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PVTx measurements for H2O+D2O mixtures in the near-critical and supercritical regions
J. of Supercritical Fluids 26 (2003) 115 /128 www.elsevier.com/locate/supflu
PVTx measurements for H2OD2O mixtures in the nearcritical and supercritical regions /
Akhmed R. Bazaev a, Ilmutdin M. Abdulagatov a,b,*, Joseph W. Magee b, Emil A. Bazaev a, Asbat E. Ramazanova a a
Institute for Geothermal Problems of the Dagestan Scientific Center of the Russian Academy of Sciences, Shamilya Str. 39, Dagestan, 367030 Makhachkala, Russia b Physical and Chemical Properties Division, National Institute of Standards and Technology, 325 Broadway, Boulder, CO 80305, USA Received 16 April 2002; received in revised form 20 September 2002; accepted 2 October 2002
Abstract The PVTx relationships of H2O/D2O mixtures were measured in the near-critical and supercritical regions. Measurements were made using a constant-volume piezometer immersed in a precision thermostat. The volume of the piezometer VPT was corrected for both temperature and pressure expansions. The uncertainty of the density measurements was estimated to be 0.05 /0.10%, depending on the experimental pressure and temperature. The uncertainty of the temperature, pressure, and composition measurements were respectively 10 mK, 0.05%, and 0.001 mol fraction. Measurements were made along various near-critical and supercritical isotherms between 517 and 680 K at pressures from 3 to 38 MPa and densities from 97 to 466 kg m 3 for two compositions, namely 0.5 and 0.6 mol fraction of H2O. The measured PVTx data for H2O/D2O mixtures were compared with values calculated from a sixterm Landau expansion and from parametric crossover equations of state. The accuracy of the method was confirmed by PVT measurements for pure water in the critical and supercritical regions. The values of partial molar volumes for dilute H2O/D2O mixtures near the critical point of pure H2O and the Krichevskii parameter were calculated using a parametric crossover equation of state and the critical locus data. Using measured values of molar volumes and corresponding values for pure components, we derived the values of excess molar volumes as a function of pressure for the supercritical isotherms. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Critical point; Crossover equation of state; Heavy water; Krichevskii parameter; Light water; Partial molar volume; Supercritical water
Contribution of the National Institute of Standards and Technology, not subject to copyright in the United State. * Corresponding author. Tel.: /1-303-497-4027; fax: /1303-497-5224. E-mail address: [email protected] (I.M. Abdulagatov).
1. Introduction Thermodynamic properties of isotopic mixtures provide insight to our understanding of the difference between isotopes and mixture critical behavior [1]. There are other theoretical problems
0896-8446/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 8 9 6 - 8 4 4 6 ( 0 2 ) 0 0 2 4 2 - 5
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connected with the behavior of isotopic mixtures, for example, how isotope-exchange reactions may affect near-critical and supercritical properties [1]. Despite their usefulness, measurements of the thermodynamic properties of H2O/D2O mixtures in the near-critical and the supercritical regions are scarce. Simonson [2] has reported the excess enthalpy HE of H2O/D2O mixtures in the temperature range from 310 to 673 K at pressures from 7 to 37 MPa with a flow calorimeter. Marshall and Simonson [3] have measured the liquid/vapor critical temperatures of H2O/D2O mixtures over the entire composition range with a precision of 9/0.1 K using an improved semi-micro phase equilibrium apparatus. The critical temperature locus TC(x ) of H2O/D2O mixtures was found to be close to a linear function of concentration x . Marshall and Simonson [3] studied the isotope-exchange reaction effect on the critical temperature locus of H2O/D2O mixtures. There were no published experimental data for either the critical pressure or the critical density. In a work in progress, Polikhronidi et al. [4] measured the isochoric heat capacity for the equimolar 0.5H2O/0.5D2O mixture in the critical and supercritical regions. They also reported the saturated properties (rS, TS, CVS) and the critical temperature and the critical density for this mixture. Crossover equations of state of the H2O/D2O mixture for the Helmholtz-energy density have been developed by Abdulkadirova et al. [1] and by Kiselev et al. [5]. Both crossover models incorporate a crossover from fluctuation-induced scaling behavior near the critical curve to regular (classical or mean-field, Landau expansion) behavior outside the critical region. A generalization of a crossover equation of state, previously developed by Kostrowicka Wyczalkowska et al. [6] for pure H2O and pure D2O, has been recently extended to the mixtures on the basis of the principle of isomorphism of the critical behavior by Abdulkadirova et al. [1]. This crossover model represents the thermodynamic properties of H2O/D2O mixtures in the same range of temperatures and densities as for the pure components, i.e., reduced susceptibility x¯ 1 B2; density r /165 kg m 3, and temperature T /630 K. An alternative parametric crossover equation was developed for the
H2O/D2O mixture by Kiselev et al. [5]. They applied the isomorphic generalization of the law of corresponding states (LCS) to the prediction of thermodynamic properties and phase behavior of H2O/D2O mixtures in a wide region around the locus of vapor /liquid critical points. This equation yields a good representation of available thermodynamic properties data in the range of temperature 0.8TC(x )B/T B/1.5TC(x ) and densities 0.35rC(x )B/r B/1.65rC(x). The predictive capabilities of the crossover models of both Abdulkadirova et al. [1] and Kiselev et al. [5] will be tested using the present experimental PVTx data for the H2O/D2O mixture. The order of the discussion is as follows. In Section 2, a brief description is given for the apparatus and calibration procedures. Section 3 starts with a discussion of experimental results for H2O/D2O mixtures and pure light water. The measurements are compared with values calculated from crossover models. Then, a discussion is given of value the Krichevskii parameter for H2O/D2O mixture and how it is estimated from the critical line and vapor-pressure data and with a crossover model. These results are compared with the data reported by other authors. Then, we discuss how the Krichevskii function is used to calculate values of the partial molar volume for an infinitely dilute H2O/D2O solution. Finally, a discussion is given about how the values of the excess molar volumes are calculated from PVTx for the H2O/D2O mixture and the pure components.
2. Experimental procedure A detailed description of the apparatus and the experimental procedure has been given previously [7 /13]. Only essential information will be given here. The measurements were made using the constant-volume method, with an extraction of the sample from the piezometer under isothermal conditions. The high-pressure piezometer is constructed of heat- and corrosion-resistant highstrength alloy EI-43BU-VD and has a volume of 32.689/0.01 cm3 at a temperature of 673.15 K and a pressure of 40.32 MPa. It was calibrated by
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filling it with pure water and then withdrawing the water in small amounts and weighing them. The weight of the water withdrawn yielded the volume of the piezometer from the well-established density of water [14] at the temperature and pressure of the calibration. The fluid under study was thermostated in a double-wall air bath. Stirring of the mixture was accomplished with the aid of a steel ball that was moved by the oscillation of the thermostat. The fluid temperature was measured with a 10 V platinum resistance thermometer (PRT-10). This PRT was calibrated by VNIIFTRI (Moscow) on ITS-90. The maximum uncertainty in the measured temperature was 10 mK. The temperature inside the thermostat was maintained uniform within 5 mK with the aid of guard heaters located between the thermostat walls and regulating heaters, which were mounted inside the thermostat. The temperature inside the thermostat and the fluid temperature were controlled automatically [15]. The pressure was measured by a dead-weight oil gauge with an estimated uncertainty of 0.05%. Corrections were made for the volume of the stainless steel stirrer and the variation of the volume with temperature and pressure. The thermostat has double-walls with an inside volume of 65 dm3. The heating elements were arranged between the walls. To minimize temperature gradients in the air thermostat, two electrically driven high-speed fans were used. The inner volume of the piezometer was calculated by taking into consideration the corrections of the elastic pressure deformation and thermal expansion. The volume of the piezometer was previously calibrated from the known density of a standard fluid (pure water) with well-established PVT information (IAPWS-95 formulation, Wagner and Pruß [14]) at a temperature of T0 /673.15 K and a pressure of P0 /38.54 MPa. The volume at these conditions was VP0T0 /(32.2109/0.003) cm3. The volume of the piezometer was measured with an uncertainty of 0.01 /0.03%. The uncertainty of the density calculations from the IAPWS-95 formulation is 0.1%. The present experimental apparatus had no dead volumes. All masses were determined with an uncertainty of 0.05 mg. The composition of the reported values are accurate to 0.001 mol
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fraction. Taking into account the uncertainties of measurements of temperature, pressure, and concentration, the total experimental uncertainty of density was estimated to be 0.05 /0.10%, depending on the temperature T and pressure P . The commercial supplier of the D2O provided a purity analysis of 99.9 mol%.
3. Results and discussion Measurements of the PVTx relationships for H2O/D2O mixtures (x /0.5 and 0.6 mol fraction of H2O) were performed along five near-critical and supercritical isotherms, namely: 646, 648, 660, 670, and 680 K. The density ranged between 97 and 466 kg m 3 and the pressure ranged from 3 to 38 MPa. Measurements for a composition of 0.6 mol fraction of H2O were made along two quasiisochores 438 and 439 kg m3 in the temperature range from 520 to 680 K. For the equimolar H2O/D2O mixture, PVTx was also measured along two quasi-isochores 463 and 465 kg m 3 for temperatures between 517 and 645 K. The experimental temperatures, densities, and pressures for near-critical and supercritical H2O/D2O mixtures are presented in Table 1 and in Figs. 1/4. The critical parameters for measured compositions are: x/0.5 mol fraction of H2O: TC /645.4715 K, PC /21.868 MPa, and rC /339.0 kg m 3; and for x/0.6 mol fraction H2O: TC /645.7964 K, PC / 21.907 MPa, and rC /335.6 kg m 3. To check and confirm the accuracy of the method, PVT was measured with pure water along one near-critical and one supercritical isotherm, 653.15 and 673.15 K. The measured values are presented in Table 2. The data were compared with values calculated from the IAPWS-95 formulation [14]. Deviation statistics for the 14 measured densities of pure light water are: AAD /0.44%, Bias /0.16%, S.D. /0.46%, S.E. /0.12%, and Max. Dev/0.7%. All of the data show good agreement with the existing reliable PVT data sets for light water [16 /20]. The experimental isothermal PVT data measured for the equimolar H2O/D2O mixture are plotted in Figs. 1 and 2 together with values calculated from the crossover equations of state
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Table 1 Experimental values of the PVTx properties of light and heavy water mixtures along the near-critical and supercritical isotherms 0.5H2O/0.5D2O T (K)
r (kg m
3
)
P (MPa)
646 463.138 281.403 193.293 150.175 98.001
22.459 21.923 21.430 20.323 17.177
463.087 373.567 281.375 193.275 150.161 97.993
23.243 22.648 22.453 21.771 20.601 17.323
462.781 373.335 281.210 193.168 150.081 97.943
27.921 26.322 25.311 23.784 22.121 18.216
462.523 373.139 281.073 193.080 150.015 97.902
32.018 29.407 27.697 25.447 23.343 18.941
462.266 372.949 280.936 192.992 149.950 97.861
36.038 32.514 30.024 26.992 24.512 19.651
517 520 550 580 620 643 645
465.846 465.787 465.195 464.582 463.731 463.212 463.163
3.578 3.766 6.138 9.520 16.015 21.418 22.049
0.6H2O/0.4D2O 520.0 550.0 580.0 620.0 630.0
441.709 441.148 440.566 439.761 439.550
3.702 5.984 9.454 15.803 17.986
Table 1 (Continued ) 0.6H2O/0.4D2O T (K)
r (kg m 3)
P (MPa)
643.0 645.0 645.5 646.0 648.0 650.0 660.0 670.0 680.0
439.269 439.220 439.208 439.195 439.145 439.095 438.838 438.593 438.343
21.188 21.989 22.219 22.489 23.360 24.220 28.924 33.076 37.489
648
660
670
680
[1,5]. Fig. 1 also compares the present measurements and the values calculated from the parametric crossover model by Kiselev et al. [5] with parameters calculated by fitting the present data and by using LCS (without using present data). As one can see from Fig. 1, agreement of the fitted model is excellent (pressure AAD /0.2%). A detailed comparison between present data and the data calculated from a six-term Landauexpansion crossover model by Abdulkadirova et al. [1] is depicted in Fig. 2 together with values calculated from an alternative parametric crossover model by Kiselev et al. [5]. Present data were not used to fit the parameters of this model. Both crossover models represent the pressure data shown in Fig. 2 with comparable accuracy (S.D. of 0.2%). The pressure dependence of the compressibility factor, Z /PVm/RT, for the 0.5H2O/0.5D2O mixture along a supercritical isotherm 660 K is shown in Fig. 3. Fig. 3 shows that there is good internal consistency between measurements of temperature T , pressure P , and specific volume V for each thermodynamic state (PVT ) and the results of the prediction of the crossover equation [5]. The dependence of pressure on density for the 0.5H2O/0.5D2O mixture and for both pure components are shown in Fig. 4 together with values calculated from crossover equations [5,21] and literature data for pure H2O and D2O along the near-critical isotherm 648 K. It is remarkable that the mixture has a very broad flat region like pure components because this isotherm (648 K) is very close to the critical isotherm of the mixture 645.5 K and to the critical isotherms of the pure
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Fig. 1. Measured pressures of 0.5H2O/0.5D2O mixtures as a function of density along different sub-critical and supercritical isotherms: (k) present data; ( */) from crossover equation of state by Kiselev et al. [5] (which fit our present data); ( / / / /), from crossover equation of state by Kiselev et al. [5] (LCS); (w) the critical point of mixture.
Fig. 2. Measured pressures of 0.5H2O/0.5D2O mixtures as a function of density along different near-critical and supercritical isotherms: (k) 646 K; (j) 648 K; (D) 660 K; (%) 670 K; (’) 680 K; ( */), from the six-term Landau-expansion crossover equation of state by Abdulkadirova et al. [1] (without using present data); ( / / / /), from the crossover equation of state by Kiselev et al. [5] (LCS, without using present data).
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Fig. 3. Compressibility factors, Z /PVm/RT , of the 0.5H2O/0.5D2O mixture and their pure components as a function of pressure along the supercritical isotherm 660 K.
components 647.096 K (for light water) and 643.847 K (for heavy water). Fig. 5 depicts deviations of the experimental pressure from the values calculated using the parametric crossover equation of state developed by Kiselev et al. [5] (LCS model). This figure does not include one data point for which the deviation is larger than 1% (1.1%). As one can see from Fig. 5, most of the measured pressure data show good agreement to within 9/1.0% with values calculated from the crossover model by Kiselev et al. [5] with parameters determined from LCS. Most of the data at densities higher than 150 kg m 3 show deviations within 9/0.5%. Only the data in the low density range r B/100 kg m 3 exhibit deviations larger than 0.5%. Most of the data at high densities show negative deviations (experimental values slightly lower than calculated values). It should be emphasized that the current data were not used to fit the parameters of the LCS crossover model by Kiselev et al. [5]. The deviation statistics for the 30 measured pressures are (Kiselev et al. [5] LCS model): AAD/0.35%, Bias //0.13%, S.D. /
0.4%, S.E. /0.08%, and Max. Dev /1.1%. The maximum deviation (1.1%) was observed at the temperature 680 K and at a density of 97.86 kg m 3. The AAD for the optimized crossover model by Kiselev et al. [5], with parameters determined from the present measurements, is 0.2% (Fig. 1 solid curves). Several aspects of dilute mixture behavior are of considerable theoretical interest. For example, there is strong interest in negatively or positively diverging solute partial molar volume in mixtures in the immediate vicinity of the solvent’s critical point and path-dependence solvent properties in near-critical systems [22 /24]. The thermodynamic behavior of dilute near-critical mixtures depends on microscopic phenomena involving density perturbations induced by the presence of the solute and propagation of this density perturbation to a distance given by the pure solvent’s correlation length, for which j2 is proportional to the isothermal compressibility KT of the pure solvent [25 /27]. The thermodynamics of high-temperature aqueous solutions is a subject of wide-ranging
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Fig. 4. Pressures of the 0.5H2O/0.5D2O mixture and their pure components H2O and D2O as a function of density along the supercritical isotherm 648 K. Table 2 Experimental values of the PVT of pure water T / 653.15 K
T/ 673.15 K
P (MPa)
r (kg m 3)
P (MPa)
r (kg m 3)
29.25 25.18 23.48 22.99 20.25 16.61 12.58
527.79 460.42 290.71 205.73 125.34 82.440 54.390
40.30 33.87 28.65 26.53 22.32 17.90 13.32
527.13 459.89 290.42 205.24 125.59 82.680 54.340
interest because of the dominant role played by coexistence of both short-ranged (solvation) and long-ranged (compressibility-driven) phenomena [23,25,27] in fluid systems. The thermodynamic
behavior of the near-critical dilute solutions is extremely important to better understand molecular interactions and microscopic structures of the solutions. In the limit of infinite dilution, many partial molar properties of the solute ¯ ¯ (V¯ 2 ; H 2 ; C P2 ) diverge strongly at the pure solvent’s critical point [28 /30]. In general, the thermodynamic behavior of infinitely dilute mixtures near the solvent’s critical point can be completely characterized by the so-called Krichevskii parameter, which is equal to the derivative (@P=@x) TC VC calculated at the critical point of the pure solvent [22,31]. Using the concept of the Krichevskii parameter, Levelt Sengers [32] proposed a description of thermodynamic behavior of dilute near-critical solutions based on the derivative (@P=@x) VT ; and Japas et al. [33] denoted J (@P=@x) VT as the
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Fig. 5. Percentage pressure deviations, dP/100((Pexp/Pcal)/Pexp) of the experimental pressures for the 0.5H2O/0.5D2O mixture from the values calculated with the crossover equation of state by Kiselev et al. [5] (LCS).
Krichevskii function. The Krichevskii function J for dilute aqueous solutions (water/n -hexane) was derived from experimental PVTx data along the measured pure solvent’s (light water) nearcritical and supercritical isotherms as a function of density in a previous paper [11]. The value of the Krichevskii function at the critical point of the pure solvent is exactly the Krichevskii parameter [34,35]. The Krichevskii function J has a simple physical meaning and a straight-connection to total correlation function integrals [25,27,36 /38]. The Krichevskii parameter also governs the critical behavior of the thermodynamic properties of dilute solutions, particularly, the parameter K1 (Abdulkadirova et al. [1]), K1
x(1 x) dPC rC RTC
dx
@P
@T
C j;rrC
dTC dx
;
which is responsible for strongly divergent properties such as the isothermal compressibility and isobaric heat capacity, and is directly related to the Krichevskii parameter. The value of the Krichevskii parameter can be estimated from the initial slopes of the critical lines data TC(x ) and PC(x ) for dilute mixtures and the values of the vaporpressure PS /TS slope of the pure solvent at the
critical point. The initial slopes of the critical lines and vapor-pressure curve are related to the Krichevskii parameter as follows [23,24,39]: C @P @PC C dPS C @x VC TC @x CRL dT CXC @TC C (1) @x CRL or, equivalently, C @P dPC C dPS C @x VC TC dTC CRL dT CXC C dTC ; dx CRL
(2)
where (@PC =@x)CRL and (@TC =@x)CCRL are the initial slopes of the PC(x) and TC(x) critical lines, and (dPS/dT )C CXC is the slope of the solvent’s vapor-pressure curve evaluated at the critical point of the solvent (always positive). Qualitatively and quantitatively, the regimes of the near-critical dilute solutions behavior depends strongly on the signs and magnitudes of the initial slopes of the critical locus TC(x ), PC(x ), and on the slope of vapor-pressure curves PS(T ) at the
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critical point of pure solvent, i.e., magnitude and sign of the Krichevskii parameter (@P=@x) TC VC [40]. Therefore, the Krichevskii parameter is a good test for the thermodynamic consistency of the models for critical and vapor-pressure curves. We calculated the Krichevskii parameter for the dilute H2O/D2O mixtures with Eq. (1), where the derivative (dPS/dT )C CXC was calculated with the vapor-pressure equation for H2O given by Levelt Sengers [41] and by Harvey and Lemmon [42] for D2O and the slope of the temperature critical line TC(x ) was calculated with equations reported by Marshall and Simonson [3]. The slope of the pressure critical line PC(x ) was calculated from linear interpolation between critical pressures for pure components. The results are: dilute solutions of H2O (as solute) in D2O (as solvent) (dPS/ 1 dT )C (Harvey and Lemmon CXC /0.266 MPa K [42], experimental values of (dPS/dT )C CXC is 0.276 MPa K 1, Polikhronidi et al. [43]; (@PC =@x)CRL 0:393 MPa mol1 ; (@TC =@x)CCRL 3:250 K/ 1 /mol (if it is assumed that TC is a linear function of x, TC(x )/ 647.14/3.25(1/x ) [3]); (dPC/ 1 dTC)C ; and the Krichevskii CRL /0.121 MPa K parameter is (@P=@x)TC VC 0:471 MPa mol1 : For dilute solutions of D2O (as solute) in H2O 1 (as solvent) (dPS/dT )C (LeCXC /0.268 MPa K velt Sengers [41]); (@PC =@x)CRL 0:393/ 1 /MPa mol ; (@TC =@x)CCRL 3:250 K mol1 ; C (dPC/dTC)CRL /0.121 MPa K 1; and the Krichevskii parameter is (@P=@x) TC VC 0:478/ 1 /MPa mol :/ Using the crossover equation of state by Kiselev et al. [5] with parameters fitted with the present PVTx measurements, the values of the Krichevskii parameter was also calculated for dilute H2O/ D2O mixtures. The values of the Krichevskii parameter calculated from this crossover equation are /0.475 MPa (if we assume that D2O is the solvent) and 0.472 MPa (is we assume that H2O is the solvent). The absolute values of the Krichevskii parameters for both cases are very close but the sign is opposite. Good agreement (within 0.84%) is found between the values of the Krichevskii parameters calculated from the critical locus and vapor-pressure data (Eq. (1)) and the values calculated from the crossover model [5]. The values of the Krichevskii parameter reported by
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Abdulkadirova et al. [1] for H2O/D2O mixtures (H2O as a solvent) are: (@P=@x) TC VC 0:468 MPa (if we assume that TC is linear function of x ) and (@P=@x) TC VC 0:428 MPa (if taking into account the isotope-exchange reaction effect on the TC(x ) curve, see Marshall and Simonson [3]). The agreement between our results and the values reported by Abdulkadirova et al. [1] is reasonably good (deviation is /0.85%). A new value (0.478 MPa) of the Krichevskii parameter was calculated by Japas et al. [44] for the H2O/D2O mixture by using Krichevskii relations (1) and (2) and assuming linearity in the critical line. A value of the Krichevskii parameter was also estimated by these authors by using the distribution coefficient of solute (D2O) at infinite dilution between vapor and liquid solvent, RT In KD //2J/r2C1(r (l )/ rC1). The limiting slope of RT In KD as a function of the liquid phase density as the system approaches the critical point is given by 2(@P=@x) TC VC =rC1 : The value of the Krichevskii parameter estimated in this way is 0.466 MPa. Therefore, both values of the Krichevskii parameter reported by Japas et al. [44] agree closely with the present result (0.478 MPa). Plyasunov et al. [45] reported an estimate of the Krichevskii parameter for 30 aqueous nonelectrolytes from various types of experimental information on the thermodynamic properties. They estimated the value of the Krichevskii parameter for the H2O/D2O mixture to be 11 MPa by using a linear correlation equation (@P=@x) TC VC 88:7 4:11Dh G0 ; where DhG0 //18.9 kJ mol 1 is the Gibbs energy of hydration of D2O. It is not surprising that the agreement is not good because the authors claimed their correlation would predict the Krichevskii parameter for aqueous nonelectrolytes, except H2O/D2O mixture. Fig. 6 shows the pressure P as a function of concentration x calculated from the crossover equation of state of Kiselev et al. [5] and the result of the present work at the critical isotherm / isochore of pure light water. The slope of this experimental P/x curve for dilute H2O/D2O mixtures (x 0/0) defines the value of the Krichevskii parameter (@P=@x) TC VC which is determined to be 0.472 MPa. Debenedetti and Mohamed [40] introduced a new parameter d which is related to
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Fig. 6. Pressures of the H2O/D2O mixtures as a function of concentration along the pure light water’s critical isotherm /isochore.
the Krichevskii function J by d /J/KT1. In the immediate vicinity of the pure solvent critical point, d is the Krichevskii parameter because KT1 0/0. Therefore, near the critical point of pure light water d / 0.47 MPa and the values of rkT /d and (@P=@x) TC VC 0: According to Debenedetti and Mohamed [40] classification, this means that the H2O/D2O mixture near the critical point of pure H2O belongs to the weakly attractive systems (the partial molar volumes is positive, V¯ 2 0) while the D2O/H2O mixture near the critical point of pure D2O belongs to attractive systems (the partial molar volume is negative, V¯ 2 B0; see below Fig. 7). The partial molar volumes V¯ for dilute 2 mixtures can be calculated using the Krichevskii function (@P=@x) TV as [22,25]
@P 1 r 1 ; K V¯ T 2 @x TV
(3)
where KT is the compressibility of pure solvent and r is the density of the pure solvent. In the vicinity of the solvent’s (pure H2O or D2O) critical point T 0/TC, KT 8/(T/TC) g 0//, therefore, the partial molar volume V¯ 2 /becomes infinite [46]. The sign of the divergence depends on values of (@P=@x) TV (KT /0 and values of KT (@P=@x)TV 1 near the critical point). Therefore, the infinitely dilute partial molar volume of D2O in H2O diverges to / near the critical point of H2O, and H2O in D2O diverges to /. The mechanism that induces these infinite values has been explained in previous work [22,23]. This anomaly is caused by the critical effects due to the divergence
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Fig. 7. Partial molar volumes of H2O/D2O dilute mixtures from Eq. (3), by use of the crossover equation of state by Kiselev et al. [5] along the various supercritical isotherms.
of the isothermal compressibility KT of the pure solvent and is common to all dilute near-critical mixtures [22,23]. Fig. 7 shows the behavior of partial molar volume of infinitely dilute mixtures of H2O/D2O near the critical point of the pure H2O along various near-critical and supercritical isotherms, calculated with Eq. (3) by using the crossover equation of state of Kiselev et al. [5]. Recently several correlating equations for the infinite dilution partial molar volume of aqueous solutions near the critical point were proposed in Refs. [47,48]. The approach is fundamentally based on the parameter A12 V¯ 2 =KT RT which is related to the infinite dilute solute/water direct correlation function integral C12, by means of A12 /1/C12. Using experimental values of molar volumes Vm(P ,T,x ) for the 0.5H2O/0.5D2O mixture and
the corresponding values for the pure components Vm(P ,T ,1) and Vm(P ,T ,0), the values of the excess molar volumes VEm(P ,T ,x ) are derived. Fig. 8 show the results for VEm(P ,T ,x) at two supercritical isotherms 660 and 680 K as a function of pressure. Fig. 8 shows that near the critical pressure, experimental values of VEm(P ,T ,x )show a weak positive maximum for the isotherm 660 K and a large negative anomaly for the isotherm 680 K.
4. Conclusion By means of a constant-volume piezometer, PVTx properties for H2O/D2O mixtures, with compositions of 0.5 and 0.6 mol fraction of H2O, have been measured in a range of temperatures from 517 to 680 K, pressures from 3 to 38 MPa,
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Fig. 8. Excess molar volumes of the 0.5H2O/0.5D2O mixture as a function of pressure along the supercritical isotherms.
and densities from 97 to 466 kg m 3 with an estimated uncertainty of 0.05 /0.10%. The measured data show good agreement within 9/1% with calculations with crossover equations of state (AAD /0.2%). The present PVTx data together with isochoric heat capacity data reported in our companion paper by Polikhronidi et al. [4] were used to test the accuracy of the prediction of new crossover equations of state for H2O/D2O mixtures in a broad range around the critical locus. The accuracy of the method was confirmed by measurements of the PVT properties of pure water in the critical and supercritical regions. Values of the Krichevskii parameter were estimated from the crossover equation of state by Kiselev et al. [5] for dilute H2O/D2O mixtures, and from the critical locus TC(x ), PC(x ) data for the mixture, and the vapor-pressure data for pure H2O or D2O. According to the Debenedetti and Mohamed [40] classification of near-critical dilute mixtures, the
H2O/D2O mixture near the critical point of pure H2O belongs to systems of weakly attractive mixtures, while the D2O/H2O mixture near the critical point of pure D2O belongs to systems of attractive mixtures. The values of partial molar volumes for dilute H2O/D2O mixtures near the pure H2O critical point were calculated using a parametric crossover equation of state. The anomaly of the pressure dependence of the excess molar volume VEm(P ,T ,x) along supercritical isotherms was observed.
Acknowledgements The authors thank Prof. J.V. Sengers and Prof. S.B. Kiselev for providing programs to calculate the thermodynamic properties of H2O/D2O mixtures from their crossover equations of state and their interest in this work.
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References [1] Kh.S. Abdulkadirova, A. Kostrowicka Wyczalkowska, M.A. Anisimov, J.V. Sengers, Thermodynamic properties of mixtures of H2O and D2O in the critical region, J. Chem. Phys. 116 (2002) 4597. [2] J.M. Simonson, The enthalpy of the isotope-exchange reaction: D2O/H2O/2HDO at temperatures to 673 K and at pressures to 40 MPa, J. Chem. Thermodyn. 23 (1991) 613. [3] W.L. Marshall, J.M. Simonson, The (liquid/vapor) critical temperature curve for {x D2O/(1/x )H2O}, J. Chem. Thermodyn. 23 (1991) 613. [4] N.G. Polikhronidi, I.M. Abdulagatov, J.W. Magee, G.V. Stepanov, Isochoric heat capacity measurements for equimolar light and heavy water mixtures (50 mol% H2O/50 mol% D2O) in the near-critical and supercritical regions, Int. J. Thermophys. (2003) in press. [5] S.B. Kiselev, I.M. Abdulagatov, A.H. Harvey, Equation of state and thermodynamic properties of pure D2O and D2O/H2O mixtures in and beyond the critical region, Int. J. Thermophys. 20 (1999) 563. [6] A. Kostrowicka Wyczalkowska, Kh.S. Abdulkadirova, M.A. Anisimov, J.V. Sengers, Thermodynamic properties of H2O and D2O in the critical region, J. Chem. Phys. 113 (2000) 4985. [7] I.M. Abdulagatov, A.R. Bazaev, A.E. Ramazanova, PVTx measurements of aqueous mixtures at supercritical conditions, Int. J. Thermophys. 14 (1993) 231. [8] I.M. Abdulagatov, A.R. Bazaev, R.K. Gasanov, A.E. Ramazanova, Measurements of the PVTx properties of n heptane in supercritical water, J. Supercritical Fluids 10 (1997) 149. [9] I.M. Abdulagatov, A.R. Bazaev, E.A. Bazaev, M.B. Saidakhmedova, A.E. Ramazanova, Volumetric properties of near-critical and supercritical water/pentane mixtures: molar, excess, partial, and apparent volumes, J. Chem. Eng. Data 43 (1998) 451. [10] I.M. Abdulagatov, A.R. Bazaev, E.A. Bazaev, M.B. Saidakhmedova, A.E. Ramazanova, PVTx measurements and partial molar volumes for water /hydrocarbon mixtures in the near-critical and supercritical conditions, Fluid Phase Equilibrium 150 (1998) 537. [11] I.M. Abdulagatov, E.A. Bazaev, A.R. Bazaev, M.G. Rabezkii, PVTx Measurements for dilute water/n -hexane mixtures in the near-critical and supercritical regions, J. Supercritical Fluids 19 (2001) 219. [12] M.G. Rabezkii, A.R. Bazaev, I.M. Abdulagatov, J.W. Magee, E.A. Bazaev, PVTx measurements for water/ toluene mixtures in the near-critical and supercritical regions, J. Chem. Eng. Data 46 (2001) 1610. [13] A.R. Bazaev, I.M. Abdulagatov, J.W. Magee, E.A. Bazaev, M.G. Rabezkii, PVT measurements for toluene in the near-critical and supercritical regions, J. Chem. Eng. Data 46 (2001) 1089. [14] W. Wagner, A. Pruß, The IAPWS formulation for the thermodynamic properties of ordinary water substance for
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[31] T. Furuya, A.S. Teja, Krichevskii parameters and solubility of heavy n -alkanes in supercritical carbon dioxide, Ind. Eng. Chem. Res. 39 (2000) 4828. [32] J.M.H. Levelt Sengers, in: E. Kiran, J.M.H. Levelt Sengers (Eds.), Supercritical Fluids: Fundamentals for Applications, Kluwer, Dordrecht, 1994, pp. 13 /37. [33] M.L. Japas, J.L. Alvarez, K. Gutkowski, R. Ferna´ndezPrini, Determination of the Krichevskii function in the near-critical dilute solutions of I2(s) and CHI3(s), J. Chem. Thermodyn. 30 (1998) 1603. [34] I.R. Krichevskii, Thermodynamic of the critical phenomena in infinitely dilute binary solutions, Russ. J. Phys. Chem. 41 (1967) 1332. [35] A.M. Rozen, The unusual properties of solutions in the vicinity of the critical point of solvent, Russ. J. Phys. Chem. 50 (1976) 837. [36] R. Ferna´ndez-Prini, M.L. Japas, Critical behavior of fluid binary mixtures: intermolecular parameters and thermodynamic properties, J. Phys. Chem. 96 (1992) 5115. [37] J.P. O’Connell, Y. Hu, K.A. Marshall, Aqueous strong electrolyte solution activity coefficients and density from fluctuation solution theory, Fluid Phase Equilibrium 158 (1999) 583. [38] H.D. Cochran, L.L. Lee, D.M. Pfund, Structure and properties of supercritical mixtures from Kirkwood-Buff fluctuation theory and integral equation methods, in: E. Matteoli, G.A. Manssori (Eds.), Fluctuation Theory of Mixtures, Taylor and Francis, New York, 1990, pp. 69 / 93. [39] M.T. Gude, A.S. Teja, Near-critical phase-behavior of dilute mixtures, Mol. Phys. 81 (1994) 599. [40] P.G. Debenedetti, R.S. Mohamed, Attractive, weakly attractive, and repulsive near-critical systems, J. Chem. Phys. 90 (1989) 4528.
[41] J.M.H. Levelt Sengers, Saturation properties of ordinary water substance, in: Physical Chemistry of Aqueous Systems. Proceedings Twelfth International Conference on the Properties of Water and Steam, in: H.J. White, J.V. Sengers, D.B. Neumann, J.C. Bellows. (Eds.), New York, Begell House, 1995, p. A143. [42] A.H. Harvey, E.W. Lemmon, Correlation for the vapor pressure of heavy water from the triple point to the critical point, J. Phys. Chem. Ref. Data 31 (2002) 173. [43] N.G. Polikhronidi, I.M. Abdulagatov, J.W. Magee, G.V. Stepanov, Isochoric heat capacity measurements for light and heavy water near the critical point, Int. J. Thermophys. 22 (2001) 189. [44] M.L. Japas, R. Fernandez-Pirini, J. Horita, D.J. Wesolowski, Fractioning of isotopic species between coexisting liquid and vapor water: complete temperature range, including the asymptotic critical behavior, J. Phys. Chem. 99 (1995) 5171. [45] A.V. Plyasunov, E.L. Shock, Estimation of the Krichevskii parameter for aqueous nonelectrolytes, J. Supercritical Fluids 20 (2001) 91. [46] G. Morrison, Modelling aqueous solutions near the critical point of water, J. Solution Chem. 17 (1986) 887. [47] J. Sedlbauer, J.P. O’Connell, R.H. Wood, A new equation of state for correlation and prediction of standard molar thermodynamic properties of aqueous species at high temperatures and pressures, Chem. Geol 163 (2000) 43. [48] A.V. Plyasunov, J.P. O’Connell, R.H. Wood, Infinite dilution partial molar properties of aqueous solutions of nonelectrolytes. I. Equations for partial molar volumes at infinite dilution and standard thermodynamic functions of hydration of volatilenonelectrolytes over wide ranges of conditions, Geochim. Cosmochim. Acta 64 (2000) 495.
PVTx measurements for H2OD2O mixtures in the nearcritical and supercritical regions /
Akhmed R. Bazaev a, Ilmutdin M. Abdulagatov a,b,*, Joseph W. Magee b, Emil A. Bazaev a, Asbat E. Ramazanova a a
Institute for Geothermal Problems of the Dagestan Scientific Center of the Russian Academy of Sciences, Shamilya Str. 39, Dagestan, 367030 Makhachkala, Russia b Physical and Chemical Properties Division, National Institute of Standards and Technology, 325 Broadway, Boulder, CO 80305, USA Received 16 April 2002; received in revised form 20 September 2002; accepted 2 October 2002
Abstract The PVTx relationships of H2O/D2O mixtures were measured in the near-critical and supercritical regions. Measurements were made using a constant-volume piezometer immersed in a precision thermostat. The volume of the piezometer VPT was corrected for both temperature and pressure expansions. The uncertainty of the density measurements was estimated to be 0.05 /0.10%, depending on the experimental pressure and temperature. The uncertainty of the temperature, pressure, and composition measurements were respectively 10 mK, 0.05%, and 0.001 mol fraction. Measurements were made along various near-critical and supercritical isotherms between 517 and 680 K at pressures from 3 to 38 MPa and densities from 97 to 466 kg m 3 for two compositions, namely 0.5 and 0.6 mol fraction of H2O. The measured PVTx data for H2O/D2O mixtures were compared with values calculated from a sixterm Landau expansion and from parametric crossover equations of state. The accuracy of the method was confirmed by PVT measurements for pure water in the critical and supercritical regions. The values of partial molar volumes for dilute H2O/D2O mixtures near the critical point of pure H2O and the Krichevskii parameter were calculated using a parametric crossover equation of state and the critical locus data. Using measured values of molar volumes and corresponding values for pure components, we derived the values of excess molar volumes as a function of pressure for the supercritical isotherms. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Critical point; Crossover equation of state; Heavy water; Krichevskii parameter; Light water; Partial molar volume; Supercritical water
Contribution of the National Institute of Standards and Technology, not subject to copyright in the United State. * Corresponding author. Tel.: /1-303-497-4027; fax: /1303-497-5224. E-mail address: [email protected] (I.M. Abdulagatov).
1. Introduction Thermodynamic properties of isotopic mixtures provide insight to our understanding of the difference between isotopes and mixture critical behavior [1]. There are other theoretical problems
0896-8446/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 8 9 6 - 8 4 4 6 ( 0 2 ) 0 0 2 4 2 - 5
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connected with the behavior of isotopic mixtures, for example, how isotope-exchange reactions may affect near-critical and supercritical properties [1]. Despite their usefulness, measurements of the thermodynamic properties of H2O/D2O mixtures in the near-critical and the supercritical regions are scarce. Simonson [2] has reported the excess enthalpy HE of H2O/D2O mixtures in the temperature range from 310 to 673 K at pressures from 7 to 37 MPa with a flow calorimeter. Marshall and Simonson [3] have measured the liquid/vapor critical temperatures of H2O/D2O mixtures over the entire composition range with a precision of 9/0.1 K using an improved semi-micro phase equilibrium apparatus. The critical temperature locus TC(x ) of H2O/D2O mixtures was found to be close to a linear function of concentration x . Marshall and Simonson [3] studied the isotope-exchange reaction effect on the critical temperature locus of H2O/D2O mixtures. There were no published experimental data for either the critical pressure or the critical density. In a work in progress, Polikhronidi et al. [4] measured the isochoric heat capacity for the equimolar 0.5H2O/0.5D2O mixture in the critical and supercritical regions. They also reported the saturated properties (rS, TS, CVS) and the critical temperature and the critical density for this mixture. Crossover equations of state of the H2O/D2O mixture for the Helmholtz-energy density have been developed by Abdulkadirova et al. [1] and by Kiselev et al. [5]. Both crossover models incorporate a crossover from fluctuation-induced scaling behavior near the critical curve to regular (classical or mean-field, Landau expansion) behavior outside the critical region. A generalization of a crossover equation of state, previously developed by Kostrowicka Wyczalkowska et al. [6] for pure H2O and pure D2O, has been recently extended to the mixtures on the basis of the principle of isomorphism of the critical behavior by Abdulkadirova et al. [1]. This crossover model represents the thermodynamic properties of H2O/D2O mixtures in the same range of temperatures and densities as for the pure components, i.e., reduced susceptibility x¯ 1 B2; density r /165 kg m 3, and temperature T /630 K. An alternative parametric crossover equation was developed for the
H2O/D2O mixture by Kiselev et al. [5]. They applied the isomorphic generalization of the law of corresponding states (LCS) to the prediction of thermodynamic properties and phase behavior of H2O/D2O mixtures in a wide region around the locus of vapor /liquid critical points. This equation yields a good representation of available thermodynamic properties data in the range of temperature 0.8TC(x )B/T B/1.5TC(x ) and densities 0.35rC(x )B/r B/1.65rC(x). The predictive capabilities of the crossover models of both Abdulkadirova et al. [1] and Kiselev et al. [5] will be tested using the present experimental PVTx data for the H2O/D2O mixture. The order of the discussion is as follows. In Section 2, a brief description is given for the apparatus and calibration procedures. Section 3 starts with a discussion of experimental results for H2O/D2O mixtures and pure light water. The measurements are compared with values calculated from crossover models. Then, a discussion is given of value the Krichevskii parameter for H2O/D2O mixture and how it is estimated from the critical line and vapor-pressure data and with a crossover model. These results are compared with the data reported by other authors. Then, we discuss how the Krichevskii function is used to calculate values of the partial molar volume for an infinitely dilute H2O/D2O solution. Finally, a discussion is given about how the values of the excess molar volumes are calculated from PVTx for the H2O/D2O mixture and the pure components.
2. Experimental procedure A detailed description of the apparatus and the experimental procedure has been given previously [7 /13]. Only essential information will be given here. The measurements were made using the constant-volume method, with an extraction of the sample from the piezometer under isothermal conditions. The high-pressure piezometer is constructed of heat- and corrosion-resistant highstrength alloy EI-43BU-VD and has a volume of 32.689/0.01 cm3 at a temperature of 673.15 K and a pressure of 40.32 MPa. It was calibrated by
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filling it with pure water and then withdrawing the water in small amounts and weighing them. The weight of the water withdrawn yielded the volume of the piezometer from the well-established density of water [14] at the temperature and pressure of the calibration. The fluid under study was thermostated in a double-wall air bath. Stirring of the mixture was accomplished with the aid of a steel ball that was moved by the oscillation of the thermostat. The fluid temperature was measured with a 10 V platinum resistance thermometer (PRT-10). This PRT was calibrated by VNIIFTRI (Moscow) on ITS-90. The maximum uncertainty in the measured temperature was 10 mK. The temperature inside the thermostat was maintained uniform within 5 mK with the aid of guard heaters located between the thermostat walls and regulating heaters, which were mounted inside the thermostat. The temperature inside the thermostat and the fluid temperature were controlled automatically [15]. The pressure was measured by a dead-weight oil gauge with an estimated uncertainty of 0.05%. Corrections were made for the volume of the stainless steel stirrer and the variation of the volume with temperature and pressure. The thermostat has double-walls with an inside volume of 65 dm3. The heating elements were arranged between the walls. To minimize temperature gradients in the air thermostat, two electrically driven high-speed fans were used. The inner volume of the piezometer was calculated by taking into consideration the corrections of the elastic pressure deformation and thermal expansion. The volume of the piezometer was previously calibrated from the known density of a standard fluid (pure water) with well-established PVT information (IAPWS-95 formulation, Wagner and Pruß [14]) at a temperature of T0 /673.15 K and a pressure of P0 /38.54 MPa. The volume at these conditions was VP0T0 /(32.2109/0.003) cm3. The volume of the piezometer was measured with an uncertainty of 0.01 /0.03%. The uncertainty of the density calculations from the IAPWS-95 formulation is 0.1%. The present experimental apparatus had no dead volumes. All masses were determined with an uncertainty of 0.05 mg. The composition of the reported values are accurate to 0.001 mol
117
fraction. Taking into account the uncertainties of measurements of temperature, pressure, and concentration, the total experimental uncertainty of density was estimated to be 0.05 /0.10%, depending on the temperature T and pressure P . The commercial supplier of the D2O provided a purity analysis of 99.9 mol%.
3. Results and discussion Measurements of the PVTx relationships for H2O/D2O mixtures (x /0.5 and 0.6 mol fraction of H2O) were performed along five near-critical and supercritical isotherms, namely: 646, 648, 660, 670, and 680 K. The density ranged between 97 and 466 kg m 3 and the pressure ranged from 3 to 38 MPa. Measurements for a composition of 0.6 mol fraction of H2O were made along two quasiisochores 438 and 439 kg m3 in the temperature range from 520 to 680 K. For the equimolar H2O/D2O mixture, PVTx was also measured along two quasi-isochores 463 and 465 kg m 3 for temperatures between 517 and 645 K. The experimental temperatures, densities, and pressures for near-critical and supercritical H2O/D2O mixtures are presented in Table 1 and in Figs. 1/4. The critical parameters for measured compositions are: x/0.5 mol fraction of H2O: TC /645.4715 K, PC /21.868 MPa, and rC /339.0 kg m 3; and for x/0.6 mol fraction H2O: TC /645.7964 K, PC / 21.907 MPa, and rC /335.6 kg m 3. To check and confirm the accuracy of the method, PVT was measured with pure water along one near-critical and one supercritical isotherm, 653.15 and 673.15 K. The measured values are presented in Table 2. The data were compared with values calculated from the IAPWS-95 formulation [14]. Deviation statistics for the 14 measured densities of pure light water are: AAD /0.44%, Bias /0.16%, S.D. /0.46%, S.E. /0.12%, and Max. Dev/0.7%. All of the data show good agreement with the existing reliable PVT data sets for light water [16 /20]. The experimental isothermal PVT data measured for the equimolar H2O/D2O mixture are plotted in Figs. 1 and 2 together with values calculated from the crossover equations of state
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Table 1 Experimental values of the PVTx properties of light and heavy water mixtures along the near-critical and supercritical isotherms 0.5H2O/0.5D2O T (K)
r (kg m
3
)
P (MPa)
646 463.138 281.403 193.293 150.175 98.001
22.459 21.923 21.430 20.323 17.177
463.087 373.567 281.375 193.275 150.161 97.993
23.243 22.648 22.453 21.771 20.601 17.323
462.781 373.335 281.210 193.168 150.081 97.943
27.921 26.322 25.311 23.784 22.121 18.216
462.523 373.139 281.073 193.080 150.015 97.902
32.018 29.407 27.697 25.447 23.343 18.941
462.266 372.949 280.936 192.992 149.950 97.861
36.038 32.514 30.024 26.992 24.512 19.651
517 520 550 580 620 643 645
465.846 465.787 465.195 464.582 463.731 463.212 463.163
3.578 3.766 6.138 9.520 16.015 21.418 22.049
0.6H2O/0.4D2O 520.0 550.0 580.0 620.0 630.0
441.709 441.148 440.566 439.761 439.550
3.702 5.984 9.454 15.803 17.986
Table 1 (Continued ) 0.6H2O/0.4D2O T (K)
r (kg m 3)
P (MPa)
643.0 645.0 645.5 646.0 648.0 650.0 660.0 670.0 680.0
439.269 439.220 439.208 439.195 439.145 439.095 438.838 438.593 438.343
21.188 21.989 22.219 22.489 23.360 24.220 28.924 33.076 37.489
648
660
670
680
[1,5]. Fig. 1 also compares the present measurements and the values calculated from the parametric crossover model by Kiselev et al. [5] with parameters calculated by fitting the present data and by using LCS (without using present data). As one can see from Fig. 1, agreement of the fitted model is excellent (pressure AAD /0.2%). A detailed comparison between present data and the data calculated from a six-term Landauexpansion crossover model by Abdulkadirova et al. [1] is depicted in Fig. 2 together with values calculated from an alternative parametric crossover model by Kiselev et al. [5]. Present data were not used to fit the parameters of this model. Both crossover models represent the pressure data shown in Fig. 2 with comparable accuracy (S.D. of 0.2%). The pressure dependence of the compressibility factor, Z /PVm/RT, for the 0.5H2O/0.5D2O mixture along a supercritical isotherm 660 K is shown in Fig. 3. Fig. 3 shows that there is good internal consistency between measurements of temperature T , pressure P , and specific volume V for each thermodynamic state (PVT ) and the results of the prediction of the crossover equation [5]. The dependence of pressure on density for the 0.5H2O/0.5D2O mixture and for both pure components are shown in Fig. 4 together with values calculated from crossover equations [5,21] and literature data for pure H2O and D2O along the near-critical isotherm 648 K. It is remarkable that the mixture has a very broad flat region like pure components because this isotherm (648 K) is very close to the critical isotherm of the mixture 645.5 K and to the critical isotherms of the pure
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Fig. 1. Measured pressures of 0.5H2O/0.5D2O mixtures as a function of density along different sub-critical and supercritical isotherms: (k) present data; ( */) from crossover equation of state by Kiselev et al. [5] (which fit our present data); ( / / / /), from crossover equation of state by Kiselev et al. [5] (LCS); (w) the critical point of mixture.
Fig. 2. Measured pressures of 0.5H2O/0.5D2O mixtures as a function of density along different near-critical and supercritical isotherms: (k) 646 K; (j) 648 K; (D) 660 K; (%) 670 K; (’) 680 K; ( */), from the six-term Landau-expansion crossover equation of state by Abdulkadirova et al. [1] (without using present data); ( / / / /), from the crossover equation of state by Kiselev et al. [5] (LCS, without using present data).
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Fig. 3. Compressibility factors, Z /PVm/RT , of the 0.5H2O/0.5D2O mixture and their pure components as a function of pressure along the supercritical isotherm 660 K.
components 647.096 K (for light water) and 643.847 K (for heavy water). Fig. 5 depicts deviations of the experimental pressure from the values calculated using the parametric crossover equation of state developed by Kiselev et al. [5] (LCS model). This figure does not include one data point for which the deviation is larger than 1% (1.1%). As one can see from Fig. 5, most of the measured pressure data show good agreement to within 9/1.0% with values calculated from the crossover model by Kiselev et al. [5] with parameters determined from LCS. Most of the data at densities higher than 150 kg m 3 show deviations within 9/0.5%. Only the data in the low density range r B/100 kg m 3 exhibit deviations larger than 0.5%. Most of the data at high densities show negative deviations (experimental values slightly lower than calculated values). It should be emphasized that the current data were not used to fit the parameters of the LCS crossover model by Kiselev et al. [5]. The deviation statistics for the 30 measured pressures are (Kiselev et al. [5] LCS model): AAD/0.35%, Bias //0.13%, S.D. /
0.4%, S.E. /0.08%, and Max. Dev /1.1%. The maximum deviation (1.1%) was observed at the temperature 680 K and at a density of 97.86 kg m 3. The AAD for the optimized crossover model by Kiselev et al. [5], with parameters determined from the present measurements, is 0.2% (Fig. 1 solid curves). Several aspects of dilute mixture behavior are of considerable theoretical interest. For example, there is strong interest in negatively or positively diverging solute partial molar volume in mixtures in the immediate vicinity of the solvent’s critical point and path-dependence solvent properties in near-critical systems [22 /24]. The thermodynamic behavior of dilute near-critical mixtures depends on microscopic phenomena involving density perturbations induced by the presence of the solute and propagation of this density perturbation to a distance given by the pure solvent’s correlation length, for which j2 is proportional to the isothermal compressibility KT of the pure solvent [25 /27]. The thermodynamics of high-temperature aqueous solutions is a subject of wide-ranging
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121
Fig. 4. Pressures of the 0.5H2O/0.5D2O mixture and their pure components H2O and D2O as a function of density along the supercritical isotherm 648 K. Table 2 Experimental values of the PVT of pure water T / 653.15 K
T/ 673.15 K
P (MPa)
r (kg m 3)
P (MPa)
r (kg m 3)
29.25 25.18 23.48 22.99 20.25 16.61 12.58
527.79 460.42 290.71 205.73 125.34 82.440 54.390
40.30 33.87 28.65 26.53 22.32 17.90 13.32
527.13 459.89 290.42 205.24 125.59 82.680 54.340
interest because of the dominant role played by coexistence of both short-ranged (solvation) and long-ranged (compressibility-driven) phenomena [23,25,27] in fluid systems. The thermodynamic
behavior of the near-critical dilute solutions is extremely important to better understand molecular interactions and microscopic structures of the solutions. In the limit of infinite dilution, many partial molar properties of the solute ¯ ¯ (V¯ 2 ; H 2 ; C P2 ) diverge strongly at the pure solvent’s critical point [28 /30]. In general, the thermodynamic behavior of infinitely dilute mixtures near the solvent’s critical point can be completely characterized by the so-called Krichevskii parameter, which is equal to the derivative (@P=@x) TC VC calculated at the critical point of the pure solvent [22,31]. Using the concept of the Krichevskii parameter, Levelt Sengers [32] proposed a description of thermodynamic behavior of dilute near-critical solutions based on the derivative (@P=@x) VT ; and Japas et al. [33] denoted J (@P=@x) VT as the
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122
Fig. 5. Percentage pressure deviations, dP/100((Pexp/Pcal)/Pexp) of the experimental pressures for the 0.5H2O/0.5D2O mixture from the values calculated with the crossover equation of state by Kiselev et al. [5] (LCS).
Krichevskii function. The Krichevskii function J for dilute aqueous solutions (water/n -hexane) was derived from experimental PVTx data along the measured pure solvent’s (light water) nearcritical and supercritical isotherms as a function of density in a previous paper [11]. The value of the Krichevskii function at the critical point of the pure solvent is exactly the Krichevskii parameter [34,35]. The Krichevskii function J has a simple physical meaning and a straight-connection to total correlation function integrals [25,27,36 /38]. The Krichevskii parameter also governs the critical behavior of the thermodynamic properties of dilute solutions, particularly, the parameter K1 (Abdulkadirova et al. [1]), K1
x(1 x) dPC rC RTC
dx
@P
@T
C j;rrC
dTC dx
;
which is responsible for strongly divergent properties such as the isothermal compressibility and isobaric heat capacity, and is directly related to the Krichevskii parameter. The value of the Krichevskii parameter can be estimated from the initial slopes of the critical lines data TC(x ) and PC(x ) for dilute mixtures and the values of the vaporpressure PS /TS slope of the pure solvent at the
critical point. The initial slopes of the critical lines and vapor-pressure curve are related to the Krichevskii parameter as follows [23,24,39]: C @P @PC C dPS C @x VC TC @x CRL dT CXC @TC C (1) @x CRL or, equivalently, C @P dPC C dPS C @x VC TC dTC CRL dT CXC C dTC ; dx CRL
(2)
where (@PC =@x)CRL and (@TC =@x)CCRL are the initial slopes of the PC(x) and TC(x) critical lines, and (dPS/dT )C CXC is the slope of the solvent’s vapor-pressure curve evaluated at the critical point of the solvent (always positive). Qualitatively and quantitatively, the regimes of the near-critical dilute solutions behavior depends strongly on the signs and magnitudes of the initial slopes of the critical locus TC(x ), PC(x ), and on the slope of vapor-pressure curves PS(T ) at the
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critical point of pure solvent, i.e., magnitude and sign of the Krichevskii parameter (@P=@x) TC VC [40]. Therefore, the Krichevskii parameter is a good test for the thermodynamic consistency of the models for critical and vapor-pressure curves. We calculated the Krichevskii parameter for the dilute H2O/D2O mixtures with Eq. (1), where the derivative (dPS/dT )C CXC was calculated with the vapor-pressure equation for H2O given by Levelt Sengers [41] and by Harvey and Lemmon [42] for D2O and the slope of the temperature critical line TC(x ) was calculated with equations reported by Marshall and Simonson [3]. The slope of the pressure critical line PC(x ) was calculated from linear interpolation between critical pressures for pure components. The results are: dilute solutions of H2O (as solute) in D2O (as solvent) (dPS/ 1 dT )C (Harvey and Lemmon CXC /0.266 MPa K [42], experimental values of (dPS/dT )C CXC is 0.276 MPa K 1, Polikhronidi et al. [43]; (@PC =@x)CRL 0:393 MPa mol1 ; (@TC =@x)CCRL 3:250 K/ 1 /mol (if it is assumed that TC is a linear function of x, TC(x )/ 647.14/3.25(1/x ) [3]); (dPC/ 1 dTC)C ; and the Krichevskii CRL /0.121 MPa K parameter is (@P=@x)TC VC 0:471 MPa mol1 : For dilute solutions of D2O (as solute) in H2O 1 (as solvent) (dPS/dT )C (LeCXC /0.268 MPa K velt Sengers [41]); (@PC =@x)CRL 0:393/ 1 /MPa mol ; (@TC =@x)CCRL 3:250 K mol1 ; C (dPC/dTC)CRL /0.121 MPa K 1; and the Krichevskii parameter is (@P=@x) TC VC 0:478/ 1 /MPa mol :/ Using the crossover equation of state by Kiselev et al. [5] with parameters fitted with the present PVTx measurements, the values of the Krichevskii parameter was also calculated for dilute H2O/ D2O mixtures. The values of the Krichevskii parameter calculated from this crossover equation are /0.475 MPa (if we assume that D2O is the solvent) and 0.472 MPa (is we assume that H2O is the solvent). The absolute values of the Krichevskii parameters for both cases are very close but the sign is opposite. Good agreement (within 0.84%) is found between the values of the Krichevskii parameters calculated from the critical locus and vapor-pressure data (Eq. (1)) and the values calculated from the crossover model [5]. The values of the Krichevskii parameter reported by
123
Abdulkadirova et al. [1] for H2O/D2O mixtures (H2O as a solvent) are: (@P=@x) TC VC 0:468 MPa (if we assume that TC is linear function of x ) and (@P=@x) TC VC 0:428 MPa (if taking into account the isotope-exchange reaction effect on the TC(x ) curve, see Marshall and Simonson [3]). The agreement between our results and the values reported by Abdulkadirova et al. [1] is reasonably good (deviation is /0.85%). A new value (0.478 MPa) of the Krichevskii parameter was calculated by Japas et al. [44] for the H2O/D2O mixture by using Krichevskii relations (1) and (2) and assuming linearity in the critical line. A value of the Krichevskii parameter was also estimated by these authors by using the distribution coefficient of solute (D2O) at infinite dilution between vapor and liquid solvent, RT In KD //2J/r2C1(r (l )/ rC1). The limiting slope of RT In KD as a function of the liquid phase density as the system approaches the critical point is given by 2(@P=@x) TC VC =rC1 : The value of the Krichevskii parameter estimated in this way is 0.466 MPa. Therefore, both values of the Krichevskii parameter reported by Japas et al. [44] agree closely with the present result (0.478 MPa). Plyasunov et al. [45] reported an estimate of the Krichevskii parameter for 30 aqueous nonelectrolytes from various types of experimental information on the thermodynamic properties. They estimated the value of the Krichevskii parameter for the H2O/D2O mixture to be 11 MPa by using a linear correlation equation (@P=@x) TC VC 88:7 4:11Dh G0 ; where DhG0 //18.9 kJ mol 1 is the Gibbs energy of hydration of D2O. It is not surprising that the agreement is not good because the authors claimed their correlation would predict the Krichevskii parameter for aqueous nonelectrolytes, except H2O/D2O mixture. Fig. 6 shows the pressure P as a function of concentration x calculated from the crossover equation of state of Kiselev et al. [5] and the result of the present work at the critical isotherm / isochore of pure light water. The slope of this experimental P/x curve for dilute H2O/D2O mixtures (x 0/0) defines the value of the Krichevskii parameter (@P=@x) TC VC which is determined to be 0.472 MPa. Debenedetti and Mohamed [40] introduced a new parameter d which is related to
124
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Fig. 6. Pressures of the H2O/D2O mixtures as a function of concentration along the pure light water’s critical isotherm /isochore.
the Krichevskii function J by d /J/KT1. In the immediate vicinity of the pure solvent critical point, d is the Krichevskii parameter because KT1 0/0. Therefore, near the critical point of pure light water d / 0.47 MPa and the values of rkT /d and (@P=@x) TC VC 0: According to Debenedetti and Mohamed [40] classification, this means that the H2O/D2O mixture near the critical point of pure H2O belongs to the weakly attractive systems (the partial molar volumes is positive, V¯ 2 0) while the D2O/H2O mixture near the critical point of pure D2O belongs to attractive systems (the partial molar volume is negative, V¯ 2 B0; see below Fig. 7). The partial molar volumes V¯ for dilute 2 mixtures can be calculated using the Krichevskii function (@P=@x) TV as [22,25]
@P 1 r 1 ; K V¯ T 2 @x TV
(3)
where KT is the compressibility of pure solvent and r is the density of the pure solvent. In the vicinity of the solvent’s (pure H2O or D2O) critical point T 0/TC, KT 8/(T/TC) g 0//, therefore, the partial molar volume V¯ 2 /becomes infinite [46]. The sign of the divergence depends on values of (@P=@x) TV (KT /0 and values of KT (@P=@x)TV 1 near the critical point). Therefore, the infinitely dilute partial molar volume of D2O in H2O diverges to / near the critical point of H2O, and H2O in D2O diverges to /. The mechanism that induces these infinite values has been explained in previous work [22,23]. This anomaly is caused by the critical effects due to the divergence
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125
Fig. 7. Partial molar volumes of H2O/D2O dilute mixtures from Eq. (3), by use of the crossover equation of state by Kiselev et al. [5] along the various supercritical isotherms.
of the isothermal compressibility KT of the pure solvent and is common to all dilute near-critical mixtures [22,23]. Fig. 7 shows the behavior of partial molar volume of infinitely dilute mixtures of H2O/D2O near the critical point of the pure H2O along various near-critical and supercritical isotherms, calculated with Eq. (3) by using the crossover equation of state of Kiselev et al. [5]. Recently several correlating equations for the infinite dilution partial molar volume of aqueous solutions near the critical point were proposed in Refs. [47,48]. The approach is fundamentally based on the parameter A12 V¯ 2 =KT RT which is related to the infinite dilute solute/water direct correlation function integral C12, by means of A12 /1/C12. Using experimental values of molar volumes Vm(P ,T,x ) for the 0.5H2O/0.5D2O mixture and
the corresponding values for the pure components Vm(P ,T ,1) and Vm(P ,T ,0), the values of the excess molar volumes VEm(P ,T ,x ) are derived. Fig. 8 show the results for VEm(P ,T ,x) at two supercritical isotherms 660 and 680 K as a function of pressure. Fig. 8 shows that near the critical pressure, experimental values of VEm(P ,T ,x )show a weak positive maximum for the isotherm 660 K and a large negative anomaly for the isotherm 680 K.
4. Conclusion By means of a constant-volume piezometer, PVTx properties for H2O/D2O mixtures, with compositions of 0.5 and 0.6 mol fraction of H2O, have been measured in a range of temperatures from 517 to 680 K, pressures from 3 to 38 MPa,
126
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Fig. 8. Excess molar volumes of the 0.5H2O/0.5D2O mixture as a function of pressure along the supercritical isotherms.
and densities from 97 to 466 kg m 3 with an estimated uncertainty of 0.05 /0.10%. The measured data show good agreement within 9/1% with calculations with crossover equations of state (AAD /0.2%). The present PVTx data together with isochoric heat capacity data reported in our companion paper by Polikhronidi et al. [4] were used to test the accuracy of the prediction of new crossover equations of state for H2O/D2O mixtures in a broad range around the critical locus. The accuracy of the method was confirmed by measurements of the PVT properties of pure water in the critical and supercritical regions. Values of the Krichevskii parameter were estimated from the crossover equation of state by Kiselev et al. [5] for dilute H2O/D2O mixtures, and from the critical locus TC(x ), PC(x ) data for the mixture, and the vapor-pressure data for pure H2O or D2O. According to the Debenedetti and Mohamed [40] classification of near-critical dilute mixtures, the
H2O/D2O mixture near the critical point of pure H2O belongs to systems of weakly attractive mixtures, while the D2O/H2O mixture near the critical point of pure D2O belongs to systems of attractive mixtures. The values of partial molar volumes for dilute H2O/D2O mixtures near the pure H2O critical point were calculated using a parametric crossover equation of state. The anomaly of the pressure dependence of the excess molar volume VEm(P ,T ,x) along supercritical isotherms was observed.
Acknowledgements The authors thank Prof. J.V. Sengers and Prof. S.B. Kiselev for providing programs to calculate the thermodynamic properties of H2O/D2O mixtures from their crossover equations of state and their interest in this work.
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