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Ind. Eng. Chem. Res. 1996, 35, 2808-2812
CORRELATIONS Infinite Dilution Partial Molar Volumes of Aqueous Solutes over Wide Ranges of Conditions John P. O’Connell* Department of Chemical Engineering, University of Virginia, Charlottesville, Virginia 22903
Andrei V. Sharygin and Robert H. Wood Department of Chemistry and Biochemistry and Center for Molecular and Engineering Thermodynamics, University of Delaware, Newark, Delaware 19716
Data for the partial molar volume at infinite dilution for several gaseous solutes and H3BO3 in water have been correlated using a generalized Krichevskii parameter. The result is a simple function for vj ∞1 in terms of the water density and compressibility for conditions from ambient to 725 K and 40 MPa. The accuracy is much better with only two parameters than with a previous five-parameter model. Introduction The infinite dilution partial molar volume, vj ∞1 , dominates the relative contribution of a solute (1) to the volumetric properties of a solution. For a solvent (2) at its critical points, vj ∞1 diverges to either +∞ (for gases) or -∞ (for heavies including salts) and the effects of this are felt far from the critical point (Levelt Sengers, 1991). While this is common to all solvents, aqueous systems are of particular interest because of their importance in geological and proposed near-critical water oxidation processes for toxic wastes. The divergence of vj ∞1 arises from the long range correlations of the solvent which results in its infinite compressibility (Levelt Sengers, 1991). However, the collection of properties known as the generalized Krichev∞ skii parameter, A12 ≡ vj ∞1 /(κ°T)2RT ≡ limN1f0(∂(pV/RT)/ ∂N1)T,V,N2, which also contains the pure solvent isothermal compressibility, (κ°T)2, gas constant, R, and temperature, T, is well behaved under all conditions. It has been used before by Cooney and O’Connell (1987) for aqueous salt properties and by Crovetto et al. (1991) ∞ for aqueous CO2. They both found that A12 was a strong function of water density but was nearly independent of temperature. The generalized Krichevskii parameter also arises naturally in fluctuation solution theory (Kirkwood and Buff, 1951; O’Connell, 1971, 1990) with ∞ G12 ∞ ∞ A12 ) 1 - C12 ) +1 1 + G°22
(1)
∞ where C12 is the dimensionless spatial integral of the infinite dilution solute-solvent direct correlation function (DCFI) and the Gij are the dimensionless spatial integrals of the total correlation function (TCFI) related to the radial distribution function. The DCFI are related to the TCFI by the integrated Ornstein-Zernike equation; the second equality in eq 1 is an example of the connection.
Recently, volumetric data at high temperatures for aqueous solutions of nonelectrolytes have become available for CH4, CO2, H2S, and NH3 (Hnedkovsky et al., 1995b) and H3BO3 (Hnedkovsky et al., 1995a). Apparent molar volumes, vφ, were reported for CH4, CO2, and H2S at a single molality near 0.2 mol kg-1 from 298 to 705 K and 20 to 35 MPa. Values were also given over the same temperatures and pressures at molalities from 0.2 to 3.0 mol kg-1 for NH3 and 0.2 to 0.75 mol kg-1 for H3BO3. The estimated uncertainty in vφ varied from 1 to 5% (Hnedkovsky et al., 1995a,b). It is also possible to treat water as a solute and obtain its partial molar volume from the equation of state for water since it equals v°2, the molar volume of pure water. Including water in the analysis is helpful because we have very accurate values of its apparent volume. Unlike the other solutes, water molecules participate in the aqueous structure so there is no divergence in its apparent volume. Hnedkovsky and Wood (1995) have shown that the correlating equations of Shock et al. (1989) for vj ∞1 and cj∞p,φ do not have the correct shape in the critical region ∞ ∞ and that C12 and A12 are much better behaved. Filling the need for a better correlating equation for vj ∞1 is the purpose of this paper. Results and Analysis Though there is a molecular theory basis for a correlation of the generalized Krichevskii parameter, the particular form to be used depends upon accurate low-concentration data over a wide range of conditions. In the present paper, we have used only the results of Hnedkovsky et al. (1995a,b). There are no other data that meet the criteria. The only other data available are from Biggerstaff and Wood (1988) for the solutes Ar, Xe, and C2H4 in water at solute mole fractions less than 0.01. However, these are believed to be less accurate, especially in the near-critical region, because they were taken on an earlier version of Hnedkovsky’s apparatus. The estimated noise level of these data in the temperature range 650-690 K is 1 order of magni-
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Ind. Eng. Chem. Res., Vol. 35, No. 8, 1996 2809
Figure 1. Extrapolation of apparent molar volume to infinite dilution for (a) NH3(aq) and (b) H3BO3(aq) at different temperatures and a constant pressure of 28 MPa: (O) T ) 653 K; (b) T ) 657 K; (4) T ) 660 K; (2) T ) 663 K; (0) T ) 666 K; (9) T ) 668 K; (]) T ) 670 K; ([) T ) 675 K; (3) T ) 679 K; (1) T ) 685 K; (+) T ) 705 K. Experimental data are from Hnedkovsky et al. (1995a,b).
tude higher and thermal equilibrium was poorer than determined for Hnedkovsky’s measurements. Even so, the equations found here describe well the earlier data considering their error bars. But they cannot be considered as a good test of the present correlating equation. The values of vj ∞1 for NH3(aq) and H3BO3(aq) were obtained by linear extrapolation of the experimental vφ to zero molality. Figure 1 shows the variations; all experimental points were used with the exception of the highest concentration for NH3(aq). It is apparent that the experimental conditions were not so close to the water critical point that a linear extrapolation is not correct. Note that vj ∞1 for NH3 is larger than vφ for T > 668 K and smaller for T < 667 K, while the opposite is seen for H3BO3. In the latter case, the temperature of changeover is slightly different, being about 673 K. For CH4, CO2, and H2S values of vj ∞1 were assumed equal to the measured vφ at m ) 0.2 mol kg-1, the only concentration for which data were taken. The estimated increase in uncertainty due to this extrapolation was 2%, about the average of the difference between vj ∞1 and vφ at the lowest NH3 concentration. The uncertainty in vj ∞1 for water was assumed to be 0.1%.
Figure 2. Generalized Krichevskii parameter as a function of water density for all solutes at temperatures (a) from 298 to 623 K and (b) from 653 to 705 K. The points are the experimental data from Hnedkovsky et al. (1995a,b), and lines represent calculations using eq 5.
As shown in Figure 2, the Krichevskii parameters for the present aqueous solutes also behave as found previously (Cooney and O’Connell, 1987; Crovetto et al., 1991) with all of the state dependence described by the solvent density, F°2 ) 1/v°2. This means that vj ∞1 can be found from the properties of pure water and a correla∞ tion of A12 . There are a variety of correlations that might be valid. However, it was pointed out to us (Liu, 1995) that it is wise to choose a form that would be analytically integrable to obtain the infinite dilution fugacity coefficient of the solute, φ∞1 ,
ln φ∞1 ≡ lim ln x1f0
f1 x1P
∫0F°(A12∞ - 1) 2
)
dF°2
( )
- ln
F°2
p
F°2RT
(2)
Further, it is desirable to have a minimum number of empirical parameters. Finally, the equation should go to proper limits. At very low densities, the second virial coefficient equation of state is valid ∞ ) 1 + 2F°2B12(T) + ... A12
(3)
where B12 is the second cross virial coefficient between the solute and solvent. Thus, the zero-density (ideal gas) limit should be unity and the initial departure should be linear in solvent density. The present data
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2810 Ind. Eng. Chem. Res., Vol. 35, No. 8, 1996 Table 1. Regression Parameters of Equations 4 and 5 for Aqueous Solutes eq no.
104a, m3 kg-1
104b, m3 kg-1
av ∆vj ∞1 /σ
max ∆vj ∞1 /σ
4 5
8.80 48.9
CH4 4.01 0.94
0.7 1.5
2.2 3.1
4 5
-1.67 37.9
CO2 3.76 0.81
0.6 1.5
2.4 3.6
4 5
-13.3 26.6
H2S 3.98 0.93
1.1 1.4
2.8 4.4
4 5
-24.3 12.5
NH3 2.81 0.28
1.4 1.3
4.5 3.6
4 5
-71.2 -31.0
H3BO3 5.03 1.58
2.1 1.0
7.3 4.3
4 5
-38.3 0
H2O 2.87 0
471 0
1453 0
are generally consistent with eq 3 though there is no ∞ explicit temperature dependence seen for A12 . After trying several forms, we chose two models that have only two parameters plus the limiting form of eq 3. The parameters were fitted using weighted least-squares fitting of the solute data. The first correlation uses the form of eq 3 at low densities and the exponential form of Cooney and O’Connell (1987) for high densities ∞ A12 ) 1 + F°2{a + b[exp(0.0044F°2) - 1]}
(4)
where the units of F°2 are kg m-3 and the F°2 values were obtained from the equation of state of water of Hill (1990). The constant in the exponent was chosen after finding that three-parameter fits for all solutes yielded similar values; the average of 0.0044 allowed a twoparameter form with very little sacrifice of accuracy from that with three parameters. It is expected that this correlation form will hold for other solvents but the empirical value of the constant in the exponential will vary with the solvent. Table 1 gives the average and maximum deviations from eq 4 for the solutes. The agreement is generally within experimental uncertainty except for H2O and H3BO3. The second form was chosen to obtain exact results for H2O while preserving the integrability of eq 4 ∞ A12 ) A°22 + F°2[a + b exp(0.005F°2) - 1]
(5)
where A°22 ≡ 1/F°2(κT)°2RT, the solvent-reduced bulk modulus which has been studied for many substances by Huang and O’Connell (1987). The constant of 0.005 in this exponent was found in the same manner as for eq 4. In the terms of fluctuation solution theory, eq 5 becomes ∞ ) F°2{a + b[exp(0.005F°2) - 1]} C°22 - C12
(6)
Table 1 also shows the results of this correlation. There is a small sacrifice of accuracy of eq 5 compared to eq 4 for CH4, CO2, and H2S, but significant improvement for the other solutes. However, as shown in Figure 3a, the error is always within a factor of 4 of the estimated uncertainty and the average is within a factor of 1.5. Equation 6 also yields an analytic expression for φ∞1
Figure 3. Absolute error in the infinite dilution partial molar volume as a function of water density for all solutes: (a) present model, eq 5; (b) Shock et al. (1989), eq 8. The dashed lines represent two standard deviations.
ln φ∞1 /φ°2 ) (a - b)F°2 + 200b[exp(0.005F°2) - 1]
(7)
Discussion Shock et al. (1989) proposed an equation for correlating vj ∞1 for aqueous nonelectrolytes based on earlier Helgeson equations (Helgeson et al., 1981). Later, Shock and Helgeson (1990) extended this to organic species and this has been widely used by geochemists to predict the properties of aqueous nonelectrolytes at high temperatures. Their equation is
[
vj ∞1 ) cˆ a1 +
a3 a2 + + ψ+p T-Θ a4 (ψ + p)(T - Θ)
( )]
1 ∂ -ω 2 (8) ∂p T
where cˆ is a conversion factor (41.84 bar cm3 cal-1), the solvent characteristic constants are ψ ) 2600 bar and Θ ) 228 K, is the solvent dielectric constant, p is in bar, ai are solute-dependent parameters, and ω is the conventional Born coefficient for each species. The parameters of eq 8 have been related to other properties of the solutes at 25 °C (Helgeson et al., 1981; Shock and Helgeson, 1990). Thus values of vj ∞1 can be predicted using eq 8. To examine whether simplifications might be made to the method of Helgeson et al., we have fitted eq 8 to the present data. The results are given in Table
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Ind. Eng. Chem. Res., Vol. 35, No. 8, 1996 2811 Table 2. Regression Parameters of Equation 8 for Aqueous Solutes solute
a1, cal/(mol‚bar)
10-2a2, cal/mol
10-2a3, (cal‚K)/(mol‚bar)
10-4a4, (cal‚K)/mol
10-3ω, cal/mol
av ∆vj ∞1 /σ
max ∆vj ∞1 /σ
CH4 CO2 H2S NH3 H3BO3 H2O
1.63 2.75 2.80 0.57 1.06 1.55
-0.72 -36.62 -38.91 12.57 2.52 -27.22
-1.13 -1.83 -1.84 1.52 -0.16 -1.13
15.03 36.40 38.07 -54.24 0.58 28.15
-6.41 -5.32 -3.88 -2.49 3.75 -0.27
9.6 9.1 7.9 4.6 1.0 160
24.5 21.9 19.4 11.6 2.4 573
Figure 4. Infinite dilution partial molar volume for H2S(aq) as a function of water density: (a) P ) 28 MPa; (b) P ) 35 MPa.
2 and are shown in Figure 3b. Table 2 shows that for CH4, CO2, H2S, and NH3, the five-parameter eq 8 has average values of ∆vj °1/σ from 4.6 to 9.6, more than 3 times that of eq 5 with only two parameters. The fit for H3BO3 from eq 8 was somewhat better than that from eq 5. Our conclusion is that eq 5 is on average much better than the previous model and more reliable for densities in the range of these experiments than is eq 4. Our ultimate intention is to utilize the simple form and parameters of eq 5 to describe other properties of aqueous solutions. However, that is beyond the scope of this paper. Figures 4 and 5 show direct comparisons of the correlations with vj ∞1 , demonstrating the good agreement with data of the current method. ∞ ∞ and C°22 - C12 The molecular arguments for 1 - C12 being functions of only F°2 are obscure. However, it is consistent with the finding of Brelvi and O’Connell (1972) and Huang and O’Connell (1987) that, for liquids,
Figure 5. Infinite dilution partial molar volume for H3BO3(aq) as a function of water density: (a) P ) 28 MPa; (b) P ) 35 MPa.
DCFI are determined primarily by the density for virtually all classes of substances. The difference between eqs 4 and 5 is that the latter describes the volume change from replacing a water molecule with a solute molecule. To the degree that the effects of excluded volume, attraction, and aqueous structure are similar among the substances, this should be more reliable. The changes in accuracy suggest that the gaseous solutes are not very much like H2O, but that NH3 and H3BO3 are more similar in the important correlations between solute and solvent. Acknowledgment The authors thank E. M. Yezdimer for help with the computer programs and graphs. J.P.O. thanks the Office of Basic Energy Sciences, U.S. Department of Energy for financial support under Contract DEFG0588ER-13943 and the Staff of the Laboratory for Applied Thermodynamics, Technical University of Delft, for
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2812 Ind. Eng. Chem. Res., Vol. 35, No. 8, 1996
their stimulation, support, and hospitality. R.H.W. thanks the U.S. Department of Energy for financial support of this research under Grant DEFG02-89ER14080. Such support does not constitute endorsement of DOE of the views expressed in this article. Nomenclature a, b ) correlation parameters in eqs 4-7 a1, a2, a3, a4 ) correlation parameters in eq 8 A°12 ) generalized Krichevskii parameter ≡ vj ∞1 /(κ°T)2RT ≡ limN1f0(∂(pV/RT)/∂N1)T,V,N2 A°22 ) reduced bulk modulus of pure solvent ≡ v°2/(κ°T)2RT B12 ) second virial coefficient for solute-solvent pairs cˆ ) correlation parameter in eq 8 ∞ ) infinite dilution direct correlation function integral C12 for solute-solvent pairs C°22 ) direct correlation function integral for pure solvent pairs ∞ ) infinite dilution total correlation function integral G12 for solute-solvent pairs G°22 ) total correlation function integral for pure solvent pairs Ni ) number of moles of species i p ) absolute pressure R ) universal gas constant T ) absolute temperature v°2 ) pure solvent molar volume vφ ) apparent volume for solute ≡ (V - N2v°2)/N1 vj ∞1 ) infinite dilution partial molar volume of solute 1 V ) total volume x1 ) mole fraction of solute 1 Greek Letters ) solvent dielectric constant Θ ) correlation parameter in eq 8 (κ°T)2 ) isothermal compressibility of pure solvent 2 F°2 ) molar density of pure solvent 2 σ ) standard deviation of measurements φ°1 ) fugacity coefficient for pure solvent 2 φ∞2 ) infinite dilution fugacity coefficient of solute 1 ψ ) correlation parameter in eq 8 ω ) correlation parameter in eq 8 Subscripts 1,2
) solute (1), solvent (2) property
Superscripts ° ) pure ) infinite dilution
∞
Literature Cited Biggerstaff, D. R.; Wood, R. H. Apparent Molar Volumes of Aqueous Argon, Ethylene, and Xenon from 300 to 715 K. J. Phys. Chem. 1988, 92, 1988-1994. Brelvi, S. W.; O’Connell, J. P. Corresponding States Correlations for Liquid Compressibility and Partial Molar Volumes of Gases at Infinite Dilution in Liquids. AIChE J. 1972, 18, 1239-43.
Cooney, W. J.; O’Connell, J. P. Correlation of Partial Molar Volumes at Infinite Dilution of Salts in Water. Chem. Eng. Commun. 1987, 56, 341-7. Crovetto, R.; Wood, R. H.; Majer, V. Revised Densities of {xCO2 + (1-x)H2O} with x < 0.014 at Supercritical Conditions: Molar Volumes, Partial Molar Volumes of CO2 at Infinite Dilution, and Excess Molar Volumes. J. Chem. Thermodyn. 1991, 23, 1139-46. Helgeson, H. C.; Kirkham, D. H.; Flowers, D. C. Theoretical Prediction of the Thermodynamic Behavior of Aqueous Electrolytes at High Pressures and Temperatures: IV. Calculation of Activity Coefficients, Osmotic Coefficients, and Apparent Molal and Standard and Relative Partial Molal Properties to 600 °C and 5 kb. Am. J. Sci. 1981, 281, 1249-1516. Hill, P. G. A Unified Fundamental Equation for the Thermodynamic Properties of H2O. J. Phys. Chem. Ref. Data 1990, 19, 1233-1274. Hnedkovsky, L.; Wood, R. H. Apparent Molar Heat Capacities of Aqueous Solutions of CH4, CO2, H2S, and NH3 from 304 K to 704 K and pressures up to 28 MPa. 1995, J. Chem. Thermodyn. Manuscript in preparation. Hnedkovsky, L.; Majer, V.; Wood, R. H. Volumes and Heat Capacities of H3BO3(aq) at Temperatures from 298.15 K to 705 K and at Pressures to 35 MPa. J. Chem. Thermodyn. 1995a, 27, 801-14. Hnedkovsky, L.; Wood, R. H.; Majer, V. Volumes of Aqueous Solutions of CH4, CO2, H2S, and NH3 from 298 K to 705 K and Pressures up to 35 MPa. J. Chem. Thermodyn. 1995b, in press. Huang, Y-H.; O’Connell, J. P. Corresponding States Correlation for the Volumetric Properties of Compressed Liquids and Liquid Mixtures. Fluid Phase Equilib. 1987, 37, 75-84. Kirkwood, J. G.; Buff, F. P. The Statistical Mechanical Theory of Liquids. J. Chem. Phys. 1951, 19, 774-82. Levelt Sengers, J. M. H. Thermodynamics of Solutions Near the Solvent’s Critical Point. In Supercritical Fluid Technology: Reviews in Modern Theory and Applications; Bruno, T. J., Ely, J. F., Eds.; CRC Press: Boca Raton, FL, 1991; Chapter 1, pp 1-56. Liu, H. Beijing University of Chemical Technology, Beijing, P. R. China, personal communication, 1995. O’Connell, J. P. Thermodynamic Properties of Solutions Based on Correlation Functions. Mol. Phys. 1971, 20, 27-33. O’Connell, J. P. Thermodynamic Properties of Mixtures from Fluctuation Solution Theory. In Fluctuation Theory of Mixtures; Matteoli, E., Mansoori, G. A., Eds.; Taylor and Francis: New York, 1990; pp 45-67. Shock, E. L.; Helgeson, H. C. Calculation of the Thermodynamic and Transport Properties of Aqueous Species at High Pressures and Temperatures: Standard Partial Molal Properties of Organic Species. Geochim. Cosmochim. Acta 1990, 54, 915-45. Shock, E. L.; Helgeson, H. C.; Sverjensky, D. A. Calculation of the Thermodynamic and Transport Properties of Aqueous Species at High Pressures and Temperatures: Standard Partial Molal Properties of Inorganic and Neutral Species. Geochim. Cosmochim. Acta 1989, 53, 2157-83.
Received for review December 4, 1995 Revised manuscript received May 22, 1996X IE950729U
X Abstract published in Advance ACS Abstracts, July 1, 1996.
