5982
J. Phys. Chem. 1996, 100, 5982-5992
Mass Diffusion Coefficients and Thermal Diffusivity in Concentrated Hydrothermal NaNO3 Solutions Thomas J. Butenhoff, Marcel G. E. Goemans,† and Steven J. Buelow* Chemical Science and Technology DiVision, CST-6, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 ReceiVed: October 6, 1995; In Final Form: January 24, 1996X
Binary mass diffusion coefficients and thermal diffusivities for hydrothermal sodium nitrate solutions as a function of pressure (270 < P < 1000 bar), temperature (400 < T < 500 °C), and concentration (0.25 < C < 3.0 m) were measured by the laser-induced grating technique. In concentrated hydrothermal NaNO3 systems, the critical slowing down was significant as far as 300 bar from the phase-separation pressure, resulting in binary diffusion coefficients near the critical point that are comparable to values at ambient conditions. Further from the critical point the mass diffusion coefficients plateaued at their ordinary values. Ordinary binary mass diffusion was about 15 times faster than at 25 °C and atmospheric pressure. The ordinary binary mass diffusion coefficients were compared with predictions from hydrodynamic diffusion theory. Experimental results agreed well with predictions from the Stokes-Einstein equation, where the diffusing species was best represented by a hydrated contact ion pair. The Wilke-Chang correlation also yielded good predictions when the solute molar volume was defined as the volume of the hydrated contact ion pair. Predictions can be improved by about 10% if the degree of association can be calculated. Thermal diffusion coefficients (Soret effect) at 450 °C were also measured and are about 250 times faster than at ambient conditions. The laser-induced grating technique was found to be highly complementary to the Taylor dispersion technique for diffusion measurements in hydrothermal systems. The grating technique works best at higher concentrations and near critical points, the two regimes where the Taylor dispersion technique becomes increasingly more difficult.
I. Introduction Although water is by far the most studied solvent, physical transport phenomena in water at elevated temperatures and pressuressand their impact on homogeneous and heterogeneous reactions, corrosion, heat exchange, and (reactive) extractionsare not adequately understood. This situation is largely due to the limited data that are available for diffusion in high-temperature, high-pressure water mixtures. Only the molecular diffusion of iodide ions and hydroquinone in near-critical subcritical water1 and the self-diffusion coefficient of compressed supercritical water2 have been reported. The need for additional data and models to predict diffusion in near-critical and supercritical water is growing as new applications of hydrothermal and highpressure chemistry are being developed. Water at elevated temperatures and pressures is a unique solvent for conducting chemical reactions and separations. Near its critical point, the viscosity of water decreases significantly from its value at ambient conditions and the salt solubility decreases but remains high, while many organic molecules and permanent gases become miscible. This combination of mutual solubility, low viscosity, and high solvent density provides the opportunity for both rapid chemical reactions and efficient mass and heat transport. For this reason, many chemists and chemical engineers are joining geochemists in exploring reactions in hightemperature, high-pressure water. The synthesis of organic and inorganic compounds,3,4 filtration and product recovery,5 the extraction and upgrading of coal,6-8 and the destruction of hazardous chemical wastes9-12 under hydrothermal conditions are all current areas of activity. This study was conducted in † Also Environmental and Water Resources Engineering Program, Department of Civil Engineering, The University of Texas at Austin, Austin, TX 78712. X Abstract published in AdVance ACS Abstracts, March 15, 1996.
0022-3654/96/20100-5982$12.00/0
support of ongoing research on hydrothermal applications for the treatment of mixed and hazardous wastes generated at the Department of Energy site in Hanford, Washington, and Los Alamos National Laboratory. Important to the elucidation of the chemistry and to the modeling of hydrothermal systems is an understanding of the mass and heat transport phenomena in near-critical and supercritical water mixtures. In general, data are limited for diffusion in fluids near and above their critical point. During the past decade measurements of mass diffusion coefficients have been reported for several nonaqueous solvents near their critical points. Diffusion coefficients reported in the literature for organic molecules in supercritical carbon dioxide typically range from 5 × 10-9 to 2.5 × 10-8 m2 s-1.13-21 Similar results have been reported for other supercritical solvents.13,18,22-24 An extensive reference list for diffusion in near-critical solvents is given by Catchpole and King.20 For aqueous systems, the molecular diffusion of iodide ions and hydroquinone in nearcritical subcritical water was reported to be 4.9 × 10-8 and 1.9 × 10-8 m2 s-1, respectively,1 and the self-diffusion coefficient of compressed supercritical water was reported to range from 4.71 × 10-7 m2 s-1 (T ) 700 °C, P ) 397 bar) to 4.74 × 10-8 m2 s-1 (T ) 400 °C, P ) 1056 bar).2 Compared to molecular mass diffusion coefficients in liquids at ambient temperature and pressureswhich are typically on the order of 10-9 m2 s-1sand gas phase molecular diffusion coefficientsswhich are on the order of 10-4-10-5 m2 s-1sall reported diffusion coefficients in supercritical fluids are more liquidlike than they are gaslike. In this paper we report binary mass diffusion coefficients and thermal diffusivities of a 1-1 electrolyte solution, sodium nitrate, as a function of pressure, temperature, and concentration above the critical point of water and compare them to predictions © 1996 American Chemical Society
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J. Phys. Chem., Vol. 100, No. 14, 1996 5983
of hydrodynamic diffusion theory. Based on the above discussion, nitrate diffusion in supercritical water was expected to behave more liquidlike than gaslike, even more so as the solvent density and viscosity increased. II. Experimental Section A. Selection of the Experimental Technique. The Taylor dispersion method has become the most popular method for determining molecular diffusion coefficients due to its versatility and experimental simplicity.25-27 However, the Taylor dispersion technique is most suited for diffusion at infinite dilution for several reasons: (i) this technique assumes concentrationindependent diffusion, (ii) concentrated conditions would result in wall adsorption leading to asymmetric dispersion profiles or wall reactions leading to erroneous molecular diffusion measurements, and (iii) since the concentration difference between the solvent and solute needs to be minimized,25,28 the signalto-background ratio becomes a problem at concentrated conditions. As the Taylor dispersion technique will be limited to infinite dilution for most hydrothermal systems, the critical slowing down of the diffusion coefficient will not be observed.29,30 In addition, accurate measurements in the vicinity of the phase separation point are hard to obtain due to temperature control limitations.28 Research is currently underway at Los Alamos National Laboratory that employs the Taylor dispersion method to determine the diffusion coefficient at infinite dilution for a range of organic and inorganic species in sub- and supercritical water.31 The mass diffusion and thermal diffusivity measurements in this work were made via the laser-induced grating method (also called transient grating or forced Rayleigh scattering).32 This method has previously been used to measure mass diffusion rates of species that were difficult to measure by traditional methods such as transient radicals in solutions33 and excitedstate species34 as well as thermal diffusivities of reactive and corrosive liquids such as molten salts.35 Recently, it has been shown that the speed of sound and the thermal and mass diffusivities of supercritical fluids can be measured with the transient grating technique.36,37 The laser-induced grating technique is highly complementary to the Taylor dispersion method for several reasons: (i) the laser-induced grating technique performs optimally at higher solute concentration; (ii) due to the experimental layout, fluid temperature control is more accurate during a laser-induced grating experiment, allowing for accurate measurements near the plait point; and (iii) the technique performs optimally where the Soret effect is at a maximum, which occurs in the vicinity of the plait point. B. Laser-Induced Grating Method. The laser-induced grating method can be described as follows:32 Two heating laser pulses of the same wavelength λh are crossed at an angle θ in the sample; the interference of the two beams causes a sinusoidally modulated intensity in the overlap region with a fringe spacing Λ given by
Λ ) λh/[2 sin(θ/2)]
(1)
The sample absorbs the pump light, and the excited molecules quickly relax and deposit their energy to the bath molecules, which produces a spatial modulation in the temperature of the sample. This temperature modulation may produce a small concentration modulation due to the Soret effect. Both the temperature and the concentration modulation contribute to form a spatial modulation of the index of refraction in the sample. This is a volume diffraction grating and is probed by Bragg diffraction of a third laser beam with a nonabsorbed wavelength
λp. Thermal and mass diffusion causes the grating to relax with time, and the thermal and mass diffusivity can be inferred from a measurement of the diffracted signal intensity as a function of the heating pulse-probe delay time. The advantages of using the laser-induced grating technique to measure diffusivities in hydrothermal solutions are38 (i) it is a noninvasive contact-free method, (ii) the temperature and concentration jumps in the sample are small (∆T < 0.1 K, ∆c/c < 0.05%), (iii) the distance scale of the experiment is short (≈11 µm), so wall effects are minimal, and (iv) the influence of natural convection is negligible because of the short time frame of the experiment (e5 ms). C. Experimental Apparatus. The optical cell can withstand pressures to 1000 bar and temperatures to 500 °C. This cell is constructed from a commercial 316 stainless steel high-pressure “cross” fitting (HiP Inc.), machined to accept two diamond windows. The Type II-A diamonds are anvil-type windows with flat optical faces providing an aperture of 1 mm on the inside and 3 mm on the outside. The high-pressure seal is aided by a gold gasket between the diamond and the cell wall, and a spring washer is used to keep a compression between the diamond and the cell as the apparatus is temperature cycled.39,40 The two windows are used to pass the laser beams through the cell and have a 4 mm path length between the entrance and exit faces. High-pressure fittings rated to 4000 bar at room temperature are used at all of the heated plumbing ports. Heat is supplied to the cell by four 175 W cartridge heaters embedded in a brass shell surrounding the stainless steel cell. A 10 cm section of both the inlet and outlet tubes to the cell is also heated. Zetex insulating tape (Style 550) is used for insulation. The heated section of the cell has dimensions of 2 mm i.d. and 25 cm length, yielding a volume of 0.8 cm3. The fluid temperature is measured with a K-type thermocouple that has a stainless steel sheath. The thermocouple is in direct contact with the fluid about 2 mm above the optically viewed region and measures the temperature to (2 K. The temperature of the brass block is controlled and maintained with an Omega CN9000A temperature controller. The fluid pressure is obtained with a hand-operated, single-piston pressure generator (HiP Inc.) and is measured to (0.5 bar with a transducer that has been calibrated with a Heise gauge. During a measurement, the sample pressure is typically maintained to within 0.05% of the nominal pressure at P e 800 bar and to within 0.4% at P g 900 bar. The sample solutions are made with analytical grade reagents and deionized (>18 MΩ‚cm) water. The cell and pressure generator are filled with the sample salt solution at room temperature via a standard HPLC pump. The cell and pressure generator are then closed off from the HPLC pump, and the cell is heated to the operating temperature for the experiment. During heating, the pressure of the system is maintained high enough to ensure that the salt solution remains in the single phase. The solution is then brought to the desired pressure via the pressure generator, and a transient-grating decay trace is measured. Typically, two decay trace measurements are made before a new pressure is chosen. This is repeated until measurements have been made at all of the desired pressures for the isotherm. This cycle is repeated (in a different order of pressures) until 6-12 decay traces have been measured at each desired pressure. No correlation was observed between the measured decay rates and the length of time the solution had spent in the heated cell. If the pressure was accidentally lowered below the phase-separation pressuresresulting in greatly distorted transmitted laser beamssthe cell temperature was re-
5984 J. Phys. Chem., Vol. 100, No. 14, 1996
Butenhoff et al. Ko¨hler’s41 is that in this experiment the pump pulse length is very short compared to the thermal grating decay time and is treated as a δ-function. If the grating fringe spacing (Λ) is small compared to the sample absorption length, the sample thickness, and the diameter of the heated area, then the assumption of onedimensional heat conduction is valid. Under these conditions the one-dimensional heat flow equation is38,45
∂T(x,t) ∂2T(x,t) ) Dth ∂t ∂x2
(2)
The initial condition is produced by the sample absorption of the two interfering pump laser beams and is given by
T(x,0) ) T0 + Tm[1 + cos(qx)]
Figure 1. Schematic diagram of the laser-induced grating apparatus. The removable mirror is put in place to align the beams through the alignment pinhole and is removed to make a measurement.
duced, fresh solution was introduced via the HPLC pump, and the cell was heated back to the operating temperature. Figure 1 is a schematic diagram of the experimental apparatus. The heating pulses (λh ) 354.7 nm, 6 ns pulse width, ≈0.6 cm-1 bandwidth, 10 Hz) are produced from the third harmonic of a Nd:YAG laser. The beam is spatially filtered and split into two beams of equal energy, which are focused and overlapped at a small crossing angle (0.6° < θ < 2.0°) in the sample at their beam waists (e-2 intensity radius ) 430 µm). The crossing angle is determined to (0.5% by measuring the distance between the two beams using a CCD linear array sensor at a known distance from their crossing point.38 The 355 nm pump laser is on the red edge of the nitrate absorption peak. It appears that the main absorption by the sample is due to corrosion products (most likely chromate ions). Typically, 50% of the pump beam energy was absorbed in the 4 mm path length of the sample. The probe beam is produced by a CW, 1 mW HeNe laser and is focused (e-2 intensity radius ) 280 µm) into the interaction volume of the sample at the Bragg angle. The probe beam is tipped out of the plane of the grating beams by ≈1°, which spatially separates the diffracted signal beam from the heating beams. A removable mirror mounted on a kinematic mount is used for laser beam alignment purposes. The mirror redirects the laser beams to an alignment pinhole which is placed at the same distance from the mirror as the cell is. This greatly simplifies the alignment of the laser beams and allows for an easy check of a good laser beam overlap in the sample throughout the experiments. The diffracted signal beam is spatially and spectrally filtered to reduce scattered laser light, detected by a PMT, amplified, and sent to a digital oscilloscope (LeCroy 9310, 300 MHz BW, 1 × 108 samples/s), where data averaging takes place. The experiment is run at 10 Hz, and 1000-4000 waveforms are averaged per measurement. The averaged waveform is then transferred to a computer for storage and data analysis. III. Data Analysis and Results A. Theory of Experiment. Several authors have discussed the effect of thermodiffusion on the signal produced in a laserinduced grating experiment.41-44 Here we follow the arguments of Ko¨hler41 to obtain the mathematical description of the experiment. The primary difference between this analysis and
(3)
In these equations, T is the temperature, T0 is the initial temperature, t is the time after heating, x is the spatial dimension along the grating wave vector (perpendicular to the fringes), q is the wavenumber of the interference pattern (q ) 2π/Λ), and Dth is the thermal diffusivity (Dth ) λF-1Cp-1, where λ is the thermal conductivity, F is the density, and Cp is the isobaric heat capacity). Tm is the initial spatial temperature amplitude and is given by
Tm ) (R/ρcp)Ip
(4)
where R is the optical absorption coefficient of the pump wavelength in the sample and Ip/2 is the energy density (energy/ area) of each of the pump beam pulses. The solution to eq 2 subject to the initial condition (eq 3) is
δT ≡ T(x,t) - T0 ≡ Tm[1 + cos(qx)e-ktht]
(5)
where kth ) Dthq2. The effect of the temperature grating on the solute concentration due to the Soret effect is described by the one-dimensional diffusion equation
∂2c(x,t) ∂2T(x,t) ∂c(x,t) + D c(x,t)[1 c(x,t)] (6) ) D12 T ∂t ∂x2 ∂x2 where c(x,t) is the solute mole fraction, D12 is the binary mass diffusion coefficient, and DT is the thermal diffusion coefficient. The thermal diffusion coefficient should not be confused with the thermal diffusivity. The thermal diffusivity has units of a diffusion coefficient (m2 s-1) and governs the dynamics of heat dissipation. The thermal diffusion coefficient has units of m2 s-1 K-1 and is a measure of the strength of the driving force due to a temperature gradient on the mass flux; DT is equal to the Soret coefficient times the mass diffusion coefficient and, as written, is positive if the solute migration due to thermodiffusion is from regions of high temperature to low temperature. The spatial and temporal evolution of the concentration grating is calculated by substituting T(x,t) from eq 5 into eq 6. For dilute solutions and small concentration changes, c(1 - c) ) c0, and one finds
δc ≡ c(x,t) - c0 DT ) - Tm cos(qx) c [e-kmt - e-ktht] Dth - D12 0 where km ) D12q2.
(7)
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J. Phys. Chem., Vol. 100, No. 14, 1996 5985
Since the sample is transparent at the probe wavelength, there is no absorptive contribution to the signal, and only the combined phase grating is detected. For small temperature and concentration changes, the refractive grating index is given by
δn(x,t) ≡ n(x,t) - n0 )
(∂T∂n)
P0,c0
δT(x,t) +
(∂n∂c)
T0,P0
δc(x,t)
(8)
where δT and δc are given in eqs 5 and 7, and n is the index of refraction of the solution for the probe wavelength. In the limit of weak diffraction, the diffracted signal S(t) from the grating is proportional to the square of the amplitude of the refractive index modulation given in eq 8.32 The final result for the signal is
S(t) ) C[e-ktht + Ime-kmt]2
(9)
where C is a constant and Im is the ratio of the amplitudes of the concentration grating decay to the thermal grating decay and is given by
-DTc0 Im )
(∂n∂c)
T0,P0
∂n ∂n (Dth - D12) + DTc0 ∂T P0,c0 ∂c T0,P0
( )
( )
Figure 2. Diffracted light signal for the 1.0 m NaNO3 solution at 450 °C and 407.3 bar. The line in the inset semilog plot shows the best fit to eq 9 plus a base line. The grating fringe spacing is 11.15 µm.
(10)
This agrees with the result obtained by Thyagarajan and Lallemand.43 B. Sources of Errors. A study of the systematic errors inherent in laser-induced grating diffusivity measurements is performed elsewhere.38 It includes experimental deviations from the ideal conditions of the basic theory such as departures from one-dimensional heat conduction, heat losses to the walls, and using pump and probe beams with Gaussian intensity distributions instead of beams with infinitely sized, uniform intensity distributions. In the analysis, by far the largest systematic error in this experiment is due to the Gaussian laser beams and causes the measured diffusivities to be lower than the true diffusivities by ≈2%. There will be a systematic error in the diffusivity results due to the uncertainty in the measurement of the grating fringe spacing. The grating fringe spacing is measured to (0.5%, which would translate into a 1% systematic error in the measured diffusivities. This is in general smaller than the measured random errors. Corrosion products are formed throughout the course of these experiments. The concentrations of these impurities are small compared to the NaNO3 concentration, and they are not expected to affect the thermal diffusivity measurements significantly. For example, when measuring Dth of supercritical water, adding small amounts of absorbing dye did not affect the results.36 It is not known how small quantities of impurities affect the measured mass diffusion rates. No systematic effects were observed for measurements made at the beginning of a day, when the concentrations of corrosion products are at a minimum, versus the end of a day, when corrosion products are at a maximum. C. Results. Each measured signal decay trace is fit to eq 9 plus a base line to determine km, kth, and Im. Typical data and fits are shown in Figures 2 and 3. In Figure 2, the pressure is near the phase-separation pressure, the ratio of the intensities of the slow mass diffusive decay to the fast thermal diffusive decay is relatively large (Im ≈ 0.5), and the fast decay rate is 40 times larger than the slow decay rate. As can be seen in the
Figure 3. Diffracted light signal for the 1.0 m NaNO3 solution at 450 °C and 650 bar (Λ ) 11.15 µm). The line in the inset semilog plot shows the best fit to eq 9 plus a base line. Note the time scale change and the poorer S/N when compared to Figure 2.
semilog plot, the bulk of the slow mass diffusive decay is in a good signal/noise regime, and the mass diffusion coefficient can be measured to a high precision. Figure 3 shows a signal decay at a pressure farther from the phase-separation pressure, where the ratio of fast to slow decay rates is only 5.4, and Im is only 0.15. Here it is much more difficult to obtain good signal/ noise in the slow decay region, and the precision of the measurement of the mass diffusion coefficient suffers. The diminishing of Im at higher pressures limits the range of mass diffusion measurements possible with this technique. Figure 4 shows the dependence of the measured decay rates on the square of the grating wave vector. The straight line dependence with a zero intercept is good evidence that the model used to derive eq 9 is adequate; i.e., the slow decay is due only to mass diffusion, and there are no photochemical induced reactions affecting the solute concentrations (where a nonzero intercept would be expected).34 Most of the measurements are made with a fixed grating spacing of Λ ) 11.15 ( 0.06 µm, and the diffusion constants are determined from D12 ) km(Λ/2π)2 and Dth ) kth(Λ/2π)2 . Measurements were made with pump intensities that varied from 30 to 160 µJ/pulse per beam. As expected, the total diffracted signal intensity varied with the square of the pump intensity, but the thermal and mass decay rates were unchanged. Typically, measurements were made with a pump intensity of ≈150 µJ/pulse per beam. The
5986 J. Phys. Chem., Vol. 100, No. 14, 1996
Butenhoff et al. TABLE 1: NaNO3-Water Mass Diffusion and Thermal Diffusivity Resultsa c (m) T (°C) P (bar) 109D12 (m2 s-1) 109Dth (m2 s-1)
Figure 4. Dependence of the measured decay rates on the grating fringe spacing. The data are for 1.0 m NaNO3 solution at 450 °C and 450 bar. The error bars are (1 standard deviation of the average of the measurements.
resulting peak-null temperature and concentration jumps are δT < 0.05 K and |δc|/c < 0.02%. The diamond windows were damaged when pump energies significantly higher than 150 µJ/ pulse per beam were used for long periods of time. Measurements were made for 1.0 m NaNO3 solutions at 400, 450, and 500 °C and 0.25 and 3.0 m NaNO3 solutions at 450 °C; these results are shown in Table 1. The numbers in Table 1 are the average of 6-12 measurements, and the quoted uncertainties are 1 standard deviation. The values in Table 1 represent measurements made throughout a day or occasionally from different days. The standard deviation for multiple measurements made in succession at a given temperature and pressure was typically 2-3 times smaller than for measurements made at different times throughout a day or on different days. IV. Discussion of Results A. Mass Diffusion. Figure 5 shows the mass diffusion coefficients for 1.0 m NaNO3 solutions at 400, 450, and 500 °C. In all three cases, the mass diffusion coefficient is at a minimum near the phase-separation pressure and appears to plateau at higher pressures. The diminishing of the molecular diffusion coefficient near the phase-separation pressure was greatest for the 450 °C isotherm, which is postulated to be near the critical temperature for the 1.0 m NaNO3-water solution. Similar trends were observed for the 0.25 and 3.0 m solutions of NaNO3 at 450 °C (Table 1). In general, diffusion rates are expected to be inversely proportional to viscosity.27 Since the viscosity decreases as the density decreases, the mass diffusion coefficient is expected to increase as the density is decreased. However, in the vicinity of the critical point, critical phenomena result in the “critical slowing down” of the molecular diffusion process.29,30,46-48 As shown in Figure 5, diffusion data were obtained at temperatures and pressures where critical phenomena dominate as well as at conditions where “ordinary mass diffusion”sthe plateau regionswas experienced. 1. Critical Phenomena. The phase diagram for the NaNO3water system has not been thoroughly mapped out, but it appears to be qualitatively similar to the phase diagram for the NaClwater system.49 We will use features of the NaCl-water phase diagram to explain some of the NaNO3 diffusivity results. Figure 6 shows an isothermal slice of the NaCl-water phase diagram.50 The phase diagram is given at 450 °C, and the concentration ordinate is plotted on a log scale. At pressures greater than 423 bar, the solution is in a one-phase fluid region for all compositions, except for extremely high NaCl concentrations where a saturated liquid-solid equilibrium exists. When the pressure is lowered from the one-phase region to the phase
Im
0.25 0.25 0.25 0.25 0.25 0.25
450 450 450 450 450 450
409.5 425.1 450.1 500.0 550.3 600.1
13.0(8) 14.7(7) 17.0(11) 19.8(13) 18.3(14) 20.2(19)
70.9(12) 72.5(10) 75.8(16) 86.9(18) 97.7(17) 107.0(16)
0.337(28) 0.328(22) 0.252(30) 0.143(25) 0.088(21) 0.071(21)
1.0 1.0 1.0 1.0 1.0 1.0 1.0
400 400 400 400 400 400 400
271.6 276.9 300.0 350.0 399.9 500.0 600.0
10.6(3) 12.1(5) 15.8(5) 20.1(8) 21.2(13) 23.2(19) 21.7(15)
68.8(14) 73.3(7) 87.8(11) 105.6(6) 116.5(12) 132.3(17) 141.8(13)
0.422(8) 0.364(9) 0.245(12) 0.166(6) 0.134(9) 0.107(11) 0.085(6)
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
450 450 450 450 450 450 450 450 450 450 450 450 450 450
407.3 410.3 422.8 435.0 450.1 475.0 500.0 550.1 600.0 650.0 700.0 750.0 800.0 900.0
1.42(9) 2.72(8) 6.35(26) 9.04(35) 11.4(6) 14.6(6) 17.1(9) 20.3(10) 21.7(12) 23.1(12) 23.1(20) 24.8(15) 24.7(13) 24.0(21)
56.6(8) 62.3(9) 71.2(9) 76.3(11) 81.2(12) 88.7(11) 96.8(14) 107.8(10) 117.2(13) 123.9(10) 129.8(15) 135.4(10) 139.4(20) 147.4(24)
0.479(9) 0.510(9) 0.475(9) 0.425(10) 0.366(16) 0.301(10) 0.255(15) 0.201(12) 0.168(12) 0.146(10) 0.128(14) 0.118(4) 0.102(9) 0.098(9)
1.0 1.0 1.0 1.0 1.0 1.0 1.0
500 500 500 500 500 500 500
544.0 550.2 600.1 700.0 800.1 900.0 1001
4.0(3) 5.4(4) 12.2(3) 19.6(5) 22.7(9) 25.0(13) 27.1(21)
78.0(11) 81.8(6) 93.8(10) 110.2(9) 122.3(7) 133.7(13) 141.9(18)
0.295(6) 0.329(7) 0.316(7) 0.239(5) 0.184(6) 0.155(13) 0.146(12)
3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0
450 450 450 450 450 450 450 450 450 450 450
401.0 412.1 425.2 450.0 475.1 500.0 550.1 601.2 650.4 699.2 751.5
8.9(2) 11.0(6) 12.8(9) 14.7(8) 16.3(5) 17.5(9) 19.0(9) 19.6(16) 20.5(11) 19.6(11) 20.5(13)
94.3(17) 100.5(13) 105.0(8) 111.9(31) 118.2(11) 123.1(23) 129.8(18) 137.1(13) 142.1(20) 145.8(13) 149.3(20)
0.332(7) 0.290(5) 0.259(7) 0.244(20) 0.224(18) 0.196(20) 0.161(12) 0.138(10) 0.125(8) 0.118(7) 0.111(10)
a The numbers in parentheses are 1 standard deviation in the last digits.
Figure 5. NaNO3 mass diffusion coefficients for 1.0 m solutions at 400, 450, and 500 °C. The error bars are (1 standard deviations from the average.
boundary, the solution separates into two phases: a higher density liquid (or solid) phase with a larger salt concentration
Hydrothermal NaNO3 Solutions
Figure 6. Pressure-composition NaCl-water phase diagram at 450 °C.50 The filled square marks the critical point. This diagram is similar to the expected NaNO3-water phase diagram.
Figure 7. NaCl-water vapor-liquid coexistence curves for the 400, 450, and 500 °C isotherms.50 These curves are similar to the expected NaNO3-water coexistence curves. The filled squares mark the critical points, and the arrows mark regions where data have been obtained.
and a lower density vapor phase with a lower salt concentration. The critical solution point or plait point occurs where the vapor composition equals the liquid composition and is at 423 bar and 8.8 wt % NaCl at 450 °C.50 Figure 7 shows the liquid-vapor coexistence curves for the 400, 450, and 500 °C isotherms of NaCl, where the ordinate is now the log of the salt concentration in molality instead of weight percent. The arrows in Figure 7 show the compositions and temperatures where NaNO3 data have been obtained. At 450 °C, data have been obtained for concentrations above, near, and below the critical composition. For 1.0 m NaNO3, data have been obtained for temperatures above, near, and below the critical temperature. The behavior of the transport properties near the vapor-liquid critical point of binary fluids has been a topic of recent interest.51-55 In general, the transport property is separated into a background contribution and a critical enhancement. The critical enhancements are caused by long-range fluctuations in the system in the vicinity of the critical point. Very far away from the critical point, the transport properties are described by their background values, which are slowly varying functions with respect to temperature, density, and composition. Asymptotically close to the critical point, the transport properties are dominated by the critical enhancements, which satisfy power laws with universal critical exponents. The crossover region occurs in between these two limiting regions, where the transport properties cannot be described either by the background values or by asymptotic power laws alone. Crossover theories are being developed to describe this region.53-55 Figure 8 shows the 450 °C, 1.0 m NaNO3 D12 results on a log-log plot; the critical temperature for this solution is not
J. Phys. Chem., Vol. 100, No. 14, 1996 5987
Figure 8. Determination of the effective critical exponents for D12 (filled circles) and Dth (open squares) for 1.0 m NaNO3 solution at 450 °C. Pc ) 406.0 bar and the lines show the slopes as P approaches Pc.
known, but it appears to be near 450 °C. The ordinate is ) P/Pc - 1 where the critical pressure Pc is taken to be the observed phase-separation pressure of 406.0 bar. Measurements were also made by approaching the critical point from the onephase fluid region by varying the temperature with constant P ) 406 bar. As expected,56 the asymptotic behavior in the two cases is identical within experimental error. The mass diffusion coefficient approaches zero as D12 ∼ 0.60(0.02 as the pressure approaches the phase-separation pressure. This measured exponent agrees remarkably well with the theoretically expected asymptotic critical exponent ν ) 0.6346,57 and is also similar to measured critical exponents at consolute points.58,59 2. Ordinary Binary Mass Diffusion. We postulate that the plateau regions in Figure 5 occur where critical phenomena are no longer important, and D12 represents the ordinary binary diffusion coefficient. Four approaches are commonly used to model ordinary binary mass diffusion: hydrodynamic, kinetic, activation, and free volume theories.27 Activation and free volume theories have little or no theoretical basis and have been successful for only a limited number of systems.27 Diffusion in supercritical fluids is frequently correlated to some variant of hydrodynamic or kinetic (hard sphere) theory. It was shown13 that the hydrodynamic limit can also be obtained from hard sphere theory. Although both theories inadequately describe the complex thermodynamic behavior of supercritical solute/ solvent systems, they often yield accurate predictions when the solute is dilute, and strong compositional effects can be ignored.18 a. Kinetic Theory. Kinetic theory (hard sphere theory) of liquids originates from the kinetic theory of gases, modified for increased densities and thus increased molecular interactions. The theoretical background of hard sphere theory has been detailed in previous publications.20,60,61 As input to hard sphere models, the effective hard sphere diameters of the solute and solvent molecules need to be obtained through molecular dynamic simulations,62,63 cumbersome calculations,64 or semiempirical correlations.20 However, if the volume dependence of the self-diffusion coefficient65 holds for the limiting tracer diffusion coefficient, hard sphere theory can be expressed as a linear relationship D12T -0.5 ∝ Vτ, where V is the molar volume.15,18 Currently, values for the slope, intercept, and exponent τ need to be determined from experimental data which implies that the predictive capability of the hard sphere theory at this stage is limited. Recently, a correlation based on an extensive literature data base was proposed by Catchpole and King20 to estimate the self-diffusion and binary diffusion coefficients in near-critical solvents.
5988 J. Phys. Chem., Vol. 100, No. 14, 1996
Butenhoff et al.
The applicability of hard sphere theory to electrolyte systems is not clear since it does not accommodate ionic species. However, electrolytes under certain hydrothermal conditions are present as (hydrated) ion pairs,66 allowing for electrolyte species to be treated as uncharged molecules in terms of hard sphere theory. Most of the data in this research are in the region where critical phenomena are significant, resulting in insufficient ordinary mass diffusion data to correlate with hard sphere theory according to Erkey et al.15 since hard sphere theory does not accommodate critical anomalous behavior. No meaningful relation was obtained when the ordinary mass diffusion data were regressed as D12T -0.5 ∝ Vτ. Once a larger data set for ordinary mass diffusion is available, the correlation proposed by Catchpole and King20 deserves further attention. More data are needed in order to determine the nature of the systemdependent correction factor. In addition, model inputs when the diffusing species are inorganic salts need further study. Kinetic theory of diffusion in electrolytes, as represented by the Nernst or Nernst-Hartley equation,27,67 does not predict the trends in diffusion observed in this research due to the ionic nature of the electrolytes assumed in these treatments. b. Hydrodynamic Theory. In hydrodynamic theory, the solute is assumed to move through a continuum. The theory of Brownian motion shows, for the case where there is no correlation of molecular motions, that the diffusion coefficient is proportional to the temperature and inversely proportional to the friction factor:27
D12 ) kT/ζ
(11)
where k is the Boltzmann constant. The friction factor (ζ) can be calculated from classical hydrodynamics resulting in two limiting cases: the slip limit [ζ ) 4πηa] and no-slip limit [ζ ) 6πηa], where a is the effective solute radius and η the solvent viscosity. The no-slip limit corresponds to large spherical particles moving in a solvent of low molecular mass, resulting in the Stokes-Einstein relation:
D12 ) kT/6πηa
(12)
The slip limit is approached when a solute diffuses through a solvent of comparable molecular size.27 The diffusion of iodide ions in water at 240 bar and temperatures ranging from 25 to 375 °C was accurately predicted by the Stokes-Einstein equation, using the equivalent ion radius at ambient conditions, while it overpredicted the diffusion coefficient of hydroquinone in subcritical water.1 Many hydrodynamic correlations have been developed, the most well-known of which is the dimensional expression proposed by Wilke and Chang:68
D12 ) 7.4 × 10-15[(βM2)0.5T/(V1)0.6η]
(13)
where the empirical constant β corrects for solvent selfassociation (β ) 1 for nonassociated liquid, β ) 2.6 for associated liquid). Other hydrodynamic correlations include the Scheibel equation69
D12 )
( ( ))
3V2 8.2 × 10-15T 1+ 1/3 V1 ηV1 D12 )
2/3
1.75 × 10-14T V11/3η
the Reddy-Doraiswamy equation70
for V1 g 2.5V2 (14a) for V1 < 2.5V2
(14b)
TxM2 D12 ) Ω for V2/V1 e 1.5, Ω ) 10-14 1/3 1/3 ηV1 V2 for V2/V1 > 1.5, Ω ) 8.5 × 10-15 (15) and the Lusis-Ratcliff equation71
D12 )
( ( ) ( ))
V2 8.52 × 10-15T 1.40 1/3 V ηV2 1
1/3
+
V2 V1
(16)
where V1 and V2 typically are defined as the solute and solvent molar volumes (cm3/mol) at their normal boiling points, respectively. Because of the limited availability of molar volume data that reflect compositional effects, the molar volumes of the pure substances (at their normal boiling point) are typically used. The binary diffusion coefficient is expressed in m2 s-1, the temperature (T) in kelvin, the viscosity (η) in Pa‚s, and the solvent molecular mass (M2) in g/mol. In general, hydrodynamic diffusion equations are derived for molecular diffusion at infinite dilution. Hydrodynamic correlations, with the exception of the Wilke-Chang relationship, systematically overpredicted the diffusion coefficient of organic molecules in supercritical solvents such as CO2,13,14,17 CH4,13 and SF6.13 Knowledge about the effective solute size at hydrothermal conditions seems to be the limiting factor for successful implementation of hydrodynamic theory for diffusion in supercritical fluids. c. EValuation of Experimental Results in Terms of Hydrodynamic Theory. The NaNO3 binary diffusion data were evaluated in terms of hydrodynamic theory. In order to apply hydrodynamic theory, the effective radius or molar volume of the diffusing species and the solvent needed to be estimated. Molar volumes of the pure substances at the normal boiling temperature or effective ion radii at ambient conditions would not accurately reflect the structure of the solvent and solute molecules under hydrothermal conditions. Instead, the solvent molar volume was estimated from the density of the solvent (water) at the experimental temperature and pressure, thus ignoring compositional effects on the solvent molar volume. Water densities were obtained from the steam tables.72 To estimate the effective radius or molar volume of the solute, the nature of the diffusing species needed to be determined; i.e., is the salt dissociated or associated under the hydrothermal conditions, and to what degree is it hydrated? Sodium nitrate is completely ionized at 20 °C and 1 bar. However, it has been shown that for pressures ranging from 300 to 800 bar most 1-1 electrolytes are primarily present as hydrated contact ion pairs where the fraction of electrolyte that is associated increases with decreasing pressure.66,73-75 In addition, molecular dynamics simulations show strong ion pairing for sodium chloride under typical supercritical water oxidation conditions.76 The fraction of associated NaNO3 was estimated from electroconductivity data. Conductivity measurements of 0.01 m NaNO3 solutions at 500 bar and temperatures ranging from 25 to 450 °C77 appear to be in qualitative agreement with published NaCl data.78 Evaluation of the ionization constantss derived from NaCl conductivity data78sat densities and concentrations encountered in this research yielded degrees of association for NaCl between 60% and 95%. The degrees of association for NaCl increased with decreasing density and increasing temperature. Association at conditions of the plateau region in Figure 5 was about 70-80%. These results are in agreement with Spohn and Brill,75 who calculated from Raman spectroscopy data that for concentrated sodium nitrate solutions (6.8 m) about 80% of the ions are present as associated ion
Hydrothermal NaNO3 Solutions
J. Phys. Chem., Vol. 100, No. 14, 1996 5989
TABLE 2: Molecular and Ionic Radii and Corresponding Molar Volumes species
comments
configuration
equivalent radius (Å)
molar volume (cm3 mol-1)
NaNO3‚4H2O NaNO3 Na+‚4H2O + NO3-
hydrated associated ions pairs associated ion pairs hydrated dissociated ions
Na+, NO3-
dissociated ions
Na+ + NO3-
dissociated ions effective ionic radius from D12 at 20 °C and 1 bar and eq 12
oblate ellipsoid prolate ellipsoid oblate ellipsoid oblate ellipsoid average dimensions sphere oblate ellipsoid average dimensions spheres
3.12 2.16 (2.41) (1.84) 2.12 (0.99) (1.84) 1.42 1.64
76.5 24.2 (23.5) (15.0) 19.3 (2.45) (15.0) 8.72 11.1
TABLE 3: Ratios between Predicted and Measured Diffusion Coefficientsa c (m)
T (°C)
plateau D12,meas (10-9 m2 s-1)
Stokes-Einsteinb,c (no-slip limit)
Stokes-Einsteinc (slip limit)
Wilke-Changc
Scheibelc
Reddy-Doraiswamyc
Lusis-Ratcliffc
0.25 1.0 1.0 1.0 3.0
450 400 450 500 450
19.4(15) 21.6(14) 23.9(16) 26.1(17) 19.8(13)
1.42(12) 0.83(06) 0.87(07) 0.87(08) 0.86(04)
2.12(18) 1.25(10) 1.30(10) 1.31(12) 1.29(06)
1.40(12) 0.87(09) 0.86(06) 0.87(08) 0.85(04)
2.76(31) 1.70(25) 1.57(14) 1.51(16) 1.60(12)
1.75(11) 1.16(08) 1.15(06) 1.14(09) 1.11(03)
2.49(24) 1.48(19) 1.46(12) 1.49(15) 1.47(09)
a The viscosity of the solution was estimated by the viscosity of water at the same temperature and density of the solution. b Model predictions were within 5-10% of the measured diffusion coefficients if the diffusing species was assumed to be a combination of hydrated contact ion pairs, anions, and hydrated cations, determining the fraction of each from NaCl ionization constants. c The ratios are the average of the calculated values for all plateau diffusion coefficients with the numbers in parentheses being 1 standard deviation in the last digits.
pairs at 450 °C and 300 bar. However, questions remain whether these conditions correspond to the single-phase region. The reservations about some key assumptions in Quist and Marshall’s78 ionization constant calculations, as discussed by Oelkers and Helgeson,79 were not considered in this research since these reservations led to results that are not in agreement with most other ionization literature. Based on the above information, calculations were performed assuming that sodium nitrate was completely associated under the conditions of this research. In addition, the effect on the model predictions of accounting for the fraction of dissociated species was quantified. The degree of hydration of the ion pairs needs to be established. As a first approximation, the number of waters required for electrolyte solvation can be obtained from the solubility equilibrium equation.78 Dell’Orco et al.80 found that for sodium nitrate approximately 4.15 water molecules were needed to achieve solvation. These results should be used with caution under the conditions experienced in this research since the theoretical treatment of the results is limited to dilute solutions of electrolytes that form a solid precipitate and are not hydrolyzed. Since sodium nitrate has a melting point of 306.8 °C at ambient pressures, the precipitate is a molten salt. For the analysis of this work a hydration number of 4 was used. Hydration numbers can also be obtained through more comprehensive models for electrolyte solvation.81 The hydrated sodium nitrate contact ion pair (NaNO3‚4H2O) was assumed to be tetrahedral-like around the sodium ion. Effective ionic radii for Na+, O2-, H+, and N5+ were obtained from Shannon.82 The radius of a water molecule was calculated to be 1.22 Å, and the equivalent radius of an oblate ellipsoid nitrate molecule to be 1.84 Å. The shape of the tetrahedrallike NaNO3‚4H2O molecule was approximated by an oblate ellipsoid. The equivalent radius59 is about 3.12 Å, which is almost double the effective ionic radius calculated from StokesEinstein (a ) kT/(6πηD12) and ambient sodium nitrate diffusion data.83 The molar volume was also calculated for the oblate ellipsoid. Summation of the molar volumes of the sodium and nitrate ions and four water molecules did not yield an accurate estimate for the molar volume of NaNO3‚4H2O since the “dead volume” in the angle between the individual ions and molecules was not accounted for. These results are shown in Table 2.
Finally, the concentration effect had to be incorporated in the analysis since available diffusion correlations assume infinite dilution. Because hydrodynamic theory employs the continuum assumption, the concentration effect could be incorporated into the viscosity. Instead of using the solvent viscosity in the analysis, the solution viscosity was estimated. The viscosity for hydrothermal NaNO3 solutions has not been measured, but it could be approximated by the viscosity of pure water at the same temperature and mass density as the solution, which was found to be a valid approach for NaCl solutions at T < 300 °C.84,85 The solution density was interpolated from hydrothermal NaNO3 solution density data obtained by Anderson.41,42 Estimated solution viscosities were 10-15% larger at 0.25 m NaNO3 and 50-55% larger at 3.0 m NaNO3 than water viscosities at the same temperature and pressure. Diffusion coefficients at the plateau region were compared with predictions from the Stokes-Einstein expression in both the slip and no-slip limit and the Wilke-Chang, Scheibel, Reddy-Doraiswamy, and Lusis-Ratcliff equations. The solvent self-association constant (β) in the Wilke-Change expression was set equal to 1 since supercritical water is characterized by the breakup of hydrogen bonds.66 The results are shown in Table 3 as ratios of the predicted to the measured plateau mass diffusion coefficients. In general, the Stokes-Einstein relation in the no-slip limit, the Wilke-Chang correlation, and the Reddy-Doraiswamy correlation yielded the best agreement with experimental results. At more dilute conditions (0.25 m) diffusion coefficients were overestimated by 30% or more. The latter results can be explained in three ways. First, the experimental technique yielded more reliable data for the higher concentrations. As is indicated by the values for Im in Table 1, the Soret effect for mass diffusion becomes quite small for the plateau values for the 0.25 m data. The results are much more susceptible to experimental artifacts in this region. Second, apparent molar volume for hydrothermal sodium nitrate solutions became increasingly negative with decreasing concentration,40 indicating that a greater number of water molecules are associated with a NaNO3 molecule or that hydrated NaNO3 doublets or triplets might be formed. This would result in a larger molecule and thus lower predicted diffusion coefficients. Third, evaluation of NaCl conductivity data indicated thatsfor
5990 J. Phys. Chem., Vol. 100, No. 14, 1996 a given density and temperaturesthe degree of association is about 10% larger in 0.25 m NaCl solutions than in more concentrated solutions (g1 m). A similar trend was observed by Spohn and Brill75 for NaNO3. If a larger fraction of molecules are associated, the average size of the solute molecules increase and the diffusion coefficient decreases. For the 1.0 and 3.0 m data the Stokes-Einstein relation in the noslip limit, the Wilke-Chang correlation, and the ReddyDoraiswamy correlation predicted the background diffusion coefficient within about 15%. The Stokes-Einstein and Wilke-Chang equations resulted in systematically low predictions while the Reddy-Doraiswamy equation yielded predictions that were systematically high. Considering the uncertainty in calculating the solute size, these results are remarkably good. Approximating a tetrahedral-like structure by an oblate ellipsoid overestimates the equivalent radius and thus underestimates the diffusion coefficient. As a result, the Stokes-Einstein or Wilke-Change relationships yield more reliable predictions. Because a hydrated sodium nitrate molecule is significantly larger and heavier than a water molecule, the no-slip limit for the Stokes-Einstein equation is reasonable. Both equations adequately captured the temperature dependence of the diffusion coefficient. Incorporating the use of the viscosity of water at the same temperature and density as the solution showed promise for accounting for the concentration dependence of the diffusion for concentrated (g1.0 m) solutions. The StokesEinstein equation predicted the diffusion coefficient within 10% if the diffusing species was assumed to be a combination of hydrated contact ion pairs, anions, and hydrated cations, determining the fraction of each from NaCl ionization constants. As a check on the sensitivity of the structure of the diffusing species on the calculated mass diffusion coefficients, equivalent radii and molar volumes were calculated for several other potential forms of the diffusing sodium nitrate: hydrated dissociated ions, nonhydrated dissociated ions, and nonhydrated associated ion pairs. These results are also shown in Table 2. As can be seen in Table 2, the estimated equivalent radii and molar volumes are significantly lower than for the hydrated ion pair. The corresponding calculated mass diffusion coefficients are all significantly higher than the measured plateau values. It appears that the structure of the diffusing sodium nitrate is best represented as a hydrated ion pair under hydrothermal conditions. B. Thermal Diffusivity. Figure 9 shows the thermal diffusivity of 0.25, 1.0, and 3.0 m NaNO3 solutions at 450 °C along with the calculated thermal diffusivity of water. The thermal diffusivity of the salt solutions appears to be at a minimum at the phase-separation pressure. In general, for a given T and P, the thermal diffusivity increases with increasing concentration. The exception is very near the phase-separation pressure and near-critical points. For example, at 450 °C and 407 bar, Dth(1.0 m) < Dth(0.25 m) < Dth(water). For a given concentration and pressure, the thermal diffusivity decreases with increasing temperature. Unfortunately, the heat capacity has not been measured for hydrothermal NaNO3 solutions so the thermal conductivity cannot currently be determined from the thermal diffusivity measurements. Figure 8 shows the 450 °C, 1.0 m NaNO3 Dth results on a log-log plot. The thermal diffusivity is given by the equation Dth ) λF-1Cp,x-1, where λ is the thermal conductivity, F is the density, and Cp,x is the isobaric heat capacity at constant composition. Asymptotically close to the critical point, the thermal conductivity of a binary solution approaches a constant,51,52,86 and Cp,x exhibits a weak divergence;86 therefore, asymptotically close to the critical point, Dth is expected to
Butenhoff et al.
Figure 9. Thermal diffusivity of 0.25, 1.0, and 3.0 m NaNO3 solutions at 450 °C. The line is the thermal diffusivity calculated from the thermal conductivity and the equation of state for water.72 The error bars are (1 standard deviation and are smaller than the data points when not shown.
weakly converge to zero. Farther from the critical point, the critical enhancement of the thermal conductivity λc is expected to diverge as in a single-component fluid λc ∼ -γ+ν,51,52 where γ ) 1.22, ν ) 0.63, and Cp,x ∼ -γ, so Dth is expected to have a critical exponent of -γ + ν + γ ) 0.63. As the phaseseparation pressure is approached, we observe Dth ∼ 0.096(0.011, with the strength of the singularity increasing at larger values of P/Pc - 1. This measured value for the critical exponent roughly corresponds to the weak singularity expected asymptotically close to the critical point. However, a value of P/Pc - 1 (or T/Tc - 1) equal to 3 × 10-3 is normally not considered asymptotically close to the critical point, nor are the measurements made precisely on the critical isotherm. It may just be fortuitous that the observed critical exponents agree so well with the exponents expected asymptotically close to the critical point, or the observed behavior may be due to the particular properties of the salt-water system. The very dissimilar components in this binary mixture may allow the asymptotic region to be observed farther from the critical point than for the bulk of the binary systems studied so far.86 This warrants further investigation. C. Thermal Diffusion Coefficient. The measurement of Im, the ratio of the concentration grating signal amplitude to the thermal grating signal amplitude, allows the thermal diffusion coefficient to be determined if (∂n/∂c)T,P and (∂n/∂T)P,c are known for the probe wavelength: by rearranging eq 10, one has
(∂T∂n) ∂n + 1)c ( ) ∂c
-Im(Dth - D12) DT ) (Im
0
P0,c0
(17)
T0,P0
Anderson has measured the index of refraction of NaNO3 solutions at various concentrations, temperatures, and pressures.39,40 From that work (∂n/∂c)T,P is known to (5%, and (∂n/∂T)c,P is known to (10%. The calculated thermal diffusion coefficients for the 1.0 m NaNO3 solution at 450 °C are shown in Figure 10. DT is positive just as it is for aqueous salt solutions at room temperature; i.e., the Soret effect causes the salt to migrate from regions of higher to lower temperature. DT increases as the phase separation pressure is approached and then levels off or decreases very near the phase separation pressure. The typical value for DT is ≈5 × 10-10 m2 s-1 K-1, which is ≈250 times larger than the value for NaNO3 solutions at 25 °C and atmospheric pressure.87,88 It is the large magnitude
Hydrothermal NaNO3 Solutions
J. Phys. Chem., Vol. 100, No. 14, 1996 5991 cell design. This work was sponsored by the Westinghouse Hanford Corp. under the auspices of the Department of Energy. References and Notes
Figure 10. Thermal diffusion coefficient for 1.0 m NaNO3 solution at 450 °C.
of the Soret effect that allows this particular experimental technique to measure both the mass and thermal diffusivities of hydrothermal electrolyte solutions. This particular method will not work well to measure mass diffusivities of room temperature aqueous electrolyte solutions because of the small magnitude of the Soret effect. One way to possibly extend the utility of the laser-induced grating to regions farther from the plait point is to use a chopped CW laser for the pump. This would allow one to use the Soret effect to drive the concentration grating for relatively long periods of time, thereby inducing a larger magnitude concentration grating. Ko¨hler et al.41,42 have successfully used this method to measure the mass diffusion of polystyrene spheres in ethyl acetate under ambient conditions. V. Conclusions Binary mass diffusion coefficients and thermal diffusivities for hydrothermal sodium nitrate solutions as a function of pressure (270 < P < 1000 bar), temperature (400 < T < 500 °C), and concentration (0.25 < C < 3.0 m) were measured by the laser-induced grating technique. In these systems, the critical slowing down was significant as far as 300 bar from the phase separation pressure. These results imply that diffusion-limited regimes are much more likely in hydrothermal processes than commonly acknowledged. Ordinary binary mass diffusion coefficients agreed well with predictions from the Stokes-Einstein equation, where the diffusing species was best represented by a hydrated contact ion pair. Model predictions could be improved by about 10% if the degree of association was calculated. The Wilke-Chang correlation also yielded good predictions when the solute molar volume was defined as the volume of the hydrated contact ion pair. Ordinary binary mass diffusion was about 15 times faster than at 25 °C and atmospheric pressure. In addition, it was determined that the Soret effect causes the salt to migrate from regions of higher to regions of lower temperature. The thermal diffusion coefficient at 450 °C is about 250 times larger than at ambient conditions. The laser-induced grating technique is complementary to the Taylor dispersion technique for measuring mass diffusion coefficients. The grating technique works best at higher concentrations and near critical points (when the magnitude of the Soret effect is large), the two regimes where the Taylor dispersion technique becomes increasingly more difficult. Acknowledgment. The authors acknowledge Graydon K. Anderson for supplying the index of refraction and density data for hydrothermal NaNO3 solutions and for the diamond optical
(1) Flarsheim, W. M.; Tsou, Y. M.; Trachtenberg, I.; Johnston, K. P.; Bard, A. J. J. Phys. Chem. 1986, 90, 3857. (2) Lamb, W. J.; Hoffman, G. A.; Jonas, J. J. Chem. Phys. 1981, 74, 6875. (3) Xu, X.; De Almeida, C. P.; Antal, M. J. Ind. Eng. Chem. Res. 1991, 30, 1478. (4) Adschiri, T.; Kanazawa, K.; Arai, K. J. Am. Ceram. Soc. 1992, 75 (4), 1019. (5) Goemans, M. G. E.; Li, L.; Gloyna, E. F. Sep. Sci. Technol. 1995, 30 (7-9), 1491. (6) Amestica, L. A.; Wolf, E. E. Fuel 1986, 65, 1226. (7) Kershaw, J. R.; Bagnell, L. J. In Supercritical Fluids, Chemical Engineering Principles and Applications; Squires, T. G., Paulaitis, M. E., Eds.; ACS Symposium Series No. 329; American Chemical Society: Washington, DC, 1987. (8) Ross, D. S.; Green, T. K.; Mansani, R.; Hum, G. P. Energy Fuels 1987, 1 (3), 287. (9) Crain, N.; Tebbal, S.; Li, L.; Gloyna, E. F. Ind. Eng. Chem. Res. 1993, 32, 2259. (10) Tester, J. W.; Holgate, H. R.; Armellini, F. J.; Webley, P. A.; Killilea, W. R.; Hong, G. T.; Barner, H. E. In Emerging Technologies in Hazardous Waste Management; Tedder, W. D., Pohland, F. G., Eds.; ACS Symposium Series No. 518, American Chemical Society: Washington, DC, 1993. (11) Harradine, D. M.; Buelow, S. J.; Dell’Orco, P. C.; Dyer, R. B.; Foy, B. R.; Robinson, J. M.; Sanchez, J. A.; Spontarelli, T.; Wander, J. Hazard. Waste Hazard. Mater. 1993, 10 (2), 233. (12) Iyer, S. D.; Nicol, G. R.; Klein, M. T. J. Supercrit. Fluids, in press. (13) Debenedetti, P. G.; Reid, R. C. AIChE J. 1986, 32, 2034. (14) Sassiat, P. R.; Mourier, P.; Caude, M. H.; Rosset, R. H. Anal. Chem. 1987, 59, 1164. (15) Erkey, C.; Gadalla, H.; Akgerman, A. J. Supercrit. Fluids 1990, 3, 180. (16) Smith, S. A.; Shenai, V. M.; Matthews, M. A. J. Supercrit. Fluids 1990, 2, 175. (17) Liong, K. K.; Wells, P. A.; Foster, N. R. Ind. Eng. Chem. Res. 1992, 31, 390. (18) Shenai, V. M.; Hamilton, B. L.; Matthews, M. A. In Supercritical Fluid Engineering Science, Fundamental and Applications; Kiran, E., Brennecke, J. F., Eds.; ACS Symposium Series No. 514; American Chemical Society: Washington, DC, 1993; Chapter 8. (19) Sua´rez, J. J.; Bueno, J. L.; Medina, I. Chem. Eng. Sci. 1993, 48 (13), 2419. (20) Catchpole, O. J.; King, M. B. Ind. Eng. Chem. Res. 1994, 33, 1828. (21) Lai, C.-C.; Tan, C.-S. Ind. Eng. Chem. Res. 1995, 34, 674. (22) Sun, C. K. J.; Chen, S. H. AIChE J. 1985, 31, 1510. (23) Sun, C. K. J.; Chen, S. H. AIChE J. 1985, 31, 1904. (24) Noel, J. M.; Erkey, C.; Bukur, D. B.; Akgerman, A. J. Chem. Eng. Data 1994, 39, 920. (25) Alizadeh, A.; Nieto de Castro, C. A.; Wakeham, W. A. Int. J. Thermophys. 1980, 3, 243. (26) Erkey, C.; Akgerman, A. In Measurement of the Transport Properties of Fluids, Experimental Thermodynamics; Wakeham, W. A., Nagashima, A., Sengers, J. V., Eds.; Blackwell Scientific Publications: Oxford, 1991; Vol. III, Chapter 9. (27) Tyrrell, H. J. V.; Harris, K. R. Diffusion in Liquids, A Theoretical and Experimental Study; Butterworths Monographs in Chemistry; Butterworths & Co. Ltd.: London, 1984. (28) Levelt Sengers, J. M. H.; Deiters, U. K.; Klask, U.; Swidersky, P.; Schneider, G. M. Int. J. Thermophys. 1993, 14, 893. (29) Clifford, A. A.; Coleby, S. E. Proc. R. Soc. London 1991, 433A, 63. (30) Bartle, K. D.; Baulch, D. L.; Clifford, A. A.; Coleby, S. E. J. Chromatogr. 1991, 557, 69. (31) Goemans, M. G. E.; Gloyna, E. F.; Buelow, S. J. Proceedings of the Second International Symposium on EnVironmental Applications of AdVanced Oxidation Technologies, February 28-March 1, San Francisco, CA; Electrical Power Research Institute, Palo Alto, CA, 1996. (32) Eichler, H. J.; Gu¨nter, P.; Pohl, D. W. Laser-Induced Dynamic Gratings; Springer-Verlag: New York, 1986. (33) Terazima, M.; Okamoto, K.; Hirota, N. J. Chem. Phys. 1995, 102, 2506. (34) Terazima, M. Chem. Phys. Lett. 1994, 218, 574. (35) Nagasaka, Y.; Nakazawa, N.; Nagashima, A. Int. J. Thermophys. 1992, 13, 555. (36) Butenhoff, T. J. Int. J. Thermophys. 1995, 16, 1. (37) Kimura, Y.; Terazima, K. M.; Hirota, N. Ber. Bunsen-Ges. Phys. Chem. 1995, 99, 196.
5992 J. Phys. Chem., Vol. 100, No. 14, 1996 (38) Nagasaka, Y.; Hatakeyama, T.; Okuda, M.; Nagashima, A. ReV. Sci. Instrum. 1988, 59, 1156. (39) Anderson, G. K. In Physical Chemistry of Aqueous Systems: Meeting the Needs of Industry; White, H. J., Sengers, J. V., Neumann, D. B., Bellows, J. C., Eds.; Begell House: New York, 1995; p 573. (40) Anderson, G. K. Int. J. Thermophys., in press. (41) Ko¨hler, W. J. Chem. Phys. 1993, 98, 660. (42) Ko¨hler, W.; Rosenauer, C.; Rossmanith, P. Int. J. Thermophys. 1995, 16, 11. (43) Thyagarajan, K.; Lallemand, P. Opt. Commun. 1978, 26, 54. (44) Bloisi, F.; Vicari, L.; Cavaliere, P.; Martellucci, S.; Quartieri, J.; Mormile, P.; Pierattini, G. Appl. Phys. B 1987, 44, 103. (45) Eichler, H.; Salje, G.; Stahl, H. J. Appl. Phys. 1973, 44, 5383. (46) Sengers, J. V. In Critical Phenomena; Green, M. S., Ed.; Academic Press: New York, 1977; p 445. (47) Saad, H.; Gulari, E. Ber. Bunsen-Ges. Phys. Chem. 1984, 88, 834. (48) Gulari, E.; Saad, M.; Bae, Y. In Supercritical Fluids, Chemical Engineering Principles and Applications; Squires, T. G., Paulaitis, M. E., Eds.; ACS Symposium Series No. 329; American Chemical Society: Washington, DC, 1987. (49) Anderson, G. K. Unpublished results, 1994. (50) Bischoff, J. L.; Pitzer, K. S. Am. J. Sci. 1989, 289, 217. (51) Mostert, R.; Sengers, J. V. Fluid Phase Equilib. 1992, 75, 235. (52) Mostert, R.; Sengers, J. V. Fluid Phase Equilib. 1993, 85, 347. (53) Anisimov, M. A.; Kiselev, S. B. Int. J. Thermophys. 1992, 13, 873. (54) Kiselev, S. B.; Kulikov, V. D. Int. J. Thermophys. 1994, 15, 283. (55) Luettmer-Strathmann, J.; Sengers, J. V. Int. J. Thermophys. 1994, 15, 1241. (56) Griffiths, R. B.; Wheeler, J. C. Phys. ReV. A 1970, 2, 1047. (57) Sengers, J. V.; Levelt Sengers, J. M. H. Annu. ReV. Phys. Chem. 1986, 37, 189. (58) Cussler, E. L. AIChE J. 1980, 26, 43. (59) Cussler, E. L. Diffusion, Mass Transfer in Fluid Systems; Cambridge University Press: New York, 1984. (60) Dymond, J. H. Chem. Soc. ReV. 1985, 14, 317. (61) Matthews, M. A.; Akgerman, A. J. Chem. Phys. 1987, 87, 2285. (62) Esteal, A. J.; Woolf, L. A.; Jolly, D. L. Physica A 1983, 121, 286. (63) Esteal, A. J.; Woolf, L. A. Chem. Phys. 1984, 88, 101. (64) Sung, W.; Stell, G. J. Chem. Phys. 1984, 80, 3350. (65) Dymond, J. H. J. Chem. Phys. 1974, 60, 969.
Butenhoff et al. (66) Marshall, W. L. Chemistry 1975, 48, 6. (67) Horvath, A. L. Handbook of Aqueous Electrolyte Solutions, Physical Properties, Estimation and Correlation Methods; Ellis Horwood: Chichester, 1985. (68) Wilke, C. R.; Chang, O. AIChE J. 1955, 1, 264. (69) Scheibel, E. G. Ind. Eng. Chem. 1954, 46, 2007. (70) Reddy, K. A.; Doraiswamy, L. K. Ind. Eng. Chem. Fundam. 1967, 6, 77. (71) Lusis, M. A.; Ratcliff, G. A. Can. J. Chem. Eng. 1968, 46, 385. (72) Haar, L.; Gallagher, J. S.; Kell, G. S. NBS/NRC Steam Tables; Hemisphere Publishing: Washington, DC, 1984. (73) Marshall, W. L. In High Temperature, High Pressure Electrochemistry in Aqueous Solutions; Jones, D., Staehle, R. W., Eds.; NACE: Houston, TX, 1976; Vol. 4, p 117. (74) Mesmer, R. E.; Marshall, W. L.; Palmer, D. A.; Simonson, J. M.; Holmes, H. F. J. Solution Chem. 1988, 17, 699. (75) Spohn, P. D.; Brill, T. B. J. Phys Chem. 1989, 93, 6224. (76) Cui, S. T.; Harris, J. G. Int. J. Thermophys. 1995, 16, 493. (77) Goemans, M. G. E.; Funk, T. J.; Anderson, G. K. Work in progress. (78) Quist, A. S.; Marshall, W. L. J. Phys. Chem. 1968, 72, 684. (79) Oelkers, E. H.; Helgeson, J. C. Geochim. Cosmochim. Acta 1991, 55, 1235. (80) Dell’Orco, P. C.; Eaton, H. K.; Reynolds, R. T.; Buelow, S. J. J. Supercrit. Fluids 1995, 8, 217. (81) Pitzer, K. S.; Pabalan, R. T. Geochim. Cosmochim. Acta 1986, 50, 1445. (82) Shannon, R. D. Acta Crystallogr. 1976, A32, 751. (83) Lobo, V. M. M. Physical Sciences Data 41&42, Handbook of Electrolyte Solutions; Elsevier Science Publishers, B. V.: Amsterdam, 1989. (84) Pepinov, R. I.; Yusufova, V. D.; Lobkova, N. A. Teploenergetica 1977, 24, 59. (85) Rogers, P. S. Z.; Pitzer, K. S. J. Phys. Chem. Ref. Data 1982, 11, 15. (86) Anisimov, M. A.; Gorodetskii, E. E.; Shmakov, N. G. SoV. Phys. JETP 1973, 36, 1143. (87) Snowdon, P. N.; Turner, J. C. R. Trans. Faraday Soc. 1960, 56, 1409. (88) Tanner, C. C. Trans. Faraday Soc. 1923, 23, 75.
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