)/dX value at X = u from Eq. (10) to that from Eq. (11). After the differentiation of h(X) with respect to X in Eqs. (10) and (11), we obtain the following relation. b0 =

a12 a2 + 2a3 u − b1 u − b2 u2 (u + a12 )2

(12)

By substituting Eq. (12) into Eq. (11) and rearranging the resultant equation, we get the following equation.  a12 a2 a2 u h(X) = + 2a3 u (X − u) + b1 (X − u)2 + b2 (X − u)(X2 − u2 ) + + a3 u2 2 2 (u + a1 ) u + a12 (13) The regression coefficients in Eqs. (10) and (13) for both solutions are calculated by the least-square regressions with the minimization of the following function (S).  (14) S= (ln PRref − ln PRcalc )2 The referenced data points are weighted equally in those regressions. The connection point of Eqs. (10) and (13) is determined as follows. The value of u is changed in step of 0.001 and the resultant S value is computed at each step. This study chooses u giving the smallest S. It was confirmed that the calculated vapor pressures for both aqueous NaCl and CaCl2 solutions decrease with the increase in the salt concentration. After obtaining the regression coefficients, this study calculates the average absolute deviation (AAD) as follows.    1   PRcalc AAD = − 1 (15)  ref N P R

where N is the number of data points. Table 4 lists the ranges of the dilute and concentrated regions and the results of the calculations. The values of u and AAD are included in Table 4. Relative deviations are calculated from the calculated and referenced data by the following equation:
calc  PR relative deviation = 100 −1 (16) PRcalc

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49

Table 4 Computed results for Eqs. (10) and (13) H2 O + NaCl

H2 O + CaCl2

Dilute region (Eq. (10)) a1 a2 a3 Vapor pressure AAD (%)

0 ≤ X ≤ 0.024 1.28746 × 10−1 −7.31097 × 10−1 −3.15058 × 102

0 ≤ X ≤ 0.012 5.89403 × 10−2 −3.40230 × 10−1 −4.47233 × 102

Concentrated region (Eq. (13)) b1 b2 u Vapor pressure AAD (%)

0.024 < X ≤ 0.117 3.92767 × 102 −2.46440 × 103 0.024

0.34

1.19

0.27 0.012 < X ≤ 0.037 −1.24733 × 103 2.43253 × 103 0.012 1.17

The temperature–concentration regions where the relative deviations become −3 to 3% are indicated below. For aqueous NaCl solutions, that region is bounded by X ≤ 0.0826 (22.61 mass%) at T < 403.15, X ≤ 0.117 (30 mass%) at 403.15 ≤ T ≤ 573.15, and X ≤ 0.110 (28.60 mass%) at 573.15 < T ≤ 643.35. The outliers are at T = 583.15 and X = 0.0932 (25 mass%) and T = 623.15 and X = 0.0839 (22.90 mass%). For aqueous CaCl2 solutions, the temperature–concentration region where the relative deviations become −3 to 3% is as follows: X ≤ 0.0348 (18.17 mass%) at T ≤ 523.13; X ≤ 0.0263 (14.27 mass%) at 523.15 < T ≤ 642.88. One outlier is at T = 623 and X = 0.0208 (11.56 mass%). Therefore, the computed vapor pressures agree well with the referenced data on both solutions. 4. Conclusions This study modifies the Wagner–Pruss equation to express the vapor pressures of aqueous NaCl and CaCl2 solutions. When the reduced temperature is replaced by T/647.096, the differences between the ln PR values and the computed values can be expressed by the salt concentration. Simple regressions generally yield the vapor pressures fitting well with the referenced data. List of symbols a1 , a2 , a3 AAD b0 b1 , b2 f(TR ) g(T) h(X) m N P

regression coefficients of Eq. (10) average absolute deviation Eq. (12) regression coefficients of Eq. (13) right-hand side of Eq. (1) Eq. (8) Eq. (10) or Eq. (13) molality number of data points pressure

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Y. Shibue / Fluid Phase Equilibria 213 (2003) 39–51

q0 q1 , q2 , q3 , q4 q5 , q6 , q7 , q8 , q9 , q10 R S T u V X Y

critical temperature or critical pressure of pure water regression coefficients for critical temperature regression coefficients for critical pressure universal gas constant objective function for least-square regression absolute temperature mole fraction of the salt at which Eqs. (10) and (13) are connected molar volume mole fraction of the salt critical temperature or critical pressure in Eqs. (5) and (6)

Greek letters ϕ Θ

osmotic coefficient Tc /647.096

Superscripts calc o ref

calculated value pure water referenced value

Subscripts c g l R

critical property gas phase liquid phase reduced value

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

K.S. Pitzer, Thermodynamics, third ed., McGraw-Hill, New York, 1995. A. Anderko, K.S. Pitzer, Geochim. Cosmochim. Acta 57 (1993) 1657–1680. I.G. Economou, C.J. Peters, J.S. Arons, J. Phys. Chem. 99 (1995) 6182–6193. S. Jiang, K.S. Pitzer, AIChE J. 42 (1996) 585–594. J.J. Kosinski, A. Anderko, Fluid Phase Equilib. 183–184 (2001) 75–86. J.S. Gallagher, J.M.H. Levelt Sengers, Int. J. Thermophys. 9 (1988) 649–661. J.M.H. Levelt Sengers, J.S. Gallagher, J. Phys. Chem. 94 (1990) 7913–7922. J.M. Simonson, R.H. Busey, R.E. Mesmer, J. Phys. Chem. 89 (1985) 557–560. R.C. Reid, J.M. Prausnitz, B.E. Poling, The Properties of Gases and Liquids, fourth ed., McGraw-Hill, New York, 1987. W. Wagner, A. Pruss, J. Phys. Chem. Ref. Data 22 (1993) 783–787. W.L. Marshall, E.V. Jones, J. Inorg. Nucl. Chem. 36 (1974) 2313–2318. W.L. Marshall, J. Chem. Soc., Faraday Trans. 86 (1990) 1807–1814. A.A. Povodyrev, M.A. Anisimov, J.V. Sengers, J.M.H. Levelt Sengers, Phys. A 244 (1997) 298–328. A.A. Povodyrev, M.A. Anisimov, J.V. Sengers, W.L. Marshall, J.M.H. Levelt Sengers, Int. J. Thermophys. 20 (1999) 1529–1545. [15] Y. Shibue, J. Chem. Eng. Data 45 (2000) 523–529. [16] C.L. Knight, R.J. Bodnar, Geochim. Cosmochim. Acta 53 (1989) 3–8. [17] M.A. Urusova, Russ. J. Inorg. Chem. 19 (1974) 450–454.

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