Limiting Conductivities of Univalent Cations and the Chloride Ion in H₂O and D₂O Under Hydrothermal Conditions

Abstract

Frequency-dependent electrical conductivities of aqueous sodium chloride, potassium chloride, cesium chloride, potassium iodide and cesium iodide have been measured in both H₂O and D₂O between T = 298 and 598 K at p ~ 20 MPa at a ionic strength of ~10⁻³ mol·kg⁻¹ using a high-precision flow-through AC electrical conductance instrument. Experimental values for the molar conductivity, Λ, of each electrolyte were used to calculate their molar conductivities at infinite dilution, Λ°, with the Fuoss–Hsia–Fernández-Prini conductivity model. Single-ion limiting conductivities for the chloride ion in H₂O, λ°(Cl⁻), were derived from Λ° by extrapolating literature values for the transference number of Cl⁻, t°(Cl⁻), in aqueous solutions of KCl and NaCl from ~400 and ~390 K up to the experimental conditions. Values for λ°(Cl⁻) in D₂O were determined from literature values of t°(Cl⁻) for KCl in D₂O near ambient conditions, assuming the same temperature dependence as in H₂O. The results were used to calculate values for the single ion limiting conductivities λ°(Na⁺), λ°(K⁺), λ°(Cs⁺), λ°(Cl⁻), and λ°(I⁻) in both light and heavy water. The values of λ° in D₂O are the first to be reported at temperatures above 338 K. The temperature dependence of the isotopic Walden product ratio, $(\lambda^\circ \eta )_{{{\text{D}}_{2} {\text{O}}}} /(\lambda {^\circ }\eta )_{{{\text{H}}_{2} {\text{O}}}}$, indicates that differences in the hydration of Cl⁻, K⁺ and Cs⁺ ions between light and heavy water at ambient conditions associated with hydrogen-bonding, the so-called “structure breaking” effects, largely disappear at temperatures above ~400 K. The value of $(\lambda^\circ \eta )_{{{\text{D}}_{2} {\text{O}}}} /(\lambda {^\circ }\eta )_{{{\text{H}}_{2} {\text{O}}}}$ for the “structure making” ion Na⁺ rises from 0.98 at 298.15 K to ~1.04 ± 0.02 at temperatures above ~375 K and remains approximately constant up to 600 K.

1 Introduction

Ionic conductivity measurements have played a pivotal role in investigations aimed at understanding and quantifying ionic hydration effects, ion association, and acid–base ionization in high-temperature water [1–4]. The pioneering studies by Noyes [5–8], Franck [9], Marshall [10, 11] and others have been reviewed [4, 12]. Briefly, these measurements yielded ionization constants, association constants and transport properties of simple anions and cations up to temperatures as high as 800 °C and pressures up to 400 MPa. Such measurements yield limiting equivalent conductivities for electrically neutral salts, Λ°(M⁺X⁻). In order to determine the contribution of individual ions, an experimental value for the transference number must be used to derive the limiting equivalent conductivity of one ion, usually λ°(Cl⁻). In a seminal paper, Marshall [13, 14] reported a reduced state relationship to predict limiting conductivities for simple mono- and divalent anions and cations from 273 to 1,073 K and up to 400 MPa, based on the limiting conductance data available at that time, largely from his own work. This calculation was based on values of λ°(Cl⁻) obtained by extrapolating Smith and Dismukes’ values [15] for the transference number of the chloride ion in NaCl(aq) which extended only up to 398 K.

Recent developments in the design and construction of static high temperature cells [16] and the introduction of high precision flow cells by Bianchi et al. [17], Wood and his co-workers [18–20] and Ho et al. [21–23] have yielded significant improvements in the speed and accuracy of conductivity measurements for very dilute solutions in the range 323–673 K at pressures up to 40 MPa. Limiting conductivities from measurements in high-precision cells have now been reported for the salts of the univalent ions H⁺, Li⁺, Na⁺, K⁺, Cs⁺, Cl⁻, Br⁻, and OH⁻ in light water over this range, as cited in the review by Corti et al. [2]. More recent measurements have yielded values of λ° for Li⁺, Na⁺, K⁺, Ca²⁺, ${\text{HSO}}_{4}^{ - } ,{\text{SO}}_{4}^{2 - }$, acetate and their ion pairs [20, 24–27]. These are all based on Marshall’s [13, 14] values for λ°(Cl⁻). In 2012, Zimmerman et al. [28] reported a correlation of critically reviewed modern values for Λ°(NaCl), from which new values of λ°(Cl⁻) were derived using Marshall’s extrapolated values of Smith and Dismukes’ [15] transference numbers for NaCl. Our recent values of λ° for Na⁺, Sr²⁺, triflate (${\text{CF}}_{3} {\text{SO}}_{3}^{ - }$) and their ion pairs are based on this correlation [29].

The thermodynamic and transport properties of ions in D₂O have been used to probe hydration effects near ambient conditions [30–35], and they are needed to model corrosion product transport and radiolysis effects in the primary coolant circuits of CANDU heavy-water nuclear reactors [36]. Despite their importance to basic research and the nuclear industry, only one study of experimental limiting conductivities in D₂O has been reported in the literature at temperatures above 373 K [37].

This paper reports experimental values for the conductivities of cations and anions of simple salts in both D₂O and H₂O at temperatures up to 600 K and a pressure of 20 MPa. The purpose of the study was to determine the limiting single-ion conductivities of univalent cations and anions in light and heavy water, up to near-critical conditions, with sufficient precision that differences due to deuterium isotope effects could be compared. The measurements were carried out in the high-precision flow AC conductance cell constructed at the University of Delaware [18, 20] and modified at the University of Guelph [28, 29, 38]. This instrument has the ability to work at the very low concentrations (≤10⁻³ mol·kg⁻¹) required to determine accurate limiting conductivities. In addition, the flow feature permits the use of experimental designs that eliminate most systematic errors so that differences in the ionic conductivities between different salts in each solvent, and between the same salts in light and heavy water, can be measured with a high degree of precision.

2 Experimental

2.1 Chemicals and Solution Preparation

Stock solutions of ~0.1 mol·kg⁻¹ NaCl (Alfa Aesar Puratonic, 99.999 %), KCl, (Alfa Aesar 99.95 %), CsCl (Alfa Aesar 99.999 %), KI (Alfa Aesar 99.9 %) and CsI (Alfa Aesar 99.999 %) in light and heavy water were prepared by mass from their anhydrous salts. Each salt was dried at 300 °C until the mass difference between weightings was less than 0.02 %. The stock solutions were prepared by mass with Nanopure light water (resistivity 18.2 MΩ cm) and heavy water provided by Ontario Power Generation Ltd. The D₂O used in solution preparation was determined to be 99.7 mol% D by using a ¹H NMR method, whereby standard additions of glacial HAc (Sigma Aldrich, “99.85 %”) were added to D₂O as an internal reference [39]. Stock solutions in both H₂O and D₂O were prepared in a glove-bag flushed with argon, which was continually kept under positive argon pressure, then stored under an atmosphere of argon until use. The solutions measured in the conductance instrument (~10⁻³ mol·kg⁻¹) were prepared by mass dilution under argon from these stock solutions. All solutions were stored and handled under an argon atmosphere to prevent bicarbonate contamination from atmospheric CO₂. The bottles were swirled manually to mix the solution and allowed to sit at least for 24 h before the runs. These sample solutions were stored in 250 mL Pyrex glass bottles with KIMAX GL-45 gas-tight tops for no more than 15 days before being injected into the conductance instrument. Each of the gas-tight lids consisted of a septum, an inlet gas tube and a connection port to the conductivity cell.

2.2 AC Conductivity Measurements

Conductivity experiments were carried out using the high-precision AC conductivity flow cell designed by R.H. Wood at the University of Delaware, and described elsewhere [28, 29]. Briefly, the cell consists of a long temperature-controlled platinum inlet tube that leads into a platinized cup that forms the outer electrode. The inner electrode is a platinum rod (electrodeposited with platinum black) that is a direct extension of the platinum tube which carries the exiting solution away from the cell. 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. The temperature of the cell is controlled using three independent systems, capable of regulating to ±0.15 K over several hours. The first system is an air bath that controls the temperature surrounding the conductivity cell to (5 ± 0.1) K below the temperature of the cell itself. The second system uses heating elements located in the cell body to control the cell temperature to ±0.01 K. The third system consists of a co-axial heater placed alongside the incoming platinum tube, and a thermocouple whose junctions are located 1 cm before the exit of the tube and in the cell body, to ensure that the difference between the temperature of the incoming fluid and the cell body is less than ±0.08 K. The temperature of the cell body was measured to an accuracy of ±0.02 K with a platinum resistance thermometer consisting of a Hart Scientific Model 5612 Probe and a Hart Scientific Model 5707 6½ digit digital multimeter.

Solutions to be analyzed were injected into the flow conductivity cell using the high-pressure liquid chromatography (HPLC) injection system [28, 29] shown in Fig. 1. The first HPLC pump was used to supply a continuous flow of de-gassed and de-ionized water to the instrument at a pressure set by a back-pressure regulator. Two 6-port valves, controlled through a computer, determined whether water from the reservoir flowed directly through the cell, or whether it pushed solution from an injection loop through the cell. The sample to be injected was loaded into the injection loop with a peristaltic pump. The second HPLC pump 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 by-passed the cell and flowed directly to the back-pressure regulator. Pressure was measured with a digital pressure transducer (Paroscientific Inc. Model 760-6K) to an accuracy of ±0.01 MPa. Once the sample was pressurized, the computer switched the 6-port valves to push the sample into the conductivity cell. Experiments were conducted at a flow rate of 0.5 cm³·min⁻¹.

Fig. 1

Schematic of the high pressure sample injection system: (1) Pyrex or Nalgene solution bottles; (2) peristaltic pump; (3) delay loop; (4) six-port injection valves; (5) HPLC pumps; (6) deionized water reservoir; (7) back pressure regulator; (8) temperature-controlled insulated air oven; (9) high temperature conductance cell; (10) pressure transducer; (11) waste solution reservoirs; (12) Argon tank (adapted from Zimmerman et al. [28])

Our study of deuterium isotope effects employed an experimental strategy in which solutions of sodium chloride, potassium chloride, potassium iodide, cesium iodide and cesium chloride, in both H₂O and D₂O, were injected in the sequence shown in Fig. 2, at each temperature, so that systematic errors in the experimental molar conductivities of solutions in the two solvents would cancel, thus yielding more precise values of the deuterium isotope effect on transport properties. Each series of measurements was started and finished by measuring the conductivity of pure solvents (H₂O and D₂O), and solutions of NaCl in H₂O and D₂O. Measurements were made at temperatures T = (298, 373, 423, 473, 522, 548, 573 and 598) K, at a constant applied pressure of p ~ 20 MPa. For each aqueous electrolyte solution, after AC impedance data were collected, a long injection of de-ionized and degassed water from the main reservoir, typically ~75–100 mL, was made to rinse the equipment until the cell conductance had returned to its baseline value. The injection cycle was then repeated with another electrolyte solution.

Fig. 2

Experimental design: injection order for set of solutions in the high temperature conductance flow cell

The complex impedance Z(ω) of solutions in the conductance cell was measured at frequencies of (100, 200, 500, 1,000, 2,000, 4,000, 6,000, 8,000, and 10,000) Hz using a programmable automatic RCL meter (Fluke Model PM6304C). During the course of each sample injection, at least 80 measurements were taken with a computer over a time span of at least 85 to 100 min. The result of these measurements was a series of values for the real and imaginary impedance, Z _Re(ω) and Z _Im(ω) as a function of angular frequency, ω:

$$ Z\left( \omega \right) = Z_{\text{Re}} \left( \omega \right) - j \cdot Z_{\text{Im}} \left( \omega \right) $$

(1)

where Z is the complex impedance, Z _Re and Z _Im are the real and imaginary components of the impedance, j ² = −1, and ω = 2πf, where f is the frequency. The solution resistance at infinite frequency R _s was found by extrapolation of the linear portion of the Warburg line to the x-axis, as described by Hnedkovsky et al. [20] and Zimmerman and Arcis [40]. The molar conductivities Λ(NaCl), measured before and after the series of other salts at each temperature, agreed with one another to within ±1 %.

3 Experimental Results

3.1 Conductivity

The experimental conductivity (specific conductance) of the solution, $\kappa_{\text{soln}}^{\text{expt}}$, was calculated from R _s, the resistance of the solution at infinite frequency using the expression:

$$ \kappa_{\text{soln}}^{\text{expt}} = \frac{{k_{\text{cell}} }}{{k_{\text{s}} }} $$

(2)

The cell constant k _cell was determined by measuring the conductivity for a series of five KCl(aq) standard solutions (10⁻⁴ to 10⁻² mol·kg⁻¹) at 298.15 K and 20 MPa, then comparing them with reference values from the expression for Λ(KCl, aq) reported by Barthel and his co-workers [41]. Following the procedure used by previous workers [18–21, 23, 24, 26, 27], the dependence of the cell constant on temperature was calculated from the cell geometry and the thermal expansion coefficient of platinum. The experimental conductivities of the electrolyte solutions, $\kappa_{\text{soln}}^{\text{expt}}$, were corrected for impurities within the solvent and the self-ionization of water by subtracting the experimental values for H₂O, $\kappa_{\text{soln}}^{\text{expt}}$, for each run, using the method of Sharygin et al. [24]:

$$ \kappa = \kappa_{\text{soln}}^{\text{expt}} - \kappa_{\text{w}}^{\text{expt}} . $$

(3)

3.2 Molar Conductivity and Solution Density

Molar conductivities, Λ ^expt, were calculated from the corrected specific conductances, κ, using the relationship:

$$ \varLambda = \kappa /c $$

(4)

where Λ ^expt is in SI units of S·m⁻²·mol⁻¹. In this study, these were converted to units of S·cm⁻²·mol⁻¹.

The molarity (c) was calculated from the molality (m), molar mass (M) and solution density (ρ _s) using Eq. 5:

$$ c = \frac{{1000m\rho_{\text{s}} }}{mM + 1000} $$

(5)

Solution densities were determined from the method developed by Zimmerman et al. [28, 29] which makes use of the Helgeson–Kirkham–Flowers–Tanger (“HKF”) model to estimate the standard partial molar volumes V° of the aqueous species in solution [42–44]. Standard partial molar volumes were calculated using the program SUPCRT [45]. We assumed that partial molar volumes D₂O are the same as those in H₂O. Values for the density, ρ _w, and dielectric constant (relative permittivity), ε _w, of water (H₂O) were calculated from the equations of state reported by Wagner and Pruss [46] and Fernandez et al. [47], respectively. The density of heavy water was taken from Hill’s equation of state [48], using the REFPROP software distributed by NIST [49]. The dielectric constant of heavy water was calculated from light water values, using the method reported by Trevani et al. [50].

The experimental conductivities, κ, for the solutions of NaCl, KCl, CsCl, CsI and KI in H₂O and D₂O, and the molar conductivities, Λ ^expt, calculated from them, are tabulated in Tables 1 and 2, respectively. Values of the cell constant, k _cell, for each experimental temperature and also the solvent densities are reported in the tables. The molar conductivities Λ ^expt(NaCl), measured before and after the series of other salts at each temperature, agreed with one another to within ±1 % which we consider to be the precision of our measurements. These values for Λ ^exp(NaCl) in light and heavy water from T = 373–598 K are reported in Table 3. Our experimental values for Λ°(NaCl) in H₂O agree with the fitted expression for critically evaluated literature data reported by Zimmerman et al. [28] to better than ±1 % from 373 to 523 K, increasing to ±6 % at 598 K, within the combined experimental uncertainties. The larger discrepancies above 523 K may reflect the effect of ion pairing or the experimental challenges of working at very high temperatures. The results also agree with the corrected values of Λ°(NaCl) in D₂O and H₂O reported by Erickson et al. [37], using this same instrument, to within ±5 %, except for the values in D₂O at 473, 523 and 548 K which are smaller than our results by 7–10 %. The difference lies just outside the combined experimental uncertainties, which are larger for Erickson’s study. From these comparisons, we consider the absolute accuracy of the conductivity measurements from 298 to 523 K to be ±3 %, rising to ±6 % at 598 K.

Table 1 Experimental molar conductivities, Λ ^expt, and limiting molar conductivities, Λ°, for NaCl_(aq), KCl_(aq), CsCl_(aq), KI_(aq), and CsI_(aq) in H₂O from T = 298 to 598 K at p = 20 MPa

Table 2 Experimental molar conductivities, Λ ^expt, and limiting molar conductivities, Λ°, for NaCl_(aq), KCl_(aq), CsCl_(aq), KI_(aq), and CsI_(aq) in D₂O from T = 298 to 598 K at p = 20 MPa

Table 3 Experimental sodium chloride molar conductivities, Λ(NaCl), collected at the beginning and at the end of each experimental run in H₂O and D₂O, from T = 298 to 598 K at p = 20 MPa

3.3 Limiting Molar Conductivity

The limiting molar conductivities, Λ° were calculated from Λ ^expt using the Fuoss–Hsia–Fernández-Prini (“FHFP”) conductivity model [51], which is the classical polynomial expansion of the Fuoss–Hsia equation for the special case of symmetrical electrolytes:

$$ \varLambda = \varLambda^{{\circ }} - S \cdot c^{1/2} + E \cdot c \cdot \ln \left( c \right) + J_{1} \cdot c - J_{2} \cdot c^{3/2} $$

(6)

Here, S is the Onsager limiting slope, and the expressions for E, J ₁ and J ₂ are given by Fernández-Prini [51]. The terms S, E, J ₁, and J ₂ all depend on Λ°. The E and J terms also depend on the distance of closest approach, a, which we defined by the Bjerrum distance, q _B:

$$ q_{B} = 2a = \frac{{\left( {z \cdot e} \right)^{2} }}{{\varepsilon_{w} \cdot k_{B} \cdot T}} $$

(7)

This treatment for the concentration dependence of Λ has been evaluated by Bianchi et al. [52], and has been shown to yield values that are accurate to ~1 % over the range 0–0.1 mol·cm⁻³. Details can be found elsewhere [28, 29]. Our extrapolation to infinite dilution for obtaining limiting molar conductivities was done with a single experimental data point, measured at an ionic strength of ~10⁻³ mol·kg⁻¹. This method can only be applied at conditions where ion association is negligible.

The limiting molar conductivities of each salt in both H₂O and D₂O are tabulated in Tables 1 and 2. The solvent properties used to carry out our calculations are reported in Table 4.

Table 4 Density (ρ), viscosity (η), dielectric constant (ε _r) and pK of H₂O and D₂O from T = 298 to 598 K at p = 20 MPa

4 Single-Ion Limiting Molar Conductivities under Hydrothermal Conditions

4.1 Transference Number Extrapolations in H₂O

The determination of the limiting equivalent conductivities of individual ions from the molar conductivity data in Tables 1 and 2 requires a value for the transference number of the chloride ion a infinite dilution, t°(Cl⁻), according to the relationship:

$$ \lambda^{{\circ }} ({\text{Cl}}^{ - } ) = t^{{\circ }} ({\text{Cl}}^{ - } ) \cdot \varLambda^{{\circ }} ({\text{M}}^{ + } {\text{Cl}}^{ - } ), $$

(8)

where M⁺ = Na⁺ or K⁺. While many transference number measurements have been made under ambient conditions [53–56], the only transference number measurements reported at elevated temperatures are those of Smith and Dismukes [15, 57], who measured values for t(Cl⁻) in NaCl(aq) and KCl(aq) up to 398 and 388 K, respectively. These are the highest temperatures ever reported for accurate transference number measurements of 1:1 salts.

Following the treatment in their original papers, experimental transference numbers for the chloride ion in NaCl and KCl from Smith and Dismukes are plotted in Fig. 3a, b as log₁₀ {[t°(Cl⁻)]/[t°(M⁺)]} versus 1/T, along with values for NaCl from Allgood and Gordon [56] and Longsworth [53], and values for KCl from Allgood et al. [55] and MacInnes and Longsworth [54]. Least-squares fits to these data yield the following linear relationships for NaCl and KCl with regression coefficients R² = 0.998 and 0.995, respectively:

$$ \log_{10} \left( {\frac{{t_{{{\text{Cl}}^{ - } }}^{{\circ }} }}{{t_{{{\text{Na}}^{ + } }}^{{\circ }} }}} \right) = \frac{55.57}{T} - 0.01049 $$

(9)

and

$$ \log_{10} \left( {\frac{{t_{{{\text{Cl}}^{ - } }}^{{\circ }} }}{{t_{{{\text{K}}^{ + } }}^{{\circ }} }}} \right) = - \frac{25.68}{T} + 0.1027 $$

(10)

The classic compilations of single ion conductivities in hydrothermal solutions by Quist and Marshall [10] and Marshall [13, 14], and more recent studies by Zimmerman et al. [28, 29], are based on values of λ°(Cl⁻) derived by extrapolating Eq. 10 to temperatures as high as 1,073 K.

Fig. 3

Temperature dependence of transference numbers in H₂O and D₂O. a Values for NaCl: open circle, Smith and Dismukes [15]; filled circle, Swain and Evans [30]. b Values for KCl: open diamond, Smith and Dismukes [57]; filled diamond, Nakahara et al. [59]

4.2 Transference Number Extrapolations in D₂O

Measurements for transference numbers in D₂O are much more rare than those in H₂O. Values have been reported for NaCl at 298 K [30] and for KCl at 288 K [58], 298 K [59] and 318 K [60]. Because literature values for t°(Cl⁻) in D₂O are only available near ambient conditions, we adopted the assumption that their temperature dependence in aqueous NaCl and KCl solutions is the same as that in light water, as expressed by Eqs. 9 and 10. Combining this assumption with the values at 298.15 K reported by Swain and Evans [30] for NaCl and by Nakahara et al. [59] for KCl, yields the expressions:

$$ \log_{10} \left( {\frac{{t_{{{\text{Cl}}^{ - } }}^{{\circ }} }}{{t_{{{\text{Na}}^{ + } }}^{{\circ }} }}} \right) = \frac{55.57}{T} - 0.01416 $$

(11)

and

$$ \log_{10} \left( {\frac{{t_{{{\text{Cl}}^{ - } }}^{{\circ }} }}{{t_{{{\text{K}}^{ + } }}^{{\circ }} }}} \right) = - \frac{25.68}{T} + 0.09394 $$

(12)

The resulting extrapolated values t°(Cl⁻) for NaCl and KCl are plotted in Fig. 3a, b, respectively.

The values used for t°_Cl− in H₂O and in D₂O in our subsequent calculations are tabulated in Tables 5 and 6.

Table 5 Single ion limiting conductivities, λ°, for Cl⁻, K⁺, Na⁺, Cs⁺ and I⁻ in H₂O from T = 298 to 598 K at p = 20 MPa

Table 6 Single ion limiting conductivities, λ°, for Cl⁻, K⁺, Na⁺, Cs⁺ and I⁻ in D₂O from T = 298 to 598 K at p = 20 MPa

4.3 The Limiting Conductivity of the Chloride Ion, λ°(Cl⁻)

The extrapolated expressions for transference numbers, Eqs. 9 and 10, were used to calculate the limiting molar conductivity of the chloride ion, λ°(Cl⁻), from the experimental molar conductivities Λ°(NaCl) and Λ°(KCl) in light water, listed in Table 1. The results are plotted in Fig. 4. The figure also includes a plot of the expression for λ°(Cl⁻) derived by Zimmerman et al. [28] from their critically evaluated compilation of conductivity literature data for NaCl. These were based on transference numbers reported by Marshall, which were derived from a linear fit to Smith and Dismukes’ data, analogous to Eq. 9. Figure 4 also includes values of λ°(Cl⁻) at higher temperatures, calculated from the limiting conductivities Λ°(NaCl) and Λ°(KCl) reported by Ref. [21] from 570 to 673 K. These studies were done at different pressures (NaCl, p = 32.53 MPa; KCl, p = 29.75 MPa), and were corrected to p = 29.75 MPa, using viscosity data from NIST and assuming that Walden’s rule is valid.

Fig. 4

Experimental limiting single ion conductivity for chloride in H₂O as a function of temperature: filled circle, this work p = 20 MPa, λ ^°(Cl⁻) derived using KCl transference numbers; filled diamond, this work p = 20 MPa, λ ^°(Cl⁻) derived using NaCl transference number; open circle, Ho et al. [21] p = 29.75 MPa, λ^°(Cl⁻) derived using KCl transference numbers; open diamond, Ho et al. [21] p = 29.75 MPa, λ ^°(Cl⁻) derived using NaCl transference numbers; solid line, Zimmerman et al. [28], p = 20 MPa; dotted line, Zimmerman et al. [28], p = 29.75 MPa

A comparison of these three sets of data provides a powerful means of estimating the accuracy and self-consistency of the linear extrapolation procedure since the values for NaCl and KCl are independent of one another. The two independent sets of values obtained for λ°(Cl⁻) agree to within 2.5 %. This demonstrates that the results from the two extrapolation methods (Eqs. 9, 10) agree to within the combined experimental uncertainties of ±1 % for each measurement, as defined by the precision of the duplicate measurements of Λ ^expt(NaCl) at each temperature.

Our experimental values for λ°(Cl⁻) can also be compared to the fitted equation recommended by Zimmerman et al. [28] from their critical evaluation of the literature data for NaCl, plotted as the solid curve in Fig. 4. The values for λ°(Cl⁻) from Zimmerman et al.’s expression are greater than the results from our KCl and NaCl measurements, but agree to within 2.5 and 5 %, with the greatest variance observed at high temperature. The results for λ°(Cl⁻) from the high temperature values for Λ°(NaCl) and Λ°(KCl) reported by Ho et al. [21] agree with one another to within 2 %, except near the critical point of water (T = 673 K) where the larger discrepancy may be attributed to a failure of Walden’s rule to hold true or to larger experimental uncertainties at these conditions. These values are higher than the fitted expression from Zimmerman et al. [28] by 3 and 5 %, respectively.

We have chosen to use the values for λ°(Cl⁻) from the transference numbers for KCl in H₂O, Eq. 10, to calculate single ion properties from the experimental results in Table 1. These values were selected because they lie between the results for λ°(Cl⁻) determined using t°(NaCl) from this work and those from the critical compilation by Zimmerman et al. [28]. We assign them an uncertainty of ±3 % in absolute accuracy, based on this comparison, and an uncertainty of ±1 % in precision relative to the values of Λ° reported in Table 1. It is not possible to assess the accuracy of the values for λ°(Cl⁻) in D₂O because there are no high temperature data with which to compare the assumptions used in deriving Eqs. 11 and 12. The precision relative to the values of Λ° for the salts reported in Table 2 is also ±1 %.

4.4 The Effect of Temperature on Limiting Conductivities of Univalent Cations

Single ion limiting conductivity data, λ°, for the cations Na⁺, K⁺ and Cs⁺ in light and heavy water were derived from the molar limiting conductivity data for the salts, Λ°, in Tables 1 and 2, by subtracting the contributions of λ°(Cl⁻) according to Kohlrausch’s law:

$$ \lambda^\circ ({\text{M}}^{ + } ) = \varLambda^\circ ({\text{M}}^{ + } {\text{Cl}}^{ - } ) - \lambda^\circ ({\text{Cl}}^{ - } ) $$

(13)

The resulting values of λ° for the cations in H₂O and in D₂O are tabulated in Tables 5 and 6, and plotted as a function of temperature in Fig. 5a, b, along with the values of λ°(Cl⁻) from which they were derived. The tables include estimated uncertainty limits δλ°(M⁺) calculated from the precisions of λ°(Cl⁻) and Λ°(MCl), according to the expression:

$$ \delta \lambda^\circ ({\text{M}}^{ + } ) = \left[\{ \delta \varLambda^\circ ({\text{M}}^{ + } {\text{Cl}}^{ - } )\}^{2} + \{ \delta \lambda^\circ ({\text{Cl}}^{ - } )\}^{2} \right]^{1/2} . $$

(14)

We note that these are uncertainties refer to relative precision rather than absolute accuracy. The values of λ° in both solvents follow the order Cl⁻ > Cs⁺ > K⁺ > Na⁺ at all temperatures.

Fig. 5

Experimental single ion conductivity of aqueous univalent ions in a H₂O and b D₂O from T = 298–600 K at p = 20 MPa: filled circle, Cl⁻; filled triangle, K⁺; open triangle, Na⁺; open diamond, I⁻; open circle, Cs⁺; solid line, Eq. 17

The only high temperature study with which to compare our results is that of Smolyakov and Veselova [61], who measured the limiting conductivity of the alkali metal chlorides from 178 to 473 K in H₂O at steam saturation pressure, p _sat. Their results for Λ° are consistent with the values in Tables 1 and 2. At 298.15 K their results, and the results from Refs. [30, 32], are ~2–5 % lower. The direction of these differences is consistent with the difference in viscosity between p _sat and p = 20 MPa, but the magnitude is greater.

While a number of empirical models [13, 14, 28, 62–66] have been proposed to describe the dependence of limiting molar or single ion conductivities on temperature and pressure (or density), only three apply to hydrothermal conditions. Oelkers and Helgeson [64, 65] used an approach based on Arrhenius energies of activation, similar to those of Brummer and Hills [62] and Smolyakov and Veselova [63], to predict the single ion limiting conductivities of a large number of ions for use in modeling geochemical systems. Marshall [13, 14] devised a “reduced state” relationship, based on a linear dependency of limiting single ion conductivity with density at constant temperature, relative to a hypothetical limit at zero density. More recently, Zimmerman et al. [28, 67, 68] reported that Λ° could be accurately represented by simple empirical functions of the solvent viscosity over a wide range of temperatures and pressures. We tested all of these equations and found that the best fit to the data in Tables 5 and 6 was obtained using the model by Zimmerman et al. [28], which takes the form:

$$ \log_{10} {\lambda}^{\circ} = \log_{10} A_{1} + \left( {A_{2} + \frac{{A_{3} }}{{\rho_{\text{w}} }}} \right)\log_{10} \eta $$

(15)

Here, A ₁, A ₂, and A ₃ are empirical fitting parameters, different for each ion, and ρ _w is the solvent density. The standard deviation in log₁₀ λ° of the fits were between 0.008 and 0.015, and values for the fitted parameters are listed in Table 7. The fit to the data for the iodide ion had a larger standard deviation in log₁₀ λ° of 0.019. None of the models could reproduce the small inflection point at ~550 K, with the result that all showed a systematic deviation of +5 % or more between experimental data and fitted results somewhere in the range 350–500 K.

Table 7 Fitted parameters (Eqs. 15, 17) for single ion limiting conductivities for Cl⁻, K⁺, Na⁺, Cs⁺ and I⁻ ions in H₂O and D₂O

An improved fit was obtained from an empirical fitting equation that we derived from a relationship reported by Smolyakov and Veselova [63]:

$$ \log_{10} \lambda^\circ \eta = A + \frac{{E_{\eta } - E_{\lambda^\circ } }}{RT}, $$

(16)

where E _η is the activation energy of viscosity and E _λ is the activation energy of ionic conductance. The new fitting expression takes the form:

$$ \log_{10} \lambda^\circ \eta = \frac{a}{T - 228} + \frac{{b + c\ln \rho_{\text{w}} }}{RT}, $$

(17)

where ρ _w is the density of water or heavy water. The pre-exponential factor, A = a/(T − 228), was chosen to yield a discontinuity at T = 228 K. This is the temperature usually associated with the critical point between the high and low density forms of the liquid in super-cooled water [43, 69–71], and it is intended to reproduce the strong temperature dependence of “structure-making” and “structure-breaking” effects at sub-ambient temperatures, described in the following section. The critical point in super-cooled D₂O is thought to be at T = (230 ± 5) K [72], consistent with Eq. 17, to within the margin of error. The Arrhenius energy term, [E _η  − E _λ ] = b + cln ρ _w, was taken from the expression for the standard molar internal energy of reaction from the “density” model of [73], which describes the solvent compressibility effects on chemical equilibria at elevated temperatures and pressures [74].

The standards deviations in log₁₀ λ°η of the fits were between 0.004 and 0.008 for ions in H₂O and between 0.004 and 0.014 for ions in D₂O. Values for the fitted parameters are listed in Table 7. The fits of Eq. 17 to the data for the iodide ion had a larger standards deviation in log₁₀ λ°η of 0.016 (H₂O) and 0.012 (D₂O). This was the only ion for which three parameter fits were not statistically significant. We were unable to identify a reason for the visibly larger statistical scatter in the results for λ°(I⁻), but suspect it may arise from sorption effects or redox reactions with the platinum surfaces in the cell. Finally we note that Eq. 17 yielded statistically superior fits for our values of λ° relative to those obtained from Eq. 15. We recommend its use to represent the behaviour of λ° over a wide range of density and temperature.

5 Discussion

5.1 Temperature Dependence of Limiting Conductivities

The classical interpretation of ionic conductivities is based on the model by Frank and Wen [75] for ionic hydration, which is summarized in the review paper by Kay [76]. Briefly, the effects of temperature on limiting conductivities can be discussed in terms of the Walden product λ°η. The Walden product is related to the “effective” ionic radius r _Stokes through Stokes law,

$$ \lambda^\circ \eta = \frac{{Fz_{i} e}}{{\pi fr_{\text{Stokes}} }}, $$

(18)

in which 6 ≥ f ≥ 4 is a boundary factor ranging between perfect slip and perfect sticking, and e·z _i is the ionic charge. Stokes’ law does not adequately describe limiting conductivities because it ignores ion–solvent, and solvent–solvent interactions at the molecular level [77]. More sophisticated continuum models such as the Hubbard–Onsager theory [78, 79] are only partly successful in describing these effects. The Frank–Wen model for ionic hydration [75] postulates that the perturbations of an ion on the hydrogen-bonding of bulk water can be inferred from the temperature dependence of the Walden product ratio,

$$ R_{298}^{T} = \frac{{(\lambda^\circ \eta_{\text{w}} )_{T} }}{{(\lambda^\circ \eta_{\text{w}} )_{{T = 298{\text{K}}}} }} $$

(19)

In this model, the so-called “structure-making” ions are defined as those with R ^T₂₉₈  > 1 and “structure-breaking” ions are those with R ^T₂₉₈  < 1. A modern interpretation of this classification that includes thermodynamic transfer properties and spectroscopic results is given by Marcus [80].

Figure 6 presents plots of the Walden product ratio R ^T₂₉₈ for the chloride ion and the cations Na⁺, K⁺ and Cs⁺ in H₂O and D₂O, relative to their values at 298.15 K, based on values for the limiting single ion conductivities in Table 5. The trends in R ^T₂₉₈ for the two solvents are very similar, and the discussion that follows in this section will focus on temperature-dependent ionic hydration effects in light water. The iodide ion was not included in this analysis because of the larger uncertainty limits noted above. We note that only Eq. 17 is able to reproduce the two inflection points observed in Fig. 6. Based on Kay’s interpretation of the values of R ^T₂₉₈ near ambient conditions and Marcus’ classification, Na⁺ is considered to be a borderline ion with a strongly-bound first hydration shell and a largely temperature-dependent Walden product. In both the classical and modern descriptions, K⁺ and Cs⁺ are classified as “structure-breakers” at ambient conditions. These are large cations with a lower surface charge density and less tightly bound first hydration spheres that disrupt the three dimensional hydrogen-bonding in water and lower the local viscosity. The Cl⁻ anion is also a “structure-breaker” with a loosely-bound first hydration sphere which provides excess mobility at temperatures below 298 K.

Fig. 6

Temperature dependence of the Walden product ratio {(λ ^° η) _T /(λ ^° η) _298.15K) for aqueous univalent ions in a H₂O and b D₂O from T = 298–600 K at p = 20 MPa: filled circle, Cl⁻; filled triangle, K⁺; open triangle, Na⁺; open diamond, I⁻; open circle, Cs⁺; solid line, Eq. 17

These “structure-making” and “structure breaking” effects decrease with increasing temperature as the nature of the ion–water interactions changes. As the temperature is raised from ambient conditions towards the critical point [81] (H₂O: T _c = 647.10 K, p _c = 22.06 MPa, ρ _c = 0.322 g·cm⁻³; D₂O: T _c = 643.85 K, p _c = 21.67 MPa, ρ _c = 0.356 g·cm⁻³), the hydrogen bonded coordination number decreases from ~4 to ~2, and the isothermal compressibility becomes very large [82]. At temperatures above ~400 K, the dominant effect is the long-range polarization of the bulk solvent due to charge–dipole interactions with liquid water, as described by the classical Born equation. These effects are reflected in the standard partial molar volumes of aqueous ions which rise to a maximum at ~330 to 360 K as short-range hydration effects become less important, then decrease towards large negative values at the critical point due to the electrostriction caused by long-range solvent polarization [74, 83]. As a result, the Walden product ratio, expressed by Eq. 19, would be expected to decrease with increasing temperature due to the viscous drag associated with a more densely packed “hydration” sphere of oriented solvent molecules around each of the ions. These effects have been quantified by Xiao and Wood [84], who used a compressible continuum model to describe the effects of electrostriction and electroviscosity on ionic transport in high temperature water. Their model correctly predicts (i) that Walden’s rule is not obeyed, (ii) that the limiting conductivities λ° are a strong function of density, and a very weak function of temperature at constant density, and (iii) that the values of λ° for small ions converge towards similar values under near-critical conditions where compressibility effects are large.

The plots of R ^T₂₉₈ in Fig. 6 are consistent with this qualitative description, and with the compressible continuum models of Xiao and Wood. The Walden product of Na⁺ at 600 K is lower than its value at 298 K by about 15 %, while those of K⁺ and Cs⁺ are lower by ~40 %, reflecting the more tightly bound first hydration sphere of Na⁺ and the larger effective radius of the hydrated ion. The behavior of the chloride ion is intermediate between these two results.

We note that these descriptions of ionic hydration effects from classical models are supported by molecular dynamics simulations, such as those of Balbuena et al. [85] who confirmed that the strongly bound first shell of both cations and anions persists well into the high-temperature, low density range of supercritical water. Their simulation showed that coordination number for cations decreases with increasing temperature, while the coordination number for anions increases. As an example, for Na⁺ the number of water molecules in the first hydration sphere decreased from n _w = 5.2 at 298 K to n _w = 4.5 at 673 K, while those for Cl⁻ increased from n _w = 7.4 to n _w = 8.1. Moreover, the simulations confirmed the increasing importance of long-range solvent polarization to ionic hydration in dense, near-critical water, as described by semi-continuum models based on the Born equation [43, 86] and the “density” model [74]. A more complete review is given by Seward and Driesner [87] and Driesner [88].

5.2 Deuterium Isotope Effects

Although Stokes’ law equates the Walden product λ°η _w with the “effective” ionic radius, the compressible continuum ionic transport model [84] and modern interpretations of experimental data [77] show that this interpretation is not valid at the conditions relevant to the present study. An alternative approach, initiated by Swain and Evans [30] and also pursued by Broadwater and Evans [31, 32] and Tada et al. [33–35] is to use the isotopic ratio of the Walden product in heavy and light water, $(\lambda^\circ \eta )_{{{\text{D}}_{2} {\text{O}}}} /(\lambda^\circ \eta )_{{{\text{H}}_{2} {\text{O}}}}$, as a means of examining relative differences in ionic hydration between the two solvents. We have adopted the Swain–Evans approach, as a practical tool to quantify and interpret the deuterium isotope effect on single-ion limiting conductivities. Values for the isotopic Walden product ratio:

$$ R_{\text{Walden}}^{{{\text{D}}/{\text{H}}}} = \frac{{(\lambda^\circ \eta )_{{{\text{D}}_{2} {\text{O}}}} }}{{(\lambda^\circ \eta )_{{{\text{H}}_{2} {\text{O}}}} }} = \frac{{r_{{{\text{Stokes}} - {\text{H}}_{2} {\text{O}}}} }}{{r_{{{\text{Stokes}} - {\text{D}}_{2} {\text{O}}}} }}, $$

(20)

for the chloride ion and the cations, were calculated from the limiting single-ion conductivities in Tables 5 and 6, and the physical property data in Table 4. These are plotted in Fig. 7a, b, along with the values of $R_{\text{Walden}}^{{{\text{D}}/{\text{H}}}}$ calculated from Eq. 20, using the fitted parameters listed in Table 7. The interpretation of these plots has been described by Swain and Evans [30], Broadwater and Kay [32], and Broadwater and Evans [31], based on the stronger hydrogen bonding in D₂O relative to H₂O. In their treatment, values of $R_{\text{Walden}}^{{{\text{D}}/{\text{H}}}}$ equal to unity indicate that the efficiency of ionic transport mechanism is the same in both solvents, while values $R_{\text{Walden}}^{\text{D/H}}$ > 1 indicate that the ions diffuse more readily in heavy water, consistent with “structure breaking” behavior. Values of $R_{\text{Walden}}^{\text{D/H}}$ < 1 indicate a more efficient transport mechanism in light water, and are interpreted as being due to "structure making" ions. We note that the limiting conductivities at 298.15 K at p = 20 MPa, in both H₂O and D₂O, are larger than those in Refs. [30–32], which were measured at p = 0.1 MPa.

Fig. 7

a Isotopic Walden product ratio, $R_{\text{Walden}}^{{{\text{D}}/{\text{H}}}}$, of aqueous chloride, potassium and cesium ions from T = 298–598 K at p = 20 MPa: filled circle, Cl⁻; filled triangle, K⁺; open circle, Cs⁺; solid line, Eq. 17. b Isotopic Walden product ratio, $R_{\text{Walden}}^{{{\text{D}}/{\text{H}}}}$, of aqueous sodium ion from T = 298–598 K at p = 20 MPa: open triangle, Na⁺; solid line, Eq. 17

As shown in Fig. 7a, the isotopic Walden product ratio for the chloride ion at 298.15 K is greater than unity, $R_{\text{Walden}}^{{{\text{D}}/{\text{H}}}} ({\text{Cl}}^{ - } )$ = 1.05 ± 0.02, consistent with its classification of it as a “structure breaker” under ambient conditions [30–32, 76, 89]. This value decreases to unity at ~473 K, then continues to fall to $R_{\text{Walden}}^{{{\text{D}}/{\text{H}}}} ({\text{Cl}}^{ - } )$ = 0.97 ± 0.02, indicating that the deuterium isotope effect is negligible at this temperature, to within the experimental uncertainties. Likewise, the values for potassium and cesium, $R_{\text{Walden}}^{{{\text{D}}/{\text{H}}}} ({\text{K}}^{ + } )$ = 1.07 ± 0.02 and $R_{\text{Walden}}^{{{\text{D}}/{\text{H}}}} ({\text{Cs}}^{ + } )$ = 1.05 ± 0.02, are consistent with their classification as “structure” breakers at room temperature. Like chloride, these values fall to unity at elevated temperatures, $R_{\text{Walden}}^{\text{D/H}}$ = 0.99 ± 0.02 and 1.00 ± 0.02 at 600 K. The sodium ion is considered to be a borderline “structure maker” [30–32, 76, 89] and its value at 298.15 K, $R_{\text{Walden}}^{\text{D/H}} ({\text{Na}}^{ + } )$  = 0.98 ± 0.02, reflects this assessment. This value rises above unity to approximately constant values of ~1.04 ± 0.02 at temperatures above ~375 K (see Fig. 7b). Here and for each of the ions discussed above, the uncertainty is taken to be the 95 % confidence limit for the values of $R_{\text{Walden}}^{\text{D/H}}$ above 370 K, relative to their mean value. A decrease in $R_{\text{Walden}}^{\text{D/H}}$ toward values below unity would be expected from ion–solvent polarization arguments because the critical temperature of D₂O is lower than that of H₂O. However, the departure from unity for the structure breaking ions is at the edge of being statistically significant. The decrease in $R_{\text{Walden}}^{\text{D/H}} ({\text{Na}}^{ + } )$ between 370 and 600 K is not statistically significant, suggesting that there may be a deuterium isotope effect on λ°(Na⁺) at sub-critical conditions that persists at higher temperatures but that the effect is small.

6 Conclusions

This paper reports the first measurements of accurate electrical conductivities of aqueous salts in D₂O under hydrothermal conditions. By using a high-precision flow-through AC electrical conductance instrument, it was possible to make measurements of similar nearly identical solutions in heavy and light water in sequence, so that many of the systematic errors associated with high temperature conductivity measurements cancel one another. The values for λ°(Cl⁻) in H₂O and D₂O obtained from this method appear to be precise to within ±1 %.The principal uncertainty is in the systematic uncertainty associated with extrapolation of transference numbers for NaCl and KCl in D₂O. The determination of λ°(Cl⁻) is an important result because it defines a single-ion conductivity scale for electrolyte solutions in heavy water for the first time. These self-consistent values of λ°(Cl⁻) were used to calculate the single ion limiting conductivities of three alkali metal cations, which follow the order λ°(Na⁺) < λ°(K⁺) ≤ λ°(Cs⁺) < λ°(Cl⁻) at all temperatures in both light and heavy water. The Walden product ratio of all these ions drops dramatically with increasing temperature, in the order $R_{298}^T$(Na⁺) < $R_{298}^T$(Cl⁻) < $R_{298}^T$(K⁺) ≈ $R_{298}^T$(Cs⁺), reflecting the shift towards hydration effects associated with long-range solvent polarization. The temperature dependence of the isotopic Walden product ratio, $(\lambda^\circ \eta )_{{{\text{D}}_{2} {\text{O}}}} /(\lambda^\circ \eta )_{{{\text{H}}_{2} {\text{O}}}}$ indicates that differences in the hydration of Cl⁻, K⁺ and Cs⁺ between light and heavy water at ambient conditions associated with hydrogen-bonding, the so-called “structure breaking” effects, largely disappear at temperatures above ~400 K. The value of $(\lambda^\circ \eta )_{{{\text{D}}_{2} {\text{O}}}} /(\lambda^\circ \eta )_{{{\text{H}}_{2} {\text{O}}}}$ for the “structure making” ion Na⁺ rises from 0.98 at 298.15 K to ~1.04 ± 0.02 at temperatures above ~375 K and remains approximately constant up to 600 K. These changes are consistent with the onset of long-range ion-solvent polarization as the dominant hydration effect as the critical temperature is approached.