Limiting Conductivities and Ion Association Constants of Aqueous

Article pubs.acs.org/jced

Limiting Conductivities and Ion Association Constants of Aqueous NaCl under Hydrothermal Conditions: Experimental Data and Correlations G. H. Zimmerman,†,⊥ H. Arcis,‡,§ and P. R. Tremaine*,‡ †

Department of Chemistry and Biochemistry, Bloomsburg University, Bloomsburg, Pennsylvania 17815, United States Department of Chemistry, University of Guelph, Guelph, Ontario, Canada N1G 2W1

‡

S Supporting Information *

ABSTRACT: Frequency-dependent electrical conductivities of solutions of aqueous sodium chloride have been measured from T = 298 K to T = 623 K at p = 20 MPa, over a very wide range of ionic strength (2·10−5 to 0.17 mol·kg−1) using a unique highprecision flow-through alternating current (AC) electrical conductance instrument. Experimental values for the equivalent conductivity, Λ, were used to calculate molar conductivities at infinite dilution, Λ°, using the Fuoss−Hsia−Fernandez−Prini (FHFP) and Turq−Blum−Bernard−Kunz (TBBK) ionic conductivity models. The resulting values for the limiting conductivity Λ° and the ion association constant of NaCl, from this work and critically evaluated literature data above 277 K, were represented to within the combined experimental uncertainties, as functions of viscosity and solvent density, respectively. New values and new correlations are reported for the limiting equivalent conductivities of the sodium ion, λ°(Na+), and the chloride ion, λ°(Cl−) from 277 K and 100 kPa to 1073 K and 500 MPa. Recently, Corti22 reported a very detailed literature review of conductivity measurements of aqueous solutions of electrolytes above 373 K, starting with the classic experiments by Noyes and Coolidge19 and continuing up to 2007. As part of the present study, this literature compilation of hydrothermal conductivity studies on NaCl was updated. The results1,2,4,5,15,17,19,23−37 are tabulated in Table 1. This compilation includes only the published studies extending to temperatures above 373 K and concentrations below ∼ 10−3 mol·L−1, consistent with our focus on limiting conductivities. The experimental conditions corresponding to each study are plotted in Figure 1. From Figure 1, it is apparent that there is a gap in the reported results for accurate conductivity measurements at pressures above the saturation pressure in the temperature range (298 to 523) K. The objective of the present study was to fill this gap, to compile the most accurate values for Λ°(NaCl), and to derive values for λ°(Cl−) based on modern data. This work reports conductivity measurements for a series of dilute solutions of sodium chloride [(2·10−5 to 0.12) mol·kg−1] at temperatures from 298 K up to 623 K at a constant applied pressure of ∼20 MPa, using a state-of-the-art, high-temperature, flow AC conductance cell built at the University of Delaware.1,36,38 Accurate values for the molar conductivities, limiting conductivities, and association constants for NaCl were

1. INTRODUCTION Flow alternating current (AC) conductance methods provide an attractive tool for determining the limiting ionic conductivities and ion-pair formation constants. Although conductivity techniques for measuring ion association at high temperatures and pressure have been available for many years, flow instruments sensitive enough to make measurements for very dilute solutions under extreme conditions have only recently been developed.1,2 The system {NaCl + H2O} is important because it is a simple salt and its physical properties are well-known over a very wide range of temperature and pressure.3 As a result, sodium chloride has been used as a validity check for the calibration of conductivity high-temperature thermodynamic and transport property measurements1,4,5 and for theoretical studies of ionic transport and ion pairing under hydrothermal conditions.6,7 The dependence of limiting equivalent conductivities Λ° on temperature and pressure for electrolytes has been described by a number of models.8−17 All of these models are based on values for the conductivity of the chloride ion λ°(Cl−) that were reported by Quist and Marshall in 1965.18 These are based on experimental values for Λ°(NaCl) to 573 K measured by Noyes and Coolidge in 1903,19 and the transference numbers of Smith and Dismukes.20,21 There have been significant advances in the measurement and theoretical treatment of conductivities under hydrothermal conditions since Quist and Marshall’s study, and these models need to be updated to include more recent results and our current level of understanding of ionic transport mechanisms in water. © 2012 American Chemical Society

Received: March 22, 2012 Accepted: July 14, 2012 Published: August 29, 2012 2415

dx.doi.org/10.1021/je300361j | J. Chem. Eng. Data 2012, 57, 2415−2429

Journal of Chemical & Engineering Data

Article

solutions with Nanopure water (resistivity 18.2 MΩ·cm) by mass. After correcting all masses for buoyancy, the molalities of these stock solutions were (0.05320 ± 0.00001) mol·kg−1 and (0.4938 ± 0.0001) mol·kg−1. KCl purchased from Alfa Aesar (99.995 % metals basis, Lot No. E21U0) was dried at 573 K until the mass difference between weights was less than 0.01 %. This was used to make two stock solutions by mass with Nanopure water (resistivity 18.2 MΩ·cm). These standard solutions of KCl were found to be (0.02190 ± 0.00001) mol·kg−1 and (0.3206 ± 0.0001) mol·kg−1 after buoyancy corrections were made and were used to determine the cell constant. The solutions measured in the conductance instrument were prepared by mass dilution under argon from these stock solutions in sealed Pyrex bottles following the procedure given by Zimmerman et al.1 using nanopure water and pumped from these same bottles directly into the conductivity cell via the injection system described below. 2.2. Experimental Apparatus. High-Temperature HighPressure Conductivity Cell. The high-temperature, highpressure conductance flow cell used for this work was built at the University of Delaware by Hnedkovsky et al.,38 with improvements on the original designs of Zimmerman et al.1 and Sharygin et al.36 to allow for the measurement of more corrosive solutions. The cell has been used at temperatures as high as 673 K at 28 MPa, with ionic strengths as low as 10−5 mol·kg−1.38 A schematic diagram is presented in Figure 2a.

Table 1. Literature Review of the High Temperature (T > 373.15 K) Conductivity Studies on the System {NaCl + H2O} T/K

p/MPa

c·103/(mol·L−1)

ref

299.15−580.15 299.15−579.15 291.15−579.15 291.15−579.15 651.15−666.15 298.15−613.15 663.15 533.15−643.15 T/K

sat sat sat sat 22.5−28.6 sat 24.6−32.1 25−200 p/MPa

0.4−90 0.5−100 2−100 0.5−100 0.009−2.5 1000−3000 0.1−200 10−5000 m·103/(mol·kg−1)

19 23 24 25 26 27 28 29 ref

573.15−656.15 293.15−573.15 373.15−1073.15 273.15−1085.15 298.15−523.15 373.15−873.15 579.15−677.15 603.15−674.15 523.15−678.15 624.15 651.15−670.15 581.15−731.15 578.15−646.7 368.60−548.44

8.6−41.2 10−150 sat −500 0.1−400 0.1−200 0.1−350 10−28 15−28 10−32 20 28 0.13−1.4 9.3−21.9 17.5−20.8

0.24−3 30−sat 1−100 10 1 1−100 0.0016−32 0.00019−19 0.16−2 2.58−11 0.003−1000 0.2−400 0.1−770 0.1−0.9

4 30 5 31 32 33 1 34 2 35 36 15 37 17

Figure 1. Experimental conditions for accurate sodium chloride conductivity studies at high temperatures and pressures.

determined under conditions that address the gap in the modern database identified above. Finally, we have compiled a database of accurate conductivity measurements for aqueous NaCl under hydrothermal conditions, including our new results, and used this to develop new equations for calculating the limiting conductivities of the sodium and chloride ions up to 1073 K and 500 MPa and association constants up to 678 K and 32 MPa.

Figure 2. (a) Schematic of the AC conductance cell: (1) platinum inlet tube; (2) platinum outlet tube; (3) diamond frit; (4) ceramic spacer; (5) Inconel Belleville washers; (6) sapphire insulator; (7) titanium ram; (8) steel screws; (9) platinum outer electrode; (10) platinum inner electrode. (b) Schematic of the high pressure sample injection system: (A1) HPLC pump 1; (A2) HPLC pump 2; (B) deionized water reservoir; (C) Pyrex solution bottle; (D) peristaltic pump; (E1) six-port injection valve 1; (E2) six-port injection valve 2; (F) delay loop; (G) large air oven containing the conductivity cell within the temperature-controlled insulated air oven; (H) back pressure regulator; (I) valves; (J) pressure transducer; (K) pressure release valves; (L) waste solution reservoirs; (M) Fluke PM 6304 programmable automatic RCL meter; (N) acquisition computer, (·-·-·) fourlead electrical connection.

2. EXPERIMENTAL SECTION 2.1. Chemicals and Solution Preparation. NaCl purchased from Alfa Aesar (99.99 % metals basis, Lot No. I14U009) was dried at 573 K until a constant mass was attained (0.01 % difference) and used to prepare two different stock 2416



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underneath the cell. Second, the temperature cell itself was maintained with two heating cartridges located in the titanium cell body, controlled by a Leeds & Northrup 6430 Series Electromax III PID controller that was connected to a Pt 1000 Ω sensor. The sensor formed one leg of a Wheatstone bridge circuit, which balanced a 4-decade variable resistor, which used to set the control point to increase the sensitivity to ± 0.01 K. The temperature of the cell was measured with a platinum resistance thermometer consisting of a Hart Scientific model 5612 probe and a model 5707 6 1/2 Digit DMM, accurate to ± 0.02 K. Third, to make sure the solution to be analyzed was brought as close as possible to the cell temperature, a preheater was used that consisted of a coaxial linear heating element placed in intimate with the inlet tube over a length of ∼47 cm. The dif ference in the temperature between the inlet end of the tube and the conductivity cell was monitored by two thermocouple junctions, one located on the inlet Pt tubing just before it entered the cell and the other on the titanium body cell. The output of this pair of thermocouples was connected to an Automation Direct Solo 4824 PID controller, which adjusted the heating power necessary to keep the temperature difference between the two thermocouples equal to zero. Flow Injection System. The injection system for the conductance equipment was very similar to that reported by Méndez De Leo and Wood,39 and is shown in Figure 2b.40 Briefly, solutions were injected using two Lab Alliance Series 1500 dual piston HPLC pumps. Pump A1, which was always turned on, was used to supply a continuous flow of degassed and deionized water from a large reservoir to the instrument at a pressure set by a back-pressure regulator (Circle Seal Controls No. BPR21U22542). Two six-port valves, controlled through a computer with Hewlett-Packard VEE Version 6.1 software, determined whether water from the reservoir flowed directly through the conductance cell or whether it pushed solution from the injection loop through the cell. The sample to be injected was loaded into an HPLC injection loop (3.2 mm o.d., passivated stainless steel tubing from Restek, with a capacity of 50 mL) with a peristaltic pump, using procedures described below. Pump A2 was then used to pressurize the sample loop by pumping deionized water from the reservoir into the loop, so that the downstream water displaced by the sample in the loop bypassed the cell and flowed directly to the back-pressure regulator. Once the sample was pressurized, the computer switched the six-port valves to push the sample into the conductance cell. Experiments were conducted at a flow rate of 0.5 mL·min−1. The pressure was measured with a digital pressure transducer (Paroscientific Inc. model 760-6K) to an accuracy of ± 0.01 MPa. The conductivity of aqueous solutions of sodium chloride was measured as a function of concentration at the same temperature, pressure, and flow rate. To minimize solvent corrections, solutions of NaCl at increasing concentrations were prepared in the same bottle, by injecting increments of stock solution from a weighed syringe through a septum into the solution. The procedure was repeated at each temperature. For this purpose, we used Pyrex glass bottles equipped with VAPLOCK Bottle Cap (1/4-28, GL45, 4-port). The first port was joined to a balloon full of argon to keep a positive pressure over the solution for the duration of the experimental runs, and the second one was connected to a No-Ox tubing which kept the solution degassed as it traveled from the bottle to the injection loop. The third port had a septum, and the fourth port

Briefly, the cell consists of a 47 cm long temperature-controlled platinum inlet tube (1.0 mm i.d.; 1.6 mm o.d.) that leads into a platinized cup (4.6 mm i.d.; 5.6 mm o.d.), which serves as the outer electrode for the cell. The inner electrode is a platinum rod (1.6 mm o.d.), electrodeposited with platinum black, and is a direct extension of the platinum tube, which carries the exiting solution away from the cell. A diamond shield protects the sapphire insulator from corrosion that would otherwise contaminate the solution. The entire electrode assembly is contained in a titanium cell body. A sapphire disk and a ceramic spacer provide electrical insulation between the two electrodes. The pressure seal inside the conductivity flow cell is maintained by compressing annealed thin gold disks, which sit between the sapphire insulator and a titanium end-cap, using a system of bolts and Inconel Belleville washers (Figure 2a). A four-wire measurement was used to acquire the AC impedance spectra. For the outer electrode, one of the leads was the Pt/Rh tube also used to flow the solution to the cell, and the other was a silver wire gold soldered to the tubing outside of the titanium body. Similarly, for the inner electrode, the Pt/Rh tubing served as one lead and another silver wire gold soldered to the tubing as the other lead. Although the resistance contribution in a four-leads measurement is zero, there remains a short portion of platinum/20 % rhodium tubing on both the inlet and the outlet tubes and an inner electrode that must be subtracted from the real portion of the impedance. This contribution becomes significant when measuring solutions with high concentrations, which have very low real impedance. Because some small modifications of this portion of the cell were done, the lead resistance contribution needed to be accurately redetermined. This was done as follows. The lead resistance calculated using the original dimensions of the instrument along with the resistivity of Pt/20 % Rh (124.8 Ω per circular mil ft, Platinum Labware, Johnson Matthey Catalog Company Inc., 2007) was found to be identical with that reported by Méndez De Leo and Wood (0.039 Ω).39 This confirmed that using resistivities could be used to accurately calculate the lead resistance. The total length of tubing contributing to the leads with an outer diameter (o.d.) of 1.0 mm used for the in-flow tube in the original instrument was 76.2 mm. In the present cell used, this was decreased to a total of 38.1 mm by gold soldering the silver lead wire closer to the Pt/Rh cup. Additionally, a 50.0 mm length of tubing (0.50 mm i.d. and 1.0 mm o.d.) was replaced with thicker tubing with a larger cross sectional area (1.0 mm i.d. and 1.6 mm o.d.). This further reduced this correction even further than the original instrument to a final value of 0.020 Ω with an estimated uncertainty of ± 0.004 Ω leads correction at room temperature as calculated with the resistivity of Pt/Rh 20 %. To further increase the accuracy of this correction, we also included the temperature coefficient of resistance (0.0014 Ω·K−1, Platinum Labware, Johnson Matthey Catalog Company Inc., 2007). This was calculated to be no more than 0.005 Ω at the highest temperature. Temperature Control and Measurement. The temperature of the conductivity cell was controlled using three independent systems that together were capable of controlling temperature to ± 0.15 K over several hours. First, a large air oven containing the conductivity cell and inlet and exit tubes was used to maintain a constant temperature environment (∼5 ± 0.1) K below the temperature of the conductivity cell, using the PID controller (Omega CNi3254), which was connected to a Pt 1000 Ω sensor and a heating element located ∼5 cm 2417



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3. RESULTS 3.1. Experimental Conductivities. The experimental conductivity (specific conductance44) of the solution, κsolnexp, was calculated from Rs, the resistance of the solution at infinite frequency using the expression:

was capped. When preparing the solution, degassed and deionized water was introduced into an empty bottle purged with argon through the No-Ox tubing. Then the solutions to run were prepared by injecting a well-known mass of stock solution through the septum and mixing with a magnetic stirrer. A peristaltic pump under computer control was used to fill the sample loop with each solution from the sample bottle. After AC impedance data were collected for the full series of solutions of NaCl at each temperature, the last most concentrated sample was followed by a long injection of deionized and degassed water from the main reservoir, typically 1000 mL, to rinse the equipment until the cell conductance had returned to its baseline value. A typical run for an electrolyte solution at given concentration, temperature, and pressure took about 1.5 h. 2.3. AC Impedance Measurements. The complex impedance Z(ω) of solutions in the conductance cell were measured at frequencies of (100, 200, 500, 1000, 2000, 5000, 10 000, 20 000, and 100 000) Hz using a programmable automatic RCL meter (Fluke model PM6304C) to obtain both the real ZRe(ω) and imaginary ZIm(ω) components of the impedance spectrum at each angular frequency Z(ω) = Z Re(ω) − j ·Z Im(ω)

exp κsoln = kcell /R s

The cell constant kcell was determined before kcell = 0.06529 ± 0.0002 and after kcell = 0.06659 ± 0.0002 temperature cycling by measuring the conductivity for a series of five KCl standard solutions (10−4 to 10−2 mol·kg−1) at 298.15 K and 20.00 MPa at the same frequency settings as the test solutions, using equations given by Barthel and co-workers for KCl(aq).45 The initial value of kcell was used in all calculations. Following the procedure used by previous workers,1,2,35,36,39,46,47 the dependence of the cell constant on temperature was calculated from the cell geometry and the thermal expansion coefficient of platinum. The change in kcell from (298 to 623) K is 0.3 %. The experimental conductivities of the electrolyte solutions, κsolnexp, were corrected for impurities within the solvent and the selfionization of water by subtracting the experimental values for H2O, κwexp, for each run, using the method of Sharygin et al.:35

(1)

exp κ = κsoln − κ wexp

where j2 = −1, ω = 2πf,41 ω is the angular frequency, and f is the frequency. Eighty measurements or more were taken with a computer over a time span of 50 min or more. The relative standard deviation of ZRe(ω) for the salt solutions was between (0.1 and 0.3) %; larger for the more concentrated solutions and at the higher temperatures. The relative standard deviation of ZIm(ω) for the salt solutions was between (0.1 and 0.5) %, except for a few cases at the highest concentrations and temperatures with relative standard deviations from (1 to 3) %. For the solvent measurements, the relative standard deviation for ZRe(ω) was usually between 1 and 3 % except at 623 K where the relative standard deviation for ZRe(ω) ranged between (3 and 16) % at the different frequencies. These were generally larger at the lower frequencies, but there was not a clear trend in this regard. The solvent resistance was found by linearly extrapolating the square of the three lowest frequencies, (100, 200, and 500) Hz, to zero frequency. The impedance measurements on NaCl(aq) reported here and experience in previous studies of other solutes to 623 K,38 suggest that the calculated values of true solution resistance Rs obtained from high-temperature conductivity cells with concentric cylindrical electrodes can be dependent upon how the impedance measurements were extrapolated and that the associated statistical uncertainty in Rs may become larger as the concentration is increased. These extrapolation methods fall into two general categories. The first category uses equivalent circuit models to represent the angular frequency dependence of the complex impedance.41,42 The second category is based on extrapolations involving ZRe(ω) and frequency ω. Following the detailed study by Zimmerman et al.,43 a generalized form of the expression for Warburg impedance, Z Re(ω) = R s + b1·ω−n

(3)

(4)

where the conductivities κ, κsoln , and are in SI units of S·m−1. In this study, these were converted to units of S·cm−1. Theoretical conductivity equations make use of the equivalent conductivity,44 of the solution, Λexp, which is defined as follows: κ Λexp = (5) N exp

κwexp

Here, N is the normality or equivalent concentration in moleq·L−1, giving the expression: N=

∑ cMc zMc = ∑ cXa |zXa |

(6)

where cM, cX and zM, zX are the molarities and the charge of the cations Mz+ and the anions Xz−, respectively. The molarity, ci, was calculated as follows: ci =

1000·mi ·ρs mi ·Mi + 1000

(7)

Here, ci, mi, and Mi are respectively the molarity, the molality, and the molecular weight of the species i, and ρs is the density of the solution. In the case of symmetrical 1−1 electrolytes like NaCl, the molarity and normality are identical, so there is no difference between equivalent and molar conductivity. The calculation of equivalent conductivities, Λexp, requires us to convert molalities to normalities, according to eqs 5−7. This calculation requires accurate solution densities. Often this is done for dilute solutions by assuming that the density of the solution is equal to that of pure water, for which accurate pVT data formulations are known.48 However, at the high concentrations required for this study the densities deviate substantially from those of pure liquid water at temperatures above ∼570 K.49 Archer50 has reported an accurate equation of state for NaCl(aq) that includes concentrations up to the solubility limit and temperatures up to 604.4 K and pressures to 100 MPa. The data treatment strategy we have adopted is based on two approaches to calculate solution densities. The first used the Archer equation of state for NaCl including extrapolated results

(2)

yields best agreement with accurate low temperature literature data for electrolyte solutions, over a wide range of concentrations in this type of cell. Here Rs is the solution resistance that we seek; b1 and the exponential term n are fitting parameters.38 This was the procedure adopted for the present study. 2418



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Table 2. Molality, Concentration, Conductivity, Experimental Equivalent Conductivity, and Equivalent Conductivities Calculated from the FHFP and the TBBK Equations from (298 to 623) K at p = 20 MPa for Aqueous Solutions of NaCl m·103 mol·kg

−1

c·103 mol·L

0 1.100 9.412 105.7 53.20 493.8

0 1.106 9.462 106.1 53.45 492.3

0 1.722

1.665

0 0.9597 0.9597 2.163 3.847 5.738 10.85

0 0.8705 0.8705 1.961 3.489 5.204 9.840

0 1.143 2.360 3.861 6.335 8.782

0 0.9662 1.994 3.263 5.353 7.422

0 1.084 2.299 3.462 4.768 8.782

0 0.8447 1.792 2.699 3.717 6.847

0 0.2259 0.6975 1.591 1.591 4.768 8.782 53.20 55.01 86.98 170.3

0 0.07312 0.1965 0.7210 1.837 7.110

−1

0 0.1531 0.4729 1.079 1.079 3.235 5.962 36.32 37.56 59.62 117.8

0 0.04360 0.1172130.01 0.4300 1.096 4.253

κsolnexp·106 S·cm

−1

Λexp

ΛTBBK −1

S·cm ·mol 2

T = 298.14 K, p = 19.27 MPa, ρw = 1005.53 kg·m−3 κwexp = 2.77·10−7 S·cm−1, κwth = 0.60·10−7 S·cm−1 125.9 136.35 123.0 ± 0.5 123.1 1120.2 118.4 ± 0.6 118.3 11277 106.3 ± 1.0 106.1 5903.4 110.4 ± 0.8 110.5 46504 94.5 ± 3.0 93.9 T = 374.92 K, p = 21.00 MPa, ρw = 966.65 kg·m−3 κwexp = 14.51·10−7 S·cm−1, κwth = 9.33·10−7 S·cm−1 367.3 593.60 355.7 ± 1.6 355.7 T = 445.79 K, p = 20.93 MPa, ρw = 907.01 kg·m−3 κwexp = 34.82·10−7 S·cm−1, κwth = 27.45·10−7 S·cm−1 609.2 518.34 591.5 ± 2.6 590.9 519.00 592.2 ± 2.6 590.9 1140.5 579.7 ± 2.6 581.5 1996.1 571.1 ± 2.6 571.9 2936.3 563.6 ± 2.6 563.3 5379.2 546.3 ± 2.6 545.9 T = 501.82 K, p = 20.93 MPa, ρw = 845.03 kg·m−3 κwexp = 57.71·10−7 S·cm−1, κwth = 37.76·10−7 S·cm−1 783.2 733.28 753.0 ± 3.3 753.6 1480.6 739.5 ± 3.3 739.7 2378.9 727.3 ± 3.3 726.7 3814.3 711.4 ± 3.3 710.0 5166.1 695.3 ± 3.3 696.6 T = 548.11 K, p = 20.53 MPa, ρw = 779.44 kg·m−3 κwexp = 44.06·10−7 S·cm−1, κwth = 36.18·10−7 S·cm−1 907.4 737.95 868.4 ± 3.9 868.6 1525.6 848.8 ± 3.8 849.1 2257.9 834.9 ± 3.8 834.7 3059.8 822.0 ± 3.8 821.3 5407.6 789.1 ± 3.8 789.5 T = 599.45 K, p = 20.51 MPa, ρw = 677.92 kg·m−3 κwexp = 26.21·10−7 S·cm−1, κwth = 23.20·10−7 S·cm−1 1033.8 157.56 1011.8 ± 4.6 1011.1 469.58 987.4 ± 4.4 990.7 1047.1 968.4 ± 4.3 966.3 1042.0 963.6 ± 4.3 966.3 2975.7 919.1 ± 4.2 913.7 5190.3 870.2 ± 4.2 871.8 25309 696.7 ± 4.2 700.2 26209 697.7 ± 4.2 696.7 38734 649.7 ± 4.2 647.7 68119 578.3 ± 4.2 578.2 T = 624.79 K, p = 20.41 MPa, ρw = 596.18 kg·m−3 κwexp = 14.98·10−7 S·cm−1, κwth = 12.95·10−7 S·cm−1 1135.7 50.049 1113.5 ± 5.3 1120.2 130.01 1096.7 ± 4.9 1104.8 462.09 1071.1 ± 4.7 1068.4 1129.6 1029.3 ± 4.6 1020.7 3859.3 907.1 ± 4.2 905.1 2419

ΛFHFP −1

S·cm ·mol 2

S·cm2·mol−1

κDH·qB·α1/2a

126.1 123.2 118.2 106.2 110.4

0.04 0.11 0.38 0.27

367.5 355.7

0.06

609.6 591.1 591.1 581.4 571.7 563.2 546.1

0.05 0.05 0.08 0.10 0.12 0.17

784.3 753.8 739.6 726.5 709.9 696.8

0.07 0.10 0.12 0.16 0.18

909.3 868.9 849.0 834.5 821.1 789.7

0.08 0.11 0.14 0.16 0.22

1035.9 1011.1 990.5 965.8 965.8 913.8 873.4

0.05 0.08 0.13 0.13 0.22 0.29

1135.9 1116.8 1102.6 1066.4 1019.7 908.9

0.03 0.06 0.11 0.17 0.32 0.36



Article

Table 2. continued m·103

c·103

κsolnexp·106

mol·kg−1

mol·L−1

S·cm−1 κwexp

8.782 10.40 23.16 53.20 170.3

5.257 6.235400. 13.94 32.36 107.7

Λexp S·cm2·mol−1 −7

= 14.98·10 4636.2 5400.5 10682 21510 56941

−1

κwth

S·cm , 881.7 866.9 766.3 664.6 528.8

± ± ± ± ±

= 12.95·10 4.2 4.2 4.2 4.2 4.2

−7

ΛTBBK

ΛFHFP

S·cm2·mol−1

S·cm2·mol−1

κDH·qB·α1/2a

886.5 867.7 764.2

0.38 0.55 0.03

S·cm

−1

882.6 862.9 765.6 666.8 542.2

a Values of κDH·qB·α1/2 > 0.5 may exceed the range of validity of the FHFP model,60 where κDH is the Debye−Hückel reciprocal length and qB is the Bjerrum distance.

at temperatures above 604.4 K. The second used a modified version of the Helgeson−Kirkham−Flowers−Tanger (HKF) model for standard partial molar volumes of aqueous ions and ion-pairs,51−54 together with the NIST formulation for the density of water,48 to calculate solution densities. Solution densities, ρs, were calculated from the molar volume, Vs, using the relationships, ρs =

Vs =

exp (T av , κsoln

1000 + ρw

(8)

∑ mi ·Vi0 + 1000·A v i

I ·ln(1 + b· I ) b

p )=

exp κsoln (T av , pav ) ·ηw (T , p)

ηw (T av , pav )

(10)

where ηw is the viscosity of water. The corrections were usually 0.2 % or less with the exception of two points at 598 K (0.41 and 0.34 %) and two points at 623 K (0.85 and 0.31 %). Experimental equivalent conductivities, Λexp, for aqueous av av sodium chloride, calculated from κexp soln(T , p ), are tabulated in Table 2 and plotted against molarity, expressed as c1/2, in Figure 3. As noted above, the calculation of c and Λexp at temperatures

∑i mi ·Mi Vs

av

(9)

Vi0

Here, mi, Mi, and are the molality, molar mass, and standard partial molar volume of each species i; ρw is the density of water density; Av, in units of m3·mol−1, is the Debye−Hückel limiting slope for the apparent molar volume at the experimental temperature and pressure;38,50 and Pitzer parameter b = 1.2 kg1/2·mol−1/2. Values for ρw and Av were calculated from the equation of state for the density and dielectric constant of water reported by Wagner and Pruss48 and Fernandez et al.,55 respectively. Standard partial molar volumes were calculated using the HKF parameters for Na+(aq), Cl−(aq), and NaCl0(aq) reported by Sverjensky et al.,54 except that our Born function used the water dielectric constant formulation reported by Fernandez et al.,55 as recommended by the National Institute of Standards and Technology (NIST). The calculation required knowledge of solution speciation, which we determined in an iterative process by fitting the TBBK model,56 to our experimental conductivity data. Details are presented below in Section 4. For the highest molality of NaCl measured at 599.45 K, the concentration calculated with densities from Archer’s equation is 0.1172 versus 0.1178 mol·L−1 with HKF, a difference of about 0.5 %. This becomes much larger at 624.79 K where using densities from Archer’s equation gives 0.1040 versus 0.1077 mol·L−1, a difference of 2.5 %. We chose to use the treatment based on the HKF model, for consistency with studies of the strontium salts which are ongoing in our laboratory.57 The experimental quantities resulting from the measurements described in Section 2 are conductivities and solution molalities. The values obtained for our different aqueous solutions of NaCl are reported in Table 2, along with the average temperature (IPTS-90) and pressure. The effects of small differences in temperature and pressure between av measurements on the experimental conductivities, κexp soln(T , av p ), have been corrected to the average temperature and pressure by assuming that:

Figure 3. Experimental equivalent conductivity of aqueous NaCl from (298 to 625) K at p = 20 MPa: ⧫, 298 K; □, 446 K; ▲, 502 K; △, 548 K; ◊, 599 K; ○, 625 K; , TBBK fit.

above 570 K used densities estimated from the HKF model for standard partial molar volumes and speciation obtained from the TBBK conductivity models described in Section 4. Parameters derived from the treatments in Section 4 are also reported in Table 2. 3.2. Uncertainty Analysis. There are several sources of uncertainty that were considered in estimating the total uncertainty of the conductivity measurements. These will be discussed in terms of percent of the reported molar conductivities, Λ. All values cited below are for 623 K where the uncertainties were the largest. The solution uncertainty in the solution preparation and subsequent dilutions done by mass were deemed to be negligible in comparison with the other sources of uncertainty. The uncertainty introduced by the measurement precision of the solvent correction was also negligible except at the lowest concentrations (maximally 0.15 2420



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Table 3. Experimental Association Constants, KA,m, Determined When Fitting the Data with the FHFP and TBBK Equations species 0

NaCl NaCl0 NaCl0 NaCl0 NaCl0

T/K

p/MPa

ρw/kg·m−3

pKw

445.79 501.82 548.11 599.45 624.79

20.93 20.93 20.53 20.51 20.41

907.01 845.03 779.44 677.92 596.18

11.371 11.109 11.096 11.361 11.750

%). The uncertainty in the leads correction was negligible except at the highest concentrations where it was estimated to be maximally 0.53 %, the second largest contributor. An uncertainty in solution densities results in an uncertainty in the conversion of molality to molarity, which then propagates into Λ. An uncertainty based on a linear function of molality was used to estimate this uncertainty. An uncertainty of ± 0.1 % was assigned at infinite dilution and an uncertainty of ± 0.3 % at 0.3 mol·kg−1. From the standard deviations of the impedance measurements, an error of 0.3 % was assigned to the conductance measurements. The largest uncertainty for the most concentrated solutions is from the uncertainty in the determination of Rs. A linear dependence on the solution molality based on a comparison of the 298.14 K NaCl results was used to estimate this extrapolation uncertainty.43 This linear dependence was assigned an extrapolation uncertainty of 0.1 % to infinite dilution and an uncertainty of 3.0 % to a 0.5 m solution at all temperatures. The assigned uncertainty was found by calculating the root-mean-square of all assigned uncertainties. Although other authors have observed systematic errors of up to 2 % due to cell constant changes that occurred during the temperature cycling,1,34 the drift in the cell constant during the entire course of our measurements (before and after temperature cycling) was 1.99 %.

3.4 5.9 9.4 14.9 37.7

± ± ± ± ±

+ + Cl−(aq) ⇌ NaCl0(aq) Na(aq) αc

αc

(1 − α)c

KA,m (TBBK)

1.0 1.7 1.0 3.5 3.3

3.4 5.6 8.4 12.5 33.8

KA, c =

± ± ± ± ±

1.2 1.5 0.66 0.71 2.5

1−α α ·(c /c°) ·γc,2± 2

(12)

where KA,c is the thermodynamic constant of association expressed in the molarity scale; α is the degree of dissociation; and c° is the hypothetical 1 mol·L−1 standard state. The mean activity coefficient of the electrolyte γc,± is described by the Debye−Hückel limiting law (eq 13) ln γc, ± =

−κ DH·qB ·α1/2 1 + κ DH·a ·α1/2

(13)

where κDH is the reciprocal radius of the ionic atmosphere; a is the distance of closest approach; and qB is the Bjerrum distance. When using an equilibrium association constant as an adjustable parameter, the FHFP expression for molar conductivity takes the form Λ = α{Λ° − S(α · c)1/2 + E(α ·c)ln(α ·c) + J1(α ·c) − J2 (α ·c)3/2 }

(14)

Equation 14 is usually fitted to the concentration-dependent molar conductivities either by adjusting Λ° and KA,c; or Λ°, KA,c, and J2. Once obtained, the thermodynamic association constant KA,c can be converted from the molarity scale to the molality scale through the solvent density, ρw.

4. THEORETICAL CONDUCTIVITY EQUATIONS AND DATA TREATMENT STRATEGY 4.1. FHFP Conductivity Model. In the analysis of our conductivity data, we used two treatments: (i) the Fuoss− Hsia−Fernandez−Prini (FHFP) conductivity model,58,59 and (ii) the Turq−Blum−Bernard−Kunz (TBBK) conductivity model.56 Both models are defined in the molarity scale, so we needed to convert our molalities into molarities, knowing the density of the solution as described above. Also, under hydrothermal conditions, the sodium and chloride ions are known to associate to form a neutral ion pair, NaCl0. The reason for using both approaches is to confirm that the values for the thermodynamic constant of ion-pair formation, KA,NaCl°, are model-independent, within the Bjerrum definition of ionpair formation.35,58−60 We began with the FHFP model, which is simpler to implement and does not require independent values for the limiting conductivities, λ°(Na+) and λ°(Cl−). The FHFP conductivity model58,59 is the classical polynomial expansion of the Fuoss−Hsia equation for the special case of symmetrical electrolytes: Λ = Λ° − S · c1/2 + E ·c·ln(c) + J1 ·c − J2 ·c 3/2

KA,m (FHFP)

KA,c = ρw ·KA,m

(15)

Following the recommendation of Justice,60 the distance of closest approach, a, was set equal to the Bjerrum distance, qB, and we only considered data at concentrations where κDH·qB·α1/2 was less than 0.5. This criterion was followed with only one exception: at 625 K we included one point with κDH·qB·α1/2 = 0.55. To fit the FHFP model to the conductivity data in Table 2, we converted our molalities into molarities using the density of the solutions, ρNaCl, calculated from eq 8. This was done using parameters derived from the TBBK treatment below, yielding densities that agreed with values from the Archer equation of state,50 to within 0.5 % below 600 K and 2.5 % at 625 K and the highest concentration. We then treated our experimental data with the FHFP model using Λ°,FHFP and KA as fitting parameters, except at 298.14 K where Λ°,FHFP was the only parameter needed. The nonlinear least-squares regression was done using the Levenberg−Marquardt algorithm, where all points were weighted equally. Many regressions were performed, always with Λ° as a fitting parameter, assuming complete dissociation (eq 11) or partial association (eqs 12 to 14). For preliminary calculations with the FHFP model, threeparameter fits were performed assuming no association (Λ°, J1, and J2) and association (Λ°, KA,c, and J2), but there was usually at least one of the parameters that was not statistically significant or was physically impossible. Fits were also done

(11)

Here, S is the Onsager limiting slope, and the expressions for E, J1, and J2 are given by Fernández-Prini.59 The terms S, E, J1, and J2 all depend on Λ°. The J terms also depend on the distance of closest approach, a. Ion-pairing is represented by the expressions, 2421



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using Λ° as the only adjustable parameter, but there were always systematic deviations in the residuals accompanied by a significant degradation of the fit. These results led us to follow Fernandez-Prini,61 who showed that for slightly associated electrolytes that KA,c and J2 are correlated and argued that in these cases it is best to adjust either Λ° and KA,c and use the theoretical value for J1 and J2 or to adjust Λ° and J2 with no KA,c. It was found that, when adjusting Λ° and KA,c, the standard deviations of the fit were always smaller by at least a factor of 2 for every case except one (sodium chloride at 502 K) than when using Λ° and J2 as the fit parameters, and so this was considered to be the best way to regress the present results. Λ° and KA,c converted in KA,m(FHFP) are reported in Tables 2 and 3. The residuals from the regressions calculated using the fit parameters Λ° and KA,c were found to be random. They are plotted in Figure 4.

crystallographic radii were used for the hard sphere contribution. For all neutral species, the activity coefficients were set equal to unity. To define the theoretical conductivity of the electrolyte solution, κ + κwth, we used the mixing rule recommended by Wood and co-workers:35,36,38,39 Nc

κ[Ic , Γc] = N

M=1 X=1

c xM =

a = xM

c Xa|z Xa| N

(19)

The TBBK model also requires values for the ionic radii and the limiting ionic molar conductivity of one species at the experimental temperature and pressure. For single ions, ionic radii were set equal to the crystallographic ionic radii, which were taken from Marcus.67 The radius of an ion pair was estimated using the cube root relationship adopted by Hnedkovsky et al. and others:36,38,39 3 3 1/3 + + r −) r NaCl0 = (r Na Cl

(21)

The values for the limiting equivalent conductivity at infinite dilution were taken from Marshall9 for H+ and from Ho et al.68 for OH− (except at 295 K, the value used was taken from Marshall9). As described in the section below, the value of λ°(Cl−) was fixed, while λ°(Na+) was used as a fitting parameter. As was the case for the FHFP model, all of our solutions were fitted with the TBBK equations for unassociated electrolytes before choosing whether to include an association constant. In the iterative regression, molar conductivities and/ or association constants were regressed by a nonlinear, leastsquares technique (Levenberg−Marquardt algorithm), where the weights in the fit were estimated from statistical uncertainties of the experimental impedance measurements. First, we used as a first approximation the density of pure water to convert molality into molarity to calculate the experimental molar conductivities, Λexp,app(NaCl). Then the density of each solution was obtained during the process of fitting the TBBK equation to our experimental data, following the iterative procedure of Zimmerman et al.57 Initially, we used Marshall’s formulation for the single ion molar conductivity of the chloride ion,9 λ°(Cl−)M, while the molar conductivity of the sodium ion, λ°(Na+), was used as a fitting parameter. Later, we

(16)

where λi and λi° are the ionic molar conductivities of the ion i in solution and at infinite dilution. The electrophoretic and relaxation contributions, δνiel/νi0 and δX/X, were solved using the mean spherical approximation (MSA). Equations were taken from the original paper,56 considering important misprints found by other authors.35,63 The activity coefficients of the ionic species in this model were calculated using the MSA according to:56,64 γeli

c c cM zM N

where Ic is the molar ionic strength, Γc is the MSA shielding parameter, N is the equivalent concentration, and xcM and xaM are the equivalent fractions of species in solution, and the sums are over all cations M and all anions X. The equivalent conductivity of the pure electrolyte, ΛMX[Ic,Γc], is calculated at the molar ionic strength of the mixture, Ic = (∑cMz2M + ∑cXz2X)/2. In these calculations, the properties of water (density, ρw; viscosity, ηw; and static dielectric constant, ε) were calculated from the equations of state reported by Wagner and Pruss,48 Huber et al.,65 and Fernandez et al.,55 respectively. Values for the ionization constant of water, Kw (eq 20), were calculated from Bandura and Lvov:66 a +·a − + − H 2O ⇌ H(aq) + OH(aq) K w = H OH a H 2O (20)

4.2. TBBK Conductivity Model. The TBBK conductivity model56 was used to treat our experimental conductivity data, following the procedure used by Sharygin et al.,35 and Méndez De Leo and Wood.39 The TBBK equation is derived from the Fuoss−Onsager continuity equation:62

log γi = log γiel + log γi HS

(18)

and

Figure 4. Percent difference of the residuals versus the square root of the concentration of NaCl from (298 to 625) K at p = 20 MPa from the FHFP equation: ◊, 298 K; □, 446 K; △, 502 K; ⧫, 548 K; ▲, 599 K; ●, 625 K.

⎛ δν el ⎞⎛ δX ⎞⎟ λi = λi◦⎜⎜1 + i0 ⎟⎟⎜1 + ⎝ X⎠ νi ⎠ ⎝

Na

∑ ∑ xMcx Xa ΛMX[Ic , Γc]

(17)

γHS i 35

where is the Coulombic contribution and is the hardsphere contribution. Following Sharygin et al., the Bjerrum distance was used to calculate the Coulombic contribution and 2422



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Table 4. Experimental Equivalent Single Ion Conductivities of Aqueous Na+ and Cl− and Auxiliary Parameters Used with the TBBK Equation ρw T/K

p/MPa

298.14 374.92 445.79 501.82 548.11 599.45 624.79

19.27 21.00 20.93 20.93 20.53 20.51 20.41

kg·m

λ°(Cl−)

λ°(Na+) −3

1005.53 966.65 907.01 845.03 779.44 677.92 596.18

pKw 13.928 12.151 11.371 11.109 11.096 11.361 11.750

2

−1

49.5 158.8 261.3 342.0 400.9 471.5 531.5

± 0.1

S·cm ·mol

used the improved value for λ°(Cl−) obtained by methods described in the following section. As a result of this iterative process, we obtained the density of the solution, ρNaCl, and ΛTBBK(NaCl), λ°(Na+), and the association constant KA. Results from the TBBK fits are tabulated in Tables 2, 3, and 4. The residuals from the regressions calculated using the fit parameters Λ° and KA,m, plotted in Figure 5, were also found to be random.

± ± ± ± ±

2.4 4.0 1.7 3.5 7.8

76.4 208.4 347.8 441.2 506.5 562.2 604.3

λ°(OH−)

λ°(H+) −1

S·cm ·mol 2

−1

S·cm ·mol 2

349.2 638.0 788.3 855.6 884.2 906.4 912.6

S·cm2·mol−1 197.7 440.3 603.0 706.4 770.6 819.0 840.0

⎛ A ⎞ log Λ° = log A1 + ⎜⎜A 2 + 3 ⎟⎟ ·log ηw ρw ⎠ ⎝

(22)

where ηw is the solvent viscosity in Poise; and A1, A2, and A3 are fitting parameters found by weighted least-squares regression. In a preliminary calculation, we chose to fit this model to our experimental values for Λ° as well as a selection of the literature values. These included low-pressure data at 0.1 MPa below 373 K,69−71 and data at less than 60 MPa between 373 and 673 K.1,2,4,5,15,17,34,35,37,72 It was further discovered that adding the results of Quist and Marshall5 between (723 and 1073) K (with densities of 600 kg·m−3 and greater) did not seriously degrade exploratory regressions, so these data were also included in the fit. Initially the weights assigned to the limiting molar conductivity data were 0.1 % for room temperature room pressure data; 1 % for the data of Noyes as recalculated by Wright et al.,72 and all of the flow results below 473 K; 3 % for the results of Noyes above 473 K,72 the results of Pearson et al.,4 and all other flow results; 5 % for Lukashov et al.,37 and 10 % for Quist and Marshall.5 After these preliminary regressions, it was found that the flow results of Ho et al.33 at (573 and 623) K and those of Lukashov37 were somewhat discordant with the results of Noyes,72 Pearson et al.,4 and all other flow measurements. These were subsequently reweighted in the fit to 10 %. The results of Quist and Marshall5 were chosen over those of Ho et al.33 because graphs of Λ versus c1/2 for these results have significantly less scatter and go to higher temperatures. The data used in the regression as well as the results obtained are tabulated in Table S.2 of the Supporting Information and shown in Figure 6. The fitted parameters for eq 22 are given in Table 5. We note that eq 22 is similar to that reported by Longinotti and Corti73 with the addition of the A3/ ρw term. In the fits reported here and below, the A3 parameter was found to be statistically significant. 5.2. Single-Ion Limiting Conductivities. One goal of this work was to obtain accurate expressions for the single ion limiting conductivities of the sodium and chloride ions, λ°(Na+) and λ°(Cl−), for use in subsequent studies in which the TBBK model is used to interpret conductivity data under hydrothermal conditions. Literature values for dependence of limiting equivalent conductivities λ° for ions under hydrothermal conditions are all based on the work of Smith and Dismukes,20,21 who measured the transport properties of sodium and potassium chloride solutions to 398 K. They reported that the logarithm of the ratio of the transference number for chloride t(Cl−) to sodium t(Na+) was a linear function of the reciprocal absolute temperature:

Figure 5. Percent difference of the residuals versus the square root of the concentration of NaCl from (298 to 625) K at p = 20 MPa from the TBBK model: ◊, 298 K; □, 446 K; △, 502 K; ⧫, 548 K; ▲, 599 K; ●, 625 K.

5. IONIC TRANSPORT PROPERTIES AND ASSOCIATION CONSTANTS 5.1. Electrolyte Limiting Conductivities. Molar conductivities at infinite dilution for sodium chloride, obtained by fitting both the FHFP and TBBK models to our experimental conductivities, are reported in Table 2. The values from the two models agreed with the data to within the experimental uncertainty and agreed with one another to within 1 % or better. Recently, Zimmerman et al.15,16 reported that Λ° could be represented accurately by simple empirical functions of the solvent viscosity and density. It was found that the present sodium chloride results and those available in the literature could be fitted, if the viscosity was greater than 0.00055 Poise, using an empirical equation, which takes the form:

log(tCl−/t Na+) = log[tCl−/(1 − tCl−)] = A + B /T 2423

(23)



Article

where Λ°FHFP(NaCl) is the limiting molar conductivity of NaCl obtained by fitting the FHFP model to our conductivity data. The application of the TBBK model to our experimental data, reported in Section 4.2, was carried out by using the values of λ°(Cl−) calculated from Λ°FHFP(NaCl) and treating λ°(Na+) as a fitting parameter. Limiting conductivities Λ°(NaCl) from the FHFP and TBBK models are listed in Table 2. Results from the two models agree to within 1 %. The limiting conductivities, λ°(Na+) and λ°(Cl−), are tabulated in Table 4. Two approaches were used to model these results. The first was based on Marshall’s reduced density relationship.9 Briefly, the correlation is based on a linear dependency of limiting ionic equivalent conductivity with density at constant temperature: ⎛ ρ ⎞ λ° = λ°°·⎜⎜1 + w ⎟⎟ ρh ⎠ ⎝

where λ°° and ρh are hypothetical (extrapolated) values of the limiting equivalent conductivity and solvent density ρw at a given temperature which correspond to values of ρ = 0 and λ° = 0; and ρw/ρh is the reduced density. In our treatment, values of λ°° were calculated from coefficients given by Marshall.9 The term ρh was used as an adjustable parameter to model the TBBK data in Table 4 and was found to have a linear dependency with the temperature from (445 to 625) K. Values of ρh for Na+ and Cl− for each experimental temperature are tabulated in Table 6. At temperatures below 445 K, values of λ°

Figure 6. Experimental molar conductivity of aqueous solutions of NaCl at infinite dilution as a function of the viscosity of water: ○, experimental literature data points; ⧫, this work; , the fitted viscosity−density correlation (eq 22).

Quist and Marshall18 assumed that this extrapolation for t(Cl−) was accurate at all temperatures, corrected these values to infinite dilution using the procedure of Smith and Dismukes,21 and used the resulting expression for t°(Cl−) to calculate λ°(Cl−) from the experimental values of Λ°(NaCl) reported by Noyes and Coolidge19 and their own data.5,31 Oelkers and Helgeson13 used this same extrapolation for predicting conductivity data up to 1073 K. Values of t(Cl−) from eq 23 using the parameters reported by Smith and Dismukes have also been used by Smolyakov and Veselova11,12 and Anderko and Lencka74 to obtain values of λ°(Cl−) up to (473 and 573) K, respectively. Hnedkovsky et al.38 have used this formulation for t°(Cl−) to interpret conductivity data at temperatures as high as 673 K at 28 MPa. Following Hnedkovsky et al.,38 we have assumed that the transference number t°(Cl−), as calculated from Quist and Marshall’s equation, was correct. This transference number was then used to calculate λ°(Cl−) from our measurements, λ°(Cl−) = Λ°FHFP(NaCl)·t °(Cl−)

(25)

Table 6. Fitted Parameters, ρh, for the Temperature Dependence of Limiting Ionic Equivalent Conductivities, λ°, According to the Marshall Reduced Density Model9 (eq 25) ρw

ρh(Na+)a

ρh(Cl−)b

T/K

p/MPa

kg·m−3

kg·m−3

kg·m−3

298.14 374.92 445.79 501.82 548.11 599.45 624.79

19.27 21.00 20.93 20.93 20.53 20.51 20.41

1005.53 966.65 907.01 845.03 779.44 677.92 596.18

1492 1517 1524 1489 1464 1438 1453

2023 1846 1968 1912 1905 1833 1809

ρh(Na+) = 1716 − 0.4443(T/K); (445 K ≤ T ≤ 625 K). bρh(Cl−) = 2357 − 0.8651(T/K); (445 K ≤ T ≤ 625 K).

a

(24)

Table 5. Fitted Parameters for the Temperature Dependence of Limiting Equivalent Conductivities, Λ°(NaCl), and Limiting Single-Ion Equivalent Conductivities, λ°(Na+) and λ°(Cl−), According to the Viscosity−Density Model (eqs 22, 26) species

Ntotala

Λ°(NaCl) λ°(Cl−) λ°(Na+)

7 7 7

species

Ntotala

Λ°(NaCl) λ°(Cl−) λ°(Na+)

110 110 110

A1

A2

Fit to the Experimental Results from This Work 1.447 ± 0.135 −0.990 ± 0.023 1.006 ± 0.203 −0.978 ± 0.049 0.464 ± 0.099 −1.028 ± 0.052 Validity Range: 298 K ≤ T ≤ 623 K; p = 20 MPa A1 A2 Fit to the Literature Database 1.673 ± 0.010 −0.939 ± 0.002 1.137 ± 0.006 −0.925 ± 0.002 0.553 ± 0.005 −0.965 ± 0.004 Validity Range: 278 K ≤ T ≤ 1073 K; p = (0.1 to 500) MPa

A3

ΔX/X·100b

45.34 ± 7.90 53.46 ± 16.92 37.04 ± 18.16

0.432 1.10 1.06

A3

ΔX/X·100b

22.76 ± 1.37 34.44 ± 1.24 10.18 ± 2.09

2.96 2.48 3.66

Ntotal is the total number of points included in the fit. bΔX/X is the average relative absolute deviation for the limiting conductivity data used in the fits. a

2424



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for each ion at 20 MPa were calculated by linear interpolation of the values of ρh given in Table 6. The results for λ°(Na+) and λ°(Cl−) are plotted in Figure 7.

Figure 8. Experimental values for the association constants of NaCl in H2O from (445 to 623) K at p = 20 MPa: ●, NaCl (fitted with FHFP); ○, NaCl (fitted with TBBK).

then fitted to these experimental association constants, with the selected literature data.

Figure 7. Experimental single ion conductivity at infinite dilution as a function of the viscosity of water: ○, λ°(Cl−) values from the literature database; ●, λ°(Cl−) values from this work; ◊, λ°(Na+) values from the literature database; ⧫, λ°(Na+) values from this work; , fit to the viscosity−density correlation (eq 26); ---, fit to the Marshall’s reduced density relationship (eq 25).

log KA , m = a +

(27)

Here the constants a to g are adjustable fitting parameters, and ρw is the density of water at the temperature and pressure of interest, taken from Wagner and Pruss.48 The regression of parameters a to g was done using a selection from 47 literature data points, chosen because they used apparently accurate instruments, and extended to high temperatures and low concentrations.1,2,4,26 We chose to use our results for KATBBK rather than KAFHFP in the fit, because the TBBK model is required for studies on asymmetric electrolytes that may make use of these results. Only three parameters were needed to fit the experimental data with a relative deviation in KA,m of less than ± 0.24. Adding more parameters did not lead to a statistically significant improvement. The results of four threeparameter fits are reported in Table 7. While the use of parameters a, b, and f yields a marginally higher mean relative absolute deviation, it is consistent with the simple model proposed by Anderson et al.78 which is conveniently used to calculate thermodynamic parameters for ion pair formation.77,78 The fit using a, b, and f is plotted in Figure 9, together with the experimental data points.

Our second approach was based on the single-ion form of eq 22 which describes the dependence of λ° as a function of the density and viscosity, ⎛ A ⎞ log λ° = log A1 + ⎜⎜A 2 + 3 ⎟⎟log ηw ρw ⎠ ⎝

⎡ g ⎤ f b c d + 2 + 3 + ⎢e + + 2 ⎥ ·log ρw ⎣ T T T T T ⎦

(26)

The results are compared with those of the Marshall reduced density model in Figure 7. Both models fit the data in Table 4, with similar precision. Attempts were made to extend these treatments to the entire database of literature values for λ°(Cl−) and λ°(Na+) in the Supporting Information, part 2; however, the viscosity−density relationship, eq 26, proved to be a more flexible model for reproducing low temperature results. The final fit was based on eq 26, using values from in the Supporting Information, part 2, corresponding to the same literature sources reported in Section 5.1 and the TBBK results in Table 4. The only exception was that the values for λ°(Cl−) reported by Bester et al.,71 and Robinson and Stokes75 were used for room temperature results. The same weights were used in this fit as in the fit for Λ°FHFP(NaCl). The results are plotted in Figure 7, and the fitted parameters are given in Table 5. 5.3. Association Constants. The association constants, KA, obtained from the FHFP and TBBK fits, tabulated in Table 3, are plotted in Figure 8. The values of KA from the two models were found to be almost the same, to within the combined statistical uncertainties of the fits. The small systematic difference likely results from the fact that the FHFP equation is not thought to be accurate at concentrations above 0.1 mol·L −1 , 76 so experimental data at higher concentrations were omitted from the FHFP fit, but included in the TBBK data treatment. The “density” model77 for temperature and pressure dependence of log K (eq 27) was

6. DISCUSSION AND CONCLUSIONS 6.1. Conductivities of Concentrated Solutions under Hydrothermal Conditions. The successful application of the data treatment procedures reported here serves as model for the study of other electrolyte systems over very wide range of temperatures and concentrations. First, the results reported in Table 2 demonstrate that the commonly used assumption that the solution density, ρ, can be replaced by the solvent density, ρw, in calculating the equivalent conductivity (eq 5) breaks down at concentrations above ∼10−2 mol·L−1 and temperatures above 300 °C. We have shown that the use of the speciation from the TBBK model, combined with estimates of partial molar volumes from HKF equation can yield acceptable 2425



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Table 7. Fitted Parameters for the Temperature and Density Dependence of Association Constants, KA, According to the Density Model,77 eq 27

a

fit no.

N

a

b·10−3

1 2 3 4

47 47 47 47

21.09 ± 0.54 −1.046 ± 0.647

0.825 ± 0.371 15.96 ± 0.88 15.25 ± 0.51

f·10−3

ΔK/Ka

−5.159 ± 0.195 −4.915 ± 0.130 0.281 ± 0.127

0.215 0.236 0.233 0.216

e −7.52 ± 0.27 −0.360 ± 0.219 −7.926 ± 0.422

22.29 ± 0.72

fit exp ΔK/K corresponds to 1/N∑Ni (|Kexp A,i − KA,i|)/(KA,i ).

Oelkers and Helgeson13,14 and Anderko and Lencka74 have used their equations to predict λ° for many other ions at high temperatures and pressures from room temperature measurements. The model reported here, eq 22, expresses Λ° as a simple function of the solvent viscosity and density. This is an extension of similar expressions by Zimmerman et al.15−17 These models have the advantage that they are able to reproduce the data below 523 K to within the estimated experimental uncertainties, unlike the equations of Marshall and Oelkers and Helgeson. The new values for λ°(Na+) and λ°(Cl−) tabulated in the Supporting Information, part 2, provide a basis for deconvoluting modern experimental values for Λ° of other electrolyte systems into their single-ion components. With an up-to-date set of values for λ°, more accurate parameters for the predictive models proposed by Marshall,9,10 Oelkers and Helgeson,13,14 and Anderko and Lencka74 can be developed, and the strengths and shortcomings of each approach can be critically evaluated. The viscosity− density relationship, represented by eq 26, is a promising new approach for correlating and predicting conductivities under hydrothermal conditions.

Figure 9. Experimental association constants of NaCl in water: ●, this work; ⧫, Ho et al.;2 △, Fogo et al.;26 ◊, Zimmerman et al.;1 ○, Pearson et al.;4 plain line, density model fit (eq 27).

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accurate solution densities under these conditions to address this problem if no experimental densities are available. The agreement between the association constants for NaCl° from the TBBK and FHFP models in Table 3 lies within the combined experimental uncertainties, confirming that both models yield accurate results. This confirms that our selection of ionic radii and other parameters for the TBBK model is appropriate. A similar approach will be used in studies now underway on other, more complex systems. The fitted parameters for Λ°(NaCl), λ°(Na+), and λ°(Cl−) in Table 5, together with the “density model” parameters for the association constants in Table 7, can be used with the FHFP equation to calculate the molar conductivities at concentrations up to 0.1 mol·L−1 (defined by eq 14, κDH·qB·α1/2 ≤ 0.5), at temperatures as high as 673 K, to within the estimated uncertainty of ∼ ± 3 % as listed in Table 5. 6.2. Correlations to Represent Single-Ion Limiting Conductivities. Several approaches have been used to represent and predict the temperature and pressure dependence of limiting equivalent conductivities. Fisher and Barnes8 expressed log Λ° as polynomials of the log of the solvent viscosity. Smolyakov and Veselova11,12 have used viscositybased expressions to correlate values of λ° up to 473 K. Their approach was extended by Anderko and Lencka,74 who found that, for some salts, this function accurately described conductivities to 573 K. Marshall9,10 devised a reduced state relationship, based on the observation that Λ° approaches a limiting value of zero, at a high value of density which is common to all electrolytes. Oelkers and Helgeson13,14 used an Arrhenius model for the conduction process. By using correlations involving the standard partial molar entropy,

ASSOCIATED CONTENT

S Supporting Information *

Calculation of solution density and literature values for the limiting equivalent conductivity of NaCl, Λ°, λ°(Na+), and λ°(Cl−). This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail address: [email protected]. Phone: 519-824-4120 ext. 56076. Fax: 519-766-1499. Funding

This research was supported by the National Science and Engineering Research Council of Canada (NSERC), Ontario Power Generation Ltd. (OPG), the University Network of Excellence in Nuclear Engineering (UNENE), and Bloomsburg University for sabbatical leave (G.H.Z.). G.H.Z. would like to express appreciation for the financial support provided by Fulbright Canada and gratitude for the support of the governments of Canada and the United States in making this program possible. Notes

The authors declare no competing financial interest. ⊥ E-mail address: [email protected]. § E-mail: [email protected].

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ACKNOWLEDGMENTS The authors express deep gratitude to Prof. Robert H. Wood, University of Delaware, for donating the AC conductance cell 2426



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(16) Zimmerman, G. H.; Scott, P. W.; Greynolds, W.; Mayorov, D. Conductance of Dilute Hydrochloric Acid Solutions to 458 K and 1.4 MPa. J. Solution Chem. 2009, 38, 499−512. (17) Erickson, K. M.; Arcis, H.; Raffa, D.; Zimmerman, G. H.; Tremaine, P. R. Deuterium Isotope Effects on the Ionization Constant of Acetic Acid in H2O and D2O by AC Conductance from 368 to 548 K at 20 MPa. J. Phys. Chem. B 2011, 115, 3038−3051. (18) Quist, A. S.; Marshall, W. L. Assignment of Limiting Equivalent Conductances for Single Ions to 400°. J. Phys. Chem. 1965, 69, 2984− 2987. (19) Noyes, A. A.; Coolidge, W. D. The electrical conductivity of aqueous solutions at high temperatures; I. Description of the apparatus. Results with NaCl and KCl up to 306 °C. Z. Phys. Chem. 1903, 46, 323−378. (20) Smith, J. E.; Dismukes, E. B. The Cation Transference Number in Aqueous Potassium Chloride at 70 to 115°. J. Phys. Chem. 1963, 67, 1160−1161. (21) Smith, J. E.; Dismukes, E. B. Transference Numbers in Aqueous Sodium Chloride at Elevated Temperatures. J. Phys. Chem. 1964, 68, 1603−1606. (22) Corti, H. R. Electrical Conductivity in Hydrothermal Binary and Ternary Systems. In Hydrothermal Properties of Materials; Valyashko, V., Ed.; John Wiley & Sons, Ltd.: New York, 2008; Chapter 4, pp 207−226. (23) Noyes, B. A.; Coolidge, W. D. The electrical conductivity of aqueous solutions at high temperatures. J. Am. Chem. Soc. 1904, 26, 134−170. (24) Noyes, A. A.; Melcher, A. C.; Cooper, H. C.; Eastman, G. W.; Kato, Y. The conductivity and ionization of salts, acids, and bases in aqueous solutions at high temperatures. J. Am. Chem. Soc. 1908, 30, 335−353. (25) Noyes, A. A.; Melcher, A. C.; Cooper, H. C.; Eastman, G. W. The conductivity and ionization of salts, acids, and bases in aqueous solutions at high temperatures. Z. Phys. Chem. 1910, 70, 335−337. (26) Fogo, J. K.; Benson, S. W.; Copeland, C. S. The Electrical Conductivity of Supercritical Solutions of Sodium Chloride and Water. J. Chem. Phys. 1954, 22, 212−216. (27) Rodnanskii, I. M.; Galinker, I. S. Electrical conductance of concentrated aqueous solutions of LiCl, NaCl and KCl at high temperatures. Dokl. Akad. Nauk. SSSR 1955, 105, 115−118. (28) Corwin, J. F.; Bayless, R. G.; Owen, G. E. The Conductivity of Dilute Sodium Chloride Solutions Under Supercritical Conditions. J. Phys. Chem. 1960, 64, 641−645. (29) Bannard, J. E. Effect of density on the electrical conductance of aqueous sodium chloride solutions. J. Appl. Electrochem. 1975, 5, 43− 53. (30) Polyakov, E. A. Electrical resistivity and density of aqueous salt solutions under high temperatures and pressures. Priklad. Geofiz. VNII 1965, 41, 163−180. (31) Quist, A. S.; Marshall, W. L. The Electrical Conductances of Some Alkali Metal Halides in Aqueous Solutions from 0 to 800° and at Pressures to 4000 bar. J. Phys. Chem. 1969, 73, 978−985. (32) Read, A. J. Ionization Constants of Benzoic Acid from 25 to 250 °C and to 2000 bar. J. Solution Chem. 1981, 10, 437−450. (33) Ho, P. C.; Palmer, D. A.; Mesmer, R. E. Electrical Conductivity Measurements of Aqueous Sodium Chloride Solutions to 600°C and 300 MPa. J. Solution Chem. 1994, 23, 997−1017. (34) Gruskiewicz, M. S.; Wood, R. H. Conductance of Dilute LiCl, NaCl, NaBr, and CsBr Solutions in Supercritical Water Using a Flow Conductance Cell. J. Phys. Chem. B 1997, 101, 6549−6559. (35) Sharygin, A. V.; Mokbel, I.; Xiao, C.; Wood, R. H. Tests of Equations for the Electrical Conductance of Electrolyte Mixtures: Measurements of Association of NaCl (Aq) and Na2SO4 (Aq) at High Temperatures. J. Phys. Chem. B 2001, 105, 229−237. (36) Sharygin, A. V.; Wood, R. H.; Zimmerman, G. H.; Balashov, V. N. Multiple Ion Association versus Redissociation in Aqueous NaCl and KCl at High Temperatures. J. Phys. Chem. B 2002, 106, 7121− 7134.

to the Hydrothermal Chemistry Laboratory at the University of Guelph, for providing us with the benefit of his extensive operating experience, and for many fruitful discussions. We are also grateful to Mr. Ian Renaud and Mr. Case Gielen of the electronics shop and machine shop in the College of Physical and Engineering Science at the University of Guelph, for their very considerable expertise in maintaining and modifying the instrument and its data acquisition system. Technical advice and encouragement were provided by Dr. Dave Guzonas, Atomic Energy of Canada Ltd.; Dr. Dave Evans, OPG, and Dr. Mike Upton, Bruce Power Ltd. G. H. Z. thanks Jason Derr for preliminary exploratory calculations that led to eq 26.

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Limiting Conductivities and Ion Association Constants of Aqueous

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