Ind. Eng. Chem. Res. 2009, 48, 2229–2235
2229
Estimation of Freezing Point Depression, Boiling Point Elevation, and Vaporization Enthalpies of Electrolyte Solutions Xinlei Ge*,†,‡ and Xidong Wang§ Department of Physical Chemistry, UniVersity of Science and Technology Beijing, 100083, China, Department of Materials Science and Engineering, Royal Institute of Technology, SE 10044, Stockholm, Sweden, and College of Engineering, Peking UniVersity, Beijing 100871, China
A novel approach is presented in this work for predicting the values of freezing point depression and boiling point elevation for electrolyte solutions at different concentrations on the basis of the Pitzer theories. This method treats the enthalpy change of the solution between the normal freezing point or boiling point and the real ones to be linear temperature dependence. Compared with the literature values, this method performs very well; also, the temperature-dependent parameters of some salts are incorporated to investigate temperature effects of this method. Furthermore, a method based on the Clausius-Clapeyron equation is derived for estimation of the enthalpy of vaporization of very high concentration solutions at different temperatures, and the predicted results are highly positive. Introduction Research on the thermodynamic properties of electrolyte solutions is very attractive because they are very important for many fields in chemical engineering, geochemistry, and so forth. Several colligative properties, such as freezing point depression(FPD), boiling point elevation(BPE), and enthalpies of vaporization are always recognized as the important properties of numerous industrial solutions. The freezing point of a solution always can be lowered by the presence of electrolyte particles, which is called freezing point depression (FPD). Similarly, the boiling point of a solution with a nonvolatile solute is always higher than the boiling point of the pure solvent because of the vapor pressure lowering by the solute. The difference is called boiling point elevation (BPE). The number of the solute particles, namely, its concentration, is the controlling factor of those properties. If the solution is treated as an ideal solution, the FPD or BPE depends only on the solute concentration that can be estimated by a simple linear relationship with the cryoscopic constant for FPD and the ebullioscopic constant for BPE. However, this is only effective in a diluted solution, and thus the accurate method to predict those properties is still essential. Keitaro1 proposed a detailed description of FPD for dilute concentration. Gary et al.2 described a new expression for aqueous solutions accurate up to three molal concentrations by considering the solute/solvent interaction. For BPE prediction, an empirical equation was used to correlate the experimental data of BPE.3 Du¨hring’s rule also has been applied for BPE calculation by plotting the boiling temperature of the solution versus that of the pure solvent,4 which is important in obtaining BPE and development of empirical BPR models.5 The enthalpy of vaporization, also known as the latent heat of vaporization or heat of evaporation, is the energy required to transform a given quantity of a substance into gas. Some researches are carried out for the correlation of pure substances.6-9 For the enthalpies of electrolyte solutions, Silvester et al.10 * To whom correspondence should be addressed. Tel: +86 10 62333949. Fax: +86-10-62327283. E-mail:
[email protected]. † University of Science and Technology Beijing. ‡ Royal Institute of Technology. § Peking University.
calculated the enthalpy and heat capacity of sodium chloride up to 300 °C. Srisaipet et al.11 proposed a relationship between vapor pressure and vaporization enthalpy by an extension of the Martin equation. The empirical method for estimation of FPD/BPE or enthalpy of vaporization in the literature usually has very limited applications, only effective for a specific system. Therefore, the present work aims to offer a universal method for predicting the FPD/BPE. Furthermore, a method for estimating the enthalpy of vaporization based on the ClausiusClapeyron equation is derived. In this work, Pitzer theories, including the original Pitzer equations12 and the one modified by Pe´rez-Villasen˜or et al.13-15 were incorporated to develop this method. Theoretical Modeling Derivation for FPD/BPE. When the solution reaches solid-liquid equilibrium (SLE), the chemical potential of the solvent is equal between the liquid and solid phases: µliq(T, P) ) µsol(T, P)
(1)
where µliq(T, P) and µsol(T, P) represent the chemical potentials of the liquid solvent and corresponding solid phase at the same temperature and pressure, respectively. Furthermore, the chemical potential of the liquid and solid solvent can be expressed as: µliq(T, P) ) µ0liq(T, P) + RT ln aliq(T, P, m)
(2a)
0 (T, P) + RT ln asol(T, P) µsol(T, P) ) µsol
(2b)
Where µliq0(T,P) and µsol0(T,P) are the chemical potentials of pure liquid and solid solvent at the same temperature and pressure, respectively, R is the gas constant, aliq is the activity of solvent in the solution, and asol ) 1 is the activity of solid solvent. The difference between the chemical potential of pure liquid and solid solvent is the free energy of fusion ∆Gfus0. With eqs 1 and 2, it can be arranged as 0 0 ) µ0liq - µsol ) -RT ln aliq ∆Gfus
(3) 16
According to the Gibbs-Helmholtz equation
10.1021/ie801348c CCC: $40.75 2009 American Chemical Society Published on Web 01/15/2009
2230 Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009
dT ( ∆GT ) ) - ∆H T
d
(4)
2
the activity of solvent is related with the osmotic coefficient, as shown below: ln aliq ) -(VmMS ⁄ 1000)φ
Combing eqs 3 and 4, one can obtain d ln aliq ∆Hfus T ) dT RT2
(5)
Thus, eq 5 can be integrated as ln aliq )
∫
T
∆Hfus T
dT (6) RT2 Where ∆HTfus is the enthalpy change of fusion. If the temperature range (T, TF) is very narrow, ∆HTfus can be assumed to be a constant. However, for some electrolyte solutions with high concentration, the FPD, θF ) TF - T, could be high. Thus, it is assumed to be linear temperature dependence, represented by the differences of heat capacity between the liquid and solid phases at the normal freezing point of solvent (∆Cpfus ) Cpliq - Cpsol), as shown below (7)
fus ∆H0,T F
is the enthalpy change of fusion of the pure solvent at TF. Integrating eq 6 with eq 7, one can obtain fus R ln aliq ) ∆H0,T F
(
)
1 1 + TF TF - θF
[
∆Cfus p ln
]
TF - θF θF + (8) TF TF - θF
The logarithmic function also can be approximated using a Taylor series expansion ln x )
( x -x 1 ) + 21 ( x -x 1 ) + 31 ( x -x 1 ) + ... 2
3
(9)
In this equation, the x value is less than unity. If only the first two terms are employed, eq 8 can be recast as fus ∆H0,T F
(
)
fus
∆Cp 1 1 + TF TF - θF 2
(
θF TF - θF
)
2
- R ln aliq ) 0 (10)
Then, the freezing point depression of the solution is θF )
√
2 fus fus 2 - 2RTF ln aliq- 2∆Cfus ∆H0,T p TFR ln aliq + (∆H0,TF) F fus 2(∆H0,T ⁄ TF + 0.5∆Cfus p -R F
ln aliq) (11)
The similar procedure also can be applied for analysis of the boiling point elevation. Details can be found in the Supporting Information. The equation for calculation BPE is listed below: θB )
Here, V ) VM + VX, VM and VX are the stoichiometric coefficients of cation and anion of the salt, MX, m is the molality, and Ms is the molecular weight of the solvent. There are many methods in the past century have been developed for the calculation of osmotic coefficient. The most popular one is original Pitzer theory12 with the following form
TF
fus fus ∆Hfus T ) ∆H0,TF + ∆Cp (T - TF)
√
2 vap vap 2 - 2RTB ln aliq + (∆H0,T ) - 2∆Cvap -∆H0,T p TBR ln aliq B B vap 2(∆H0,T ⁄ TB + 0.5∆Cvap p +R B
(13)
φ - 1 ) -|ZMZX|
AφI1⁄2 1 + bI
1⁄2
+
(
)
2VMVX (0) m[βMX + V
(
)
2(VMVX)3⁄2 2 φ m CMX (14) V For 2-2 type electrolytes, taking account of the association effects of ions, another additional parameter is necessary, as shown below (1) exp(-RI1⁄2)] + βMX
φ-1 ) -|ZMZX|
AφI1⁄2
+
(
)
2VMVX (0) m[βMX + V
1 + bI (1) (2) exp(-R1I1⁄2) + βMX exp(-R2I1⁄2)] + βMX 1⁄2
(
)
2(VMVX)3⁄2 2 φ m CMX (15) V (0) (1) (2) φ , βMX , βMX , and CMX , The adjustable parameters are βMX which are known as Pitzer parameters. Aφ is the Debye-Hu¨ckel constant, which only depends on the temperature. I ) 1/2 ∑miZ2i is the ionic strength. The parameters for 227 electrolytes17 have been tabulated for a concentration range below 6 mol kg-1; Kim and William18 further obtained the parameters for 304 single salts across an extended concentration range. However, Pitzer12 fixed b ) 1.2 and R ) 2.0 in eq 14 and R1 ) 1.4, R2 ) 12 in eq 15 without any explanation. Pe´rezVillasen˜or et al.13 evaluated the different cases by letting b and R float, and they obtained the following form for calculating the osmotic coefficient. φ - 1 ) -|ZMZX|
AφI1⁄2 1 + bMXI
1⁄2
+
(
(
)
2VMVX mBMX + V
)
4(VMVX)3⁄2 |ZMZX|1⁄2m2CMX (16) V In eq 16, bMX, BMX, and CMX are the adjustable parameters. However, most parameters in these equations are regressed from literature values at 298.15 K, which sometimes leads to large deviation in calculation at the freezing point or boiling point. Pe´rez-Villasen˜or et al.14 reported the temperature dependence of the parameters of some salts for their modified form. For the calculation of AΦ in water, Spencer et al.19 and Moller20 proposed the following equations Above 298.15 K: AΦ ) 3.36901532 × 10-1 -
ln aliq) (12)
vap where ∆H0,T is the enthalpy of vaporization of pure solvent at B its normal boiling point, TB. ∆Cpvap ) Cpvap - Cpliq;, is the difference in heat capacity between the vapor and liquid phases at the normal boiling point of the pure solvent. Thus, if the activity of solvent in the solution is known, the FPD/BPE can be estimated from eqs 11 and 12. In a solution,
6.32100430 × 10-4T + 9.14252359 ⁄ T1.25143986 × 10-2lnT + 2.26089488 × 10-3 ⁄ (T - 263) + 1.92118597 × 10-6T2 + 45.2586464 ⁄ (680 - T) (17a) Below 298.15 K:AΦ ) 86.6836498 + 8.48795942 × 10-2T 8.88785150 × 10-5T2 + 4.88096393 × 10-8T3 1.32731477 × 103 ⁄ T - 17.6460172 ln T (17b)
Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009 2231
In this work, the prediction is performed first by using the parameters at 298.15 K. If the temperature-dependence parameters are available, further investigations will be carried out for comparison. All of the parameters used in this work are listed in the Supporting Information. With the thermodynamic properties shown in Table 1, the FPD/BPE can be predicted. Derivation for Vaporization Enthalpy. At low pressure, the vapor pressure of the solvent in a solution is also related to the solvent activity, P ) aliqP0
(18)
where P and P0 are the vapor pressures of the solution and pure solvent at the same temperature respectively. For an electrolyte solution, the Clausius-Clapeyron equation21 can be used to describe the relationship between the enthalpy of vaporization and the temperature-dependent vapor pressure dln P -∆Hvap ) (19) d(1 ⁄ T) R Rearranging eq 19 and combining with eq 18, one can obtain ∆Hvap ) RT2(dln P0 ⁄ dT + dln aliq ⁄ dT)
(20)
The first term in the right-hand of eq 20 can be obtained from the vapor pressure equation of pure solvent. The standard equation proposed by Wagner23 can be employed, as shown below: ln P0 ) ln Pc + (Tc ⁄ T)(a1τ + a2τ1.5 + a3τ3 + a4τ6)
(21)
where Pc and Tc are the critical vapor pressure and temperature of the solvent, the ai are adjustable parameters, and τ ) 1 - Tr with Tr ) T/Tc. This equation was first proposed for argon and nitrogen but was extended for many pure liquid solvents. For water,24 Tc ) 647.3 K and Pc ) 22.12 MPa; a1 ) -7.76451, a2 ) 1.45838, a3 ) -2.77580, a4 ) -1.23303; the temperature range is (275 K to Tc). Thus, its derivative can be written as: d ln P0 Tc ) - 2 (a1τ + a2τ1.5 + a3τ3 + a4τ6) dT T 1 (a + 1.5a2τ0.5 + 3a3τ2 + 6a4τ5) (22) T 1 From eq 13, the second term of eq 20 can be expressed as VmMS dφ d ln aliq )· dT 1000 dT From eqs 14 and 16, one can obtain
)[
(
(0) |ZMZX|I1⁄2 dAφ 2VMVX dβMX dφ )+ + · m dT V dT 1 + bI1⁄2 dT
](
(23)
)
φ 2(VMVX)3⁄2 2 dCMX exp(-RI ) + (24) m dT V dT
(1) dβMX
1⁄2
(
)
|ZMZX|I1⁄2 dAφ 2VMVX dBMX dφ )+ + m 1⁄2 dT V dT 1 + b I dT MX
(
)
4(VMVX)3⁄2 dCMX |ZMZX|1⁄2m2 dT (25) V dAΦ/dT can be derived from eq 17. If the temperature dependence of the parameters are known, the vaporization enthalpy can be obtained from eqs 20-25. However, if they are not available, one can assume that they do not vary in a narrow temperature range, eq 20 can be simplified as
Table 1. Thermodynamic Properties of Pure Water at 1 Atm22 boiling point
373.15 K
specific heat capacity, Cps (0 °C) specific heat capacity, Cpl (0 °C) specific heat capacity, Cpl (100 °C) specific heat capacity, Cpv (372.76 K) enthalpy of vaporization, ∆H0vap (100 enthalpy of fusion, ∆H0fus (0 °C)
vap
∆H
°C)
2.11 J · g-1 · K-1 4.2176 J · g-1 · K-1 4.2159 J · g-1 · K-1 2.0784 J · g-1 · K-1 40.657 kJ · mol-1 333.6 J · g-1
(
d ln P0 VmMS |ZMZX|I1⁄2 dAφ + ) RT · dT 1000 1 + bI1⁄2 dT 2
)
(26)
Therefore, the vaporization enthalpy can be roughly estimated without using any adjustable parameters in the original Pitzer equation, and only one adjustable parameter, b, is needed in the modified form. Similarly, the results for the salts with available temperature-dependence parameters are also shown for comparison. It should be pointed out that the value estimated from this equation is only the value of enthalpy of vaporization at the beginning of the vaporization, without considering the variation of solution concentration during vaporization. Results and Discussion In this work, the average absolute relative deviation percentages between the calculated and literature data for an electrolyte solution are defined as: AARD% ) 100 × (
∑ |V
cal
- V ref| ⁄ V ref) ⁄ np
(27)
np
The V stands for the values of θF, θB, and ∆Hvap in this work. np is the data points in literature. The superscript cal and ref refer to the calculated results and reference data, respectively. Tables 2 and 3 showed the predicted results of FPD and BPE by eqs 11 and 12 with the original Pitzer equation and the modified one. One can see that the prediction is fairly well, which indicates this method is highly reliable. Figures 1, 2, and 3 plotted some typical calculated results of the freezing point depression or boiling point elevation compared with the experimental data, which represented the good performance of this method. However, some relative large deviations also can be found for some systems, such as Na2CO3, ZnCl2, ZnBr2, and so forth. In our opinion, these exceptions can be attributed to the following aspects: (a) The error from the present predictive method. The linearly temperature dependent change of enthalpy change of fusion or vaporization and the assumption of constant ∆Cp were employed in this work. These assumptions may bring about some errors in case of electrolyte solutions, especially in high concentration range. (b) The properties of electrolyte. The hydrolysis of some salts, such as ZnCl2, ZnBr2, and so forth can bring about the changes in the nature of the solute. The complex forms of ions existing in the solution simultaneously, such as HCO3-, CO32-. OH- in Na2CO3 solution, make the calculation of ionic strength be inaccurate. Moreover, the association effects of ions for some systems, such as MnSO4 and so forth also can not be neglected. These factors make the model perform less satisfactorily. (c)The temperature dependence of parameters. In the present method, the activity of water for predicting the FPD/BPE data was calculated directly from Pitzer parameters at 298.15 K, which maybe not accurate. Thus, for some salts, the temperature effects are considered and results are also shown in Tables 2 and 3. Details of numerical expressions for temperature
2232 Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009 Table 2. Prediction Results of Freezing Point Depression of Single-Electrolyte Solutions Mmax data Pitzer M-Pitzer (mol · kg-1) points, np (AARD%) (AARD%)
salts HCl HNO3 LiCl NaCl NaBr NaNO3 NaH2PO4 NaOH CH3COONa KCl KBr KI KNO3 KOH KH2PO4 CsCl NH4Cl NH4NO3 AgNO3 Na2SO4 Na2CO3 (NH4)2SO4 Na2S2O3 Na2HPO4 H2SO4 K2SO4 K2HPO4 BaCl2 MgCl2 SrCl2 CoCl2 CaCl2 ZnCl2 ZnBr2 CuSO4 MgSO4 MnSO4 ZnSO4 Na3PO4 La(NO3)3
3.740 3.484 4.3827 5.2017 1.851 8.22 2.084 4.07 1.206 3.2998 3.954 4.016 1.099 5.9412 0.816 1.485 4.5802 8.77 1.121 1.41 0.602 1.441 1.581 0.107 5.252 0.302 0.499 0.915 2.9409 1.992 0.4217 4.3254 3.1443 3.6331 1.020 1.582 1.656 1.180 0.156 0.1749
12 15 17 34 14 22 20 10 24 29 26 17 12 12 16 23 11 14 24 7 14 16 3 23 15 13 20 17 18 11 33 4 5 13 22 15 14 5 10
2.01/1.51a 0.76 3.01 5.40/2.78a 1.30 6.37 3.34 4.11/3.01a 0.60 2.62/1.06a 4.37 3.32 4.17 3.68 1.03 2.49 1.82 7.73 1.63 2.87 9.14 3.99 8.55 1.95 5.53 0.79/0.55a 3.76 0.64 3.08 2.64 2.16 3.48 13.20 8.59 2.70 1.68 9.28 4.63 4.26 1.57
1.02 1.63 2.76 5.50/3.21b 1.82 7.71 3.50 6.99/6.80b 0.88 2.54/4.28b 4.42/9.25b 3.23 3.99 2.89/6.46b 0.98 3.06 1.97 2.45 2.81 8.70 8.32 1.73 1.05 3.67 0.39 2.50 2.27 10.09 15.46 10.13 6.39 16.35 13.89 0.53
refs 22 22 22,25 22,25,26 22 22,27 22 22 22 22,25,28 22,25 22 22,28 22,26 22 22 22,25 27 22 22,27,29 22 22 22 22 22 22,30 22 22,30 22,25,26 22 30 22,25,26 26 26 22 22,30 22 22 22 30
a
Calculated results by using the original Pitzer parameters at 273.15 K. b Calculated results by using the modified Pitzer parameters at 273.15 K.
Figure 1. Experimental and calculated values of freezing point depression versus the ionic strength. 0, KI;22 O, NaOH;22 4, HCl.22
Figure 2. Experimental and calculated values of freezing point depression versus the ionic strength. 0, SrCl2; O, ZnSO4.
Table 3. Prediction Results of Boiling Point Elevation of Single-Electrolyte Solutions salts KCl KBr KNO3 NaCl
Mmax (mol · kg-1) 7.8000 5.0000 1.4836 1.000
data points, np 20 15 9 12
Pitzer (AARD%) a
1.23/0.67 2.02 3.22 0.70/0.09a
M-Pitzer (AARD%)
refs
1.32 2.01/1.61b 3.26 0.79/0.10b
31 32 33 34
a Calculated results by using the original Pitzer parameters at 373.15 K. b Calculated results by using the modified Pitzer parameters at 373.15 K.
dependency of those salts can be found in the Supporting Information. One can see from Table 2 that the deviations for most of the systems are reduced, except for KCl, KBr, and KOH by the modified Pitzer model, whereas the predicted results are all improved in Table 3. Probably because the temperature difference between boiling point and 298.15 K is larger than the one between freezing point and 298.15 K, the temperature dependence becomes more important. On the other hand, the enthalpies of vaporization also can be determined from eqs 20-26. However, very limited experimental data for this property have been found for electrolyte solutions, except a series of work on saturated aqueous solutions by Apelblat et al.35-39,41-47 The present method is employed to predict the values of enthalpies of vaporization to compare with them, shown in Table 4. Some good predictions are
Figure 3. Experimental and calculated values of boiling point elevation versus the ionic strength. 0, KCl;31 O, KBr.32
obtained, whereas bad predictions also can be found. It should be noted that, most of the experimental ∆Hvap values are the ones at saturated points. The concentrations are very high, such as 36.229 mol · kg-1 for NH4NO3 and 7.837mol · kg-1 for Na2S2O3. As mentioned before, the predictions deteriorate in the case of very high concentration. Accordingly, for a solution that is not saturated, such as KSCN and CaCl2 in Table 5, a good result can be obtained.
Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009 2233 Table 4. Prediction Results of Vaporization Enthalpies of Single-Electrolyte Solutions salts NaCl NaBr NaNO3 NaNO2 NaClO4 NaClO3 Na Acetate KCl KBr KI KNO3 K Acetate KClO3 KSCN CsCl CsNO3 CsI RbCl LiCl NH4NO3 NH4Cl NH4Br NH4I Na2S2O3 Na2S2O8 Na2CO3 Li2SO4 Cs2SO4 (NH4)2SO4 BaCl2 SrCl2 CaCl2 MnCl2 Mg(NO3)2 Cd(NO3)2 UO2(NO3)2 CuSO4 ZnSO4 MgSO4 CdSO4
Mmax (mol · kg-1) 6.222 11.27 13.348 14.915 21.00 9.83 8.97 5.562 6.157 10.150 8.307 35.3 2.45 15.41 13.00 3.282 3.63 8.785 21.039 36.229 8.557 9.93 13.893 7.837 3.562 4.470 3.216 5.39 6.005 1.958 4.678 3.776 6.721 5.728 10.18 5.373 1.639 4.194 4.310 3.78
temperature range(K) 278.15∼323.15 278.15∼323.15 278.15∼323.15 278.15∼323.15 278.15∼318.15 288.15∼303.15 278.15∼318.15 298.15∼318.15 283.15∼308.15 278.15∼323.15 278.15∼323.15 278.15∼318.15 298.15∼343.15 303.15 278.15∼323.15 278.15∼323.15 278.15∼303.15 278.15∼318.15 283.15∼313.15 283.15∼313.15 283.15∼313.15 278.15∼323.15 278.15∼323.15 278.15∼318.15 278.15∼318.15 278.15∼323.15 278.15∼323.15 278.15∼323.15 283.15∼308.15 283.15∼313.15 278.15∼323.15 303.15 283.15∼308.15 278.15∼323.15 278.15∼323.15 278.15∼323.15 283.15∼308.15 283.15∼308.15 278.15∼323.15 278.15∼323.15
a
data points, np
Conclusions In this work, a predictive method for calculating the freezing point depression, boiling point elevation, and vaporization enthalpies of electrolyte solutions were proposed based on the Pitzer theories. The calculation only needs the thermodynamic
a
10 10 10 10 9 4 9 21 6 10 10 9 10 6 10 10 6 9 7 7 7 10 10 9 9 10 10 10 6 7 10 11 6 10 10 10 6 6 10 10
Calculated results by using the temperature-dependent parameters of the Pitzer model. parameters of the modified Pitzer model.
In consideration of the temperature effects, the Pitzer temperature-dependent parameters for some salts have been measured by silvester et al.48 Some of them were employed in the present work and the prediction results are also listed in Table 4. From this table, one can find out that the predicted results for a few of salts become slightly better. That is because those temperature-dependent parameters have a very limited applicable concentration range, which are not reliable for calculating the vaporization enthalpy at the saturation point. From the predicted results of FPD/BPE and enthalpies of vaporization shown in Tables 2-4, generally, relative large deviations can be found in the high ionic strength range, which indicates that this method is not very suitable in this case. However, because the Pitzer parameters have been generated for hundreds of single salts, this predictive method still can be recommended as a very good first approximation for these properties, at least for the strong electrolytes or solutions in a concentration range that is not very high, which will be useful in case of scarcity of experimental data.
Pitzer (AARD%) 1.14/1.16 2.74/8.62a 4.17 3.52 15.91 6.14 10.09 3.38/3.28a 2.67/2.60a 4.82 3.09 10.38 2.42 0.59 5.00 0.71 3.56 2.87 19.47 22.04 7.02 1.59 6.52 13.12 3.38 3.87 6.09/6.51a 10.45 8.51 9.98/10.02a 9.83 0.35/0.31a 34.28 6.17 22.50 11.61 2.96 20.11 3.23 3.15 b
M-Pitzer (AARD%) b
1.12/1.12 3.40/2.22b 4.10 15.66 6.04 10.02 3.33/8.33b 2.72/2.16b 4.76 3.13 10.79 2.40 0.72 4.97 0.70 3.55 19.86 6.99 12.91 3.37 3.82 6.18 10.33 9.99 9.70 0.35 34.14 6.01 2.96 20.06
refs 35 35 35 35 36 36 36 37 38 39 39,40 36 36 40 41 41 41 35 42 42 42 43 39 39 44 45 39 41 38 46 39 40 38 39 47 39 38 46 46 47
Calculated results by using the temperature-dependent
constants for pure solvent and the well-known reported Pitzer parameters. This method was used for prediction of 40 freezing point depressions, 4 boiling point elevations, and 40 vaporization enthalpies. The predicted results are very well compared with the experimental data, representing a good performance of this method. However, the present model is not very suitable for solutions with very high concentration. Besides this shortage, this method can provide a good first approximation for the prediction of these properties, at least for the strong electrolytes or solutions within the applicable concentration range of the Pitzer parameters. Acknowledgment This work was financially supported by the National Natural Science Foundation of China (No. 50425415) and National Basic Research Program of China (973 Program: 2007CB613608). Supporting Information Available: Pitzer parameters for the electrolytes of the original equation and the modified one, temperature-dependence parameters of some salts of the original Pitzer model and the modified Pitzer one, details of the derivation of eq 12 for the calculation of boiling point elevation. This material is available free of charge via the Internet at http:// pubs.acs.org.
2234 Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009
Nomenclature aliq ) solvent activity R ) gas constant, 8.314472 J · mol-1 · K-1 T ) temperature, K ∆Gfus ) free energy change of fusion, J · mol-1 ∆Gvap ) free energy change of vaporization, J · mol-1 ∆Hfus ) enthalpy change of fusion, J · mol-1 ∆Hvap ) enthalpy change of vaporization, J · mol-1 Cp ) heat capacity, J · g-1 · K-1 m ) salt molality, mol · kg-1 of solvent Ms ) molecular weight of the solvent; Ms)18.0153 g · mol-1 for water AΦ) Debye-Hu¨ckel constant Z ) valence of ions I ) ionic strength bMX, BMX, CMX ) adjustable parameters for the modified Pitzer equation P ) vapor pressure np ) number of data points Greek Letters µ ) chemical potential θ ) change of freezing point or boiling point, K Φ ) osmotic coefficient V ) stoichiometric number of ions from dissociation R1,R2 ) nonlinear parameters for second virial coefficient in Pitzer equation (1) (2) φ β(0) MX, βMX, βMX, CMX ) adjustable parameters for the Pitzer equation Subscripts liq ) liquid state sol ) solid state 0 ) properties of the pure solvent c ) critical point F ) freezing point depression B ) boiling point elevation T ) properties at real freezing point or boiling point fus ) fusion vap ) vaporization s ) solvent M ) cation X ) anion MX ) neutral electrolyte
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ReceiVed for reView September 9, 2008 ReVised manuscript receiVed November 22, 2008 Accepted December 8, 2008 IE801348C
