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. 2022 Oct 25;126(43):8791–8803. doi: 10.1021/acs.jpcb.2c02929

Limiting Conductivities of Strong Acids and Bases in D2O and H2O: Deuterium Isotope Effects on Proton Hopping over a Wide Temperature Range

Hugues Arcis †,‡,*, Jeff Plumridge †, Peter R Tremaine †,*
PMCID: PMC9639610  PMID: 36283024

Abstract

graphic file with name jp2c02929_0011.jpg

The molar conductivity (Λ°) of hydrochloric acid, potassium hydroxide, and sodium hydroxide has been measured in both light and heavy waters from 298 to 598 K at p = 20 MPa using a high-precision flow-through alternating current (AC) conductance instrument. The results were used to explore the deuterium isotope effect on ionic transport by proton hopping mechanisms under hydrothermal conditions. Extrapolations of published transport number data to elevated temperature were used to calculate the individual ionic contributions (λ°) for H3O+, D3O+, OH–, and OD–, from which the excess molar conductivities due to proton hopping were calculated. These are the first reported values for the excess conductivities for D3O+ and OD– at temperatures above 318 K. The excess conductivities indicate a strong deuterium isotope effect whereby the transport of D3O+ by proton hopping is reduced by ∼33% relative to H3O+, and OD– is reduced by over 60% relative to OH–, over the entire temperature range. A well-defined maximum in the excess conductivities of D3O+ and H3O+ at ∼420 K suggests that the Eigen cation (H2O)4H+ and the Zundel transition-state cation (H2O)2H+ are destabilized at elevated temperatures as the three-dimensional, tetrahedrally hydrogen-bonded networks in water break down. The less pronounced maximum for OD– and OH– suggested that their Eigen and Zundel anions, (H2O)3OH– and (H2O)OH–, are less destabilized in the two-dimensional networks and chains that dominate the “structure” of liquid water under these conditions.

1. Introduction

The transport of H3O+/D3O+ and OH–/OD– ions in aqueous solutions has been the subject of many investigations.1−10 However, a complete understanding of the process over a full range of temperatures is still lacking. The mobility of hydrogen- or deuterium-containing ions H3O+, OH–, D3O+, and OD– in water (sometimes referred to as “anomalous conductivity”) is higher than that of other ions with similar crystallographic radii because there are two separate mechanisms of ionic movement: (i) the bulk ionic diffusion and (ii) proton hopping, also known as “Grotthuss” mechanism and/or “prototropic transfer”. The exact details of the underlying mechanisms remain poorly understood over a complete range of temperature and pressure.9,10 It is understood that the bulk ionic diffusion and “proton hopping” contribution must have different activation energies with different temperature-dependent rate constants because of the change in the hydrogen-bonding network of water.11 At ambient conditions, the hydrogen bonding in liquid H2O (or D2O) is strong, resulting in a tetrahedral orientation of molecules within the build solvent and an average coordination number of 4.12 The coordination number decreases with increasing temperature as water approaches supercritical conditions.12 The effects of these changes on the proton hopping mechanism are not well understood.

Alternating current (AC) conductivity has proven to be a very useful tool for quantifying the extra mobility arising from proton hopping (e.g., refs (11, 13−19)). Conductivity measurements using static cells have been widely used to study ion association, ionization constants, and ionic mobility under hydrothermal conditions up to and including near-critical and supercritical conditions, both for reactive transport modeling of industrial applications such as water-cooled nuclear reactor systems20−22 and for fundamental studies.23−27 The significant technical improvements incorporated into the novel flow cell design of Wood et al.28 have made it possible to extend these measurements to ionic strengths as low as 10–5 mol kg–1. The flow capability also improves accuracy by allowing the conductivity of standards and solutions of interest to be measured in sequence at constant temperature and pressure, so that the effects of systematic errors can be minimized.

Although several conductivity studies of deuterium isotope effects have been reported at ambient temperature (e.g.,29−32), temperature-dependent measurements33−35 are much more scarce, despite their importance to basic science and advanced nuclear reactor technologies.36 Tada et al.4,5,8 reported conductivity measurements of aqueous HCl/DCl, KOH/KOD, and KCl from 5 to 65 °C in light and heavy waters. They quantified proton hopping effects by assuming that the bulk ionic diffusion coefficients for H3O+ and OH– were equivalent to those of K+ and Cl–, respectively, based on the observed similarities in their crystallographic radii. More recently, Plumridge et al.33 have shown that ionic diffusion of simple ions such as Na+, K+, Cs+, Cl–, and I– is the same in light and heavy waters from ∼400 to 600 K if viscosity effects were considered [i.e., the Walden products are equal, (λ°η)D2O  ≈ (λ°η)H2O)], suggesting that the approach can be extended to much higher temperatures and pressures.

This study reports experimental values for the limiting molar conductivities, Λ°, of potassium hydroxide, sodium hydroxide, and hydrochloric acid from T = 298 K to T = 598 K at p ∼ 20 MPa in both H2O and D2O. Experiments were carried out sequentially using a Wood-type high precision flow-through AC electrical conductance instrument to eliminate most systematic errors so that differences in ionic conductivities in light and heavy waters could be measured with a high degree of precision. Single-ion limiting conductivities, λ°(D3O+), λ°(H3O+), λ°(OD–), and λ°(OH–), were derived from the single-ion values for λ°(Cl–) recently reported by Plumridge et al.33 The results presented below indicate that between 298 and 598 K ionic transport due to proton hopping is (i) greater for H3O+/D3O+ than for OH–/OD–, (ii) more efficient in H2O than in D2O, and (iii) that the proton hopping contributions for each of these ions increase to a maximum at temperatures in the range of 420–500 K and then decrease. These trends and the insights they yield into proton hopping mechanisms are discussed.

2. Materials and Experimental Section

2.1. Chemicals and Solution Preparation

Stock solutions (∼0.1 mol kg–1) of DCl (Sigma-Aldrich 35% w/w), NaOH (Alfa Aesar 50% w/w), NaOD (Sigma-Aldrich 40% w/w), KOH (Alfa Aesar 50% w/v), and KOD (Sigma-Aldrich 40% w/w) were prepared by dilution with Nanopure H2O (resistivity 18.2 MΩ·cm) or D2O (99.7 atom % D), which was provided by Ontario Power Generation Ltd. Solutions of HCl (Fluka, 0.1 N standard solution), DCl (Sigma-Aldrich 35 wt % solution), NaOH (Alfa Aesar 50% w/w), NaOD (Sigma-Aldrich 40% wt % solution in D2O, 99+ atom % D), KOH (Alfa Aesar 50% w/v), and KOD (Sigma-Aldrich 40% wt % solution in D2O, 99+ atom % D) were prepared by dilution to ∼0.1 mol kg–1 with Nanopure light water or heavy water. The HCl and DCl solutions were standardized by quadruplicate titration against a known mass of tris(hydroxymethyl)aminomethane (THAM), while the NaOH, NaOD, KOH, and KOD solutions were standardized by quadruplicate titration against known masses of potassium hydrogen phthalate (KHP). Stock solutions of ∼0.1 mol kg–1 NaCl (Alfa Aesar Puratonic, 99.999%) and KCl (Alfa Aesar 99.95%) in light and heavy waters 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 solutions to be measured in the conductance instrument (∼10–3 mol kg–1) were prepared by mass dilution from the stock solutions. All solutions were prepared in a glovebag flushed with argon, which was continually kept under a positive argon pressure and then stored under an argon atmosphere until use.33 High-density polyethylene Nalgene or Pyrex bottles were used for the preparation and storage of HCl/DCl solutions, while hydroxide solutions were prepared in high-density polypropylene Nalgene bottles.

2.2. Flow Conductivity Technique

The experiments were carried out using a one-of-a-kind custom-made high-precision flow-through AC conductivity cell first developed by Zimmerman and Wood28 and described in our previous studies.25−27,37 Experiments were performed at a constant flow rate of 0.5 cm3 min–1. For each solution, complex frequency-dependent impedances were measured at frequencies of 100, 200, 500, 1000, 2000, 4000, 6000, 8000, and 10000 Hz33 to yield the real solution resistance extrapolated at zero frequency, Rs, following the method described by Zimmerman and Arcis.38 The instrument was capable of regulating temperature and pressure to within ±0.15 K and ±0.10 MPa over several days, and they were measured to an accuracy of ±0.02 K and ±0.01 MPa, respectively. Sample solutions to be analyzed were injected into the cell using a high-pressure liquid chromatography (HPLC) injection system. A full description of the instrument is available elsewhere.25,37

2.3. Experimental Design

The experimental strategy involved the sequential injection of our samples in H2O and D2O at each temperature, so that systematic errors in the solution conductivities in both solvents would cancel,33−35 thus yielding more precise values of the deuterium isotope effect on transport properties. The present results were obtained in the same series of experimental runs reported by Plumridge et al.33 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. The conductivities of aqueous sodium chloride, measured before and after the series of other electrolytes at each temperature, agreed with one another to within ±1% (see Table S1), which is considered to be the precision of our measurements.

2.4. Experimental Conductivities and Molar Conductivities

Experimental conductivities, κsolnexp, for each set of solutions were calculated from Rs, and a calibration constant was measured for our instrument before and after each series of solution injections (κsolnexp = kcell/Rs). Experimental solution conductivities, κsoln, were corrected for impurities within the solvent using eq 1 for salts and eq 2 for acids and bases39,40 to yield the electrolyte conductivity (κ)

2.4. 1
2.4. 2

Here, κwexp is the experimental conductivity value for the solvent and κw was calculated from the limiting conductivities of H3O+ and OH–, and the ionization constant and density of water.41

Corrected conductivities were then normalized by dividing κ by the solution molarities, c, to yield molar conductivities, Λexp (converted to units of S cm–2 mol–1 in this study). Molarities were calculated from molalities (m), molar masses (M), and solution densities (ρs) using the exact same procedure, as described in ref (33). Standard partial molar volumes of aqueous ions required for estimating ρs were calculated using the program SUPCRT,42 in which they were assumed to be the same in both H2O and D2O. Although the apparent molar volume measurements on aqueous NaCl, NaOH, NaOD, HCl, and DCl by Trevani et al.43 do show small differences in light and heavy waters at elevated temperatures, changes of this magnitude would affect the calculated concentrations in our study by less than 1%. Results are reported in Tables 1 and 2 and plotted in Figures 1–3.

Table 1. Experimental Molar Conductivities (Λexp) and Limiting Molar Conductivities (Λ°) for KCl, HCl, and KOH in H2O from T = 298 K to T = 598 K at p = 20 MPaa,b.

  m·106 c·106 R κ·106 Λexp Λ°
solute (mol kg–1) (mol L–1) (Ω) (S cm–1) (S·cm2 mol–1) (S·cm2 mol–1)
T = 298.2 K, p = 20.5 MPa, ρw = 1.006 kg m–3, ηw = 0.00886 poise, εr = 79.16 poise, kcell = 0.0659 cm–1, κw = 0.984 × 10–6 S cm–1
NaCl (aq) 1027.11 1033.25 527.50 123.94 119.95±1.20 122.73±1.23
KCl (aq) 1031.87 1038.03 438.36 149.34 143.87±1.44 146.84±1.47
HCl (aq) 1044.43 1050.67 150.87 435.86 414.83±4.15 419.62±4.20
KOH (aq) 808.27 813.12 293.38 223.69 275.10±2.75 278.64±2.79
T = 373.2 K, p = 20.9 MPa, ρw = 0.968 kg m–3, ηw = 0.00287 poise, εr = 56.24 poise, kcell = 0.0636 cm–1, κw = 2.061 × 10–6 S cm–1
NaCl (aq) 957.86 927.00 205.85 317.80 342.83±3.43 351.52±3.52
KCl (aq) 1080.74 1045.91 162.46 403.24 385.54±3.86 395.16±3.95
HCl (aq) 1042.42 1008.84 78.93 832.96 825.67±8.26 838.40±8.38
NaOH (aq) 1462.68 1415.58 81.33 808.38 571.06±5.71 584.24±5.84
KOH (aq) 807.80 781.79 129.82 505.97 647.20±6.47 657.70±6.58
T = 423.3 K, p = 20.2 MPa, ρw = 0.928 kg m–3, ηw = 0.00187 poise, εr = 44.72 poise, kcell = 0.0658 cm–1, κw = 3.254 × 10–6 S cm–1
NaCl (aq) 1027.11 952.79 136.79 477.86 501.54±5.02 515.99±5.16
KCl (aq) 1031.87 957.19 122.82 532.60 556.42±5.56 571.48±5.71
HCl (aq) 1044.43 968.85 66.50 988.45 1020.23±10.20 1039.12±10.39
KOH (aq) 808.27 749.79 103.00 637.69 850.48±8.50 866.59±8.67
T = 473.1 K, p = 20.2 MPa, ρw = 0.878 kg m–3, ηw = 0.00139 poise, εr = 35.54 poise, kcell = 0.0658 cm–1, κw = 3.978 × 10–6 S cm–1
NaCl (aq) 998.01 876.41 114.09 572.56 653.31±6.53 679.90±6.80
KCl (aq) 955.23 838.83 109.09 598.99 714.07±7.14 740.11±7.40
HCl (aq) 1096.66 963.04 58.65 1120.90 1163.93±11.64 1190.11±11.90
NaOH (aq) 869.95 763.96 90.87 723.18 946.62±9.47 968.85±9.69
KOH (aq) 808.27 709.80 92.97 706.82 995.81±9.96 1017.78±10.18
T = 522.9 K, p = 20.2 MPa, ρw = 0.817 kg m–3, ηw = 0.00111 poise, εr = 35.54 poise, kcell = 0.0657 cm–1, κw = 4.085 × 10–6 S cm–1
NaCl (aq) 1101.58 899.62 92.26 708.49 787.54±7.88 821.78±8.22
KCl (aq) 955.23 780.10 98.43 663.86 850.99±8.51 883.03±8.83
HCl (aq) 1096.66 895.60 57.97 1304.86 1266.06±12.66 1300.07±13.00
NaOH (aq) 869.95 710.47 87.68 749.64 1055.13±10.55 1084.15±10.84
KOH (aq) 623.94 509.56 108.05 608.26 1193.70±11.94 1219.93±12.20
T = 548.1 K, p = 20.9 MPa, ρw = 0.780 kg m–3, ηw = 0.00100 poise, εr = 24.49 poise, kcell = 0.0635 cm–1, κw = 3.551 × 10–6 S cm–1
NaCl (aq) 1124.57 877.19 87.63 720.95 821.89±8.22 860.20±8.60
KCl (aq) 1080.74 842.99 84.88 744.47 883.13±8.83 920.75±9.21
HCl (aq) 1042.42 813.11 61.59 1030.99 1267.97±12.68 1305.12±13.05
NaOH (aq) 1462.68 1140.97 51.79 1226.05 1074.57±10.75 1116.31±11.16
KOH (aq) 807.80 630.10 83.71 758.59 1203.92±12.04 1237.32±12.37
T = 572.8 K, p = 20.8 MPa, ρw = 0.737 kg m–3, ηw = 0.00090 poise, εr = 21.22 poise, kcell = 0.0635 cm–1, κw = 2.944 × 10–6 S cm–1
NaCl (aq) 964.37 710.49 100.42 629.15 885.53±8.86 925.42±9.25
KCl (aq) 1091.88 804.43 83.47 757.45 941.61±9.42 983.81±9.84
HCl (aq) 1034.36 762.05 64.81 979.63 1285.51±12.86 1327.69±13.28
NaOH (aq) 1323.95 975.45 57.83 1097.73 1125.36±11.25 1171.12±11.71
KOH (aq) 669.32 493.11 102.38 620.18 1257.70±12.58 1293.06±12.93
T = 597.9 K, p = 20.7 MPa, ρw = 0.682 kg m–3, ηw = 0.00081 poise, εr = 17.87 poise, kcell = 0.0635 cm–1, κw = 2.070 × 10–6 S cm–1
NaCl (aq) 787.31 537.22 123.69 510.96 951.11±9.51 992.64±9.93
KCl (aq) 1091.88 745.06 85.02 744.32 999.00±9.99 1047.07±10.47
HCl (aq) 1034.36 705.81 70.27 903.16 1279.61±12.80 1328.39±13.28
NaOH (aq) 1182.07 806.64 67.92 934.44 1158.44±11.58 1209.04±12.09
KOH (aq) 693.48 473.19 102.43 619.67 1309.55±13.10 1352.09±13.52
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a

The reported errors for the limiting conductivity correspond to our experimental precision. The absolute accuracy from 298 to 523 K is estimated to be ±3%, increasing to ±6% at 598 K.

b

10 Poise = 1 Pa.s.

Table 2. Experimental Molar Conductivities (Λexp) and Limiting Molar Conductivities (Λ°) for KCl, DCl, and DOH in D2O from T = 298 K to T = 598 K at p = 20 MPaa,b.

  m·106 c·106 R κ·106 Λexp Λ°
solute (mol kg–1) (mol L–1) (Ω) (S cm–1) (S·cm2 mol–1) (S·cm2 mol–1)
T = 298.2 K, p = 20.5 MPa, ρD2O = 1.115 kg m–3, ηD2O = 0.01088 poise, εr = 78.96 poise, kcell = 0.0659 cm–1, κw = 0.578 × 10–6 S cm–1
NaCl (aq) 919.32 1024.84 638.99 102.55 100.06±1.00 102.34±1.02
KCl (aq) 1234.75 1376.43 383.15 171.41 124.53±1.25 127.37±1.27
DCl (aq) 821.33 915.60 234.14 281.00 306.91±3.07 310.44±3.10
KOD (aq) 1055.37 1176.51 329.39 199.62 169.68±1.70 172.58±1.73
T = 373.2 K, p = 20.9 MPa, ρD2O = 1.074 kg m–3, ηD2O = 0.00336 poise, εr = 56.00 poise, kcell = 0.0636 cm–1, κw = 1.886 × 10–6 S cm–1
NaCl (aq) 870.66 934.99 232.15 281.75 301.34±3.01 308.90±3.09
KCl (aq) 1284.94 1379.84 141.32 464.04 336.30±3.36 345.79±3.46
DCl (aq) 1002.71 1076.79 93.55 703.85 653.65±6.54 663.64±6.64
NaOD (aq) 1290.10 1385.45 114.08 577.15 416.58±4.17 425.83±4.26
KOD (aq) 1101.98 1183.40 124.40 529.31 447.27±4.47 455.86±4.56
T = 423.3 K, p = 20.2 MPa, ρD2O = 1.029 kg m–3, ηD2O = 0.00216 poise, εr = 44.51 poise, kcell = 0.0658 cm–1, κw = 2.209 × 10–6 S cm–1
NaCl (aq) 919.32 946.15 156.04 421.96 443.43±4.43 456.04±4.56
KCl (aq) 1234.75 1270.75 103.67 632.64 497.84±4.98 513.00±5.13
DCl (aq) 1022.86 1052.71 76.95 857.40 814.47±8.14 828.36±8.28
KOD (aq) 1055.37 1086.17 96.85 681.69 627.60±6.28 639.77±6.40
T = 473.1 K, p = 20.2 MPa, ρD2O = 0.973 kg m–3, ηD2O = 0.00158 poise, εr = 35.38 poise, kcell = 0.0658 cm–1, κw = 2.898 × 10–6 S cm–1
NaCl (aq) 978.43 952.44 117.12 501.96 586.63±5.87 612.86±6.13
KCl (aq) 1313.77 1278.85 81.66 802.61 627.60±6.28 657.63±6.58
DCl (aq) 1022.86 995.69 69.82 956.26 950.36±9.50 968.42±9.68
NaOD (aq) 1012.32 985.44 85.05 777.55 789.04±7.89 804.86±8.05
KOD (aq) 1055.37 1027.34 82.75 798.99 777.72±7.78 794.07±7.94
T = 522.9 K, p = 20.2 MPa, ρD2O = 0.904 kg m–3, ηD2O = 0.00123 poise, εr = 27.80 poise, kcell = 0.0657 cm–1, κw = 3.271 × 10–6 S cm–1
NaCl (aq) 9842.90 889.73 103.70 630.69 708.86±7.09 741.38±7.41
KCl (aq) 1313.77 1187.55 73.28 893.89 752.72±7.53 789.63±7.90
DCl (aq) 1068.31 965.67 65.09 1014.98 1051.06±10.51 1076.13±10.76
NaOD (aq) 1044.62 944.28 79.71 829.63 878.58±8.79 900.51±9.01
KOD (aq) 851.13 769.38 92.69 714.20 928.27±9.28 946.62±9.47
T = 548.1 K, p = 20.9 MPa, ρD2O = 0.862 kg m–3, ηD2O = 0.00110 poise, εr = 24.38 poise, kcell = 0.0635 cm–1, κw = 3.044 × 10–6 S cm–1
NaCl (aq) 997.10 859.84 98.58 641.01 745.50±7.46 781.89±7.82
KCl (aq) 1284.94 1108.04 72.39 874.05 788.82±7.89 829.42±8.29
DCl (aq) 1002.71 864.67 68.31 934.06 1080.25±10.80 1108.16±11.08
NaOD (aq) 1290.10 1112.54 64.24 992.99 892.54±8.93 922.91±9.23
KOD (aq) 1101.98 950.29 71.66 890.59 937.17±9.37 964.82±9.65
T = 572.8 K, p = 20.8 MPa, ρD2O = 0.813 kg m–3, ηD2O = 0.00099 poise, εr = 21.10 poise, kcell = 0.0635 cm–1, κw = 2.604 × 10–6 S cm–1
NaCl (aq) 943.20 766.98 102.27 618.02 805.79±8.06 845.47±8.45
KCl (aq) 1276.10 1037.68 72.51 872.80 841.11±8.41 886.31±8.86
DCl (aq) 1056.13 858.80 68.64 928.59 1081.26±10.81 1115.77±11.16
NaOD (aq) 1186.69 965.01 69.56 916.29 949.51±9.50 984.24±9.84
KOD (aq) 1036.56 842.91 76.52 833.27 988.56±9.89 1020.86±10.21
T = 597.9 K, p = 20.7 MPa, ρD2O = 0.751 kg m–3, ηD2O = 0.00088 poise, εr = 17.71 poise, kcell = 0.0635 cm–1, κw = 1.953 × 10–6 S cm–1
NaCl (aq) 754.05 566.05 127.63 495.24 874.90±8.75 916.09±9.16
KCl (aq) 1276.10 957.98 73.96 856.06 893.61±8.94 945.10±9.45
DCl (aq) 1056.13 792.83 75.24 844.39 1065.03±10.65 1108.88±11.09
NaOD (aq) 1306.39 980.77 66.20 959.49 978.31±9.78 1025.11±10.25
KOD (aq) 1094.35 821.54 75.59 840.50 1023.08±10.23 1066.82±10.67
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a

The reported errors for the limiting conductivity correspond to our experimental precision. The absolute accuracy from 298 to 523 K is estimated to be ±3%, increasing to ±6% at 598 K.

b

10 Poise = 1 Pa.s.

Figure 1.

Figure 1

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Limiting molar conductivity, log Λ°, of hydrochloric acid in H2O and D2O from 298 to 598 K: open circles, HCl (this work); solid circles, DCl (this work); open diamonds, HCl;47 open triangles, HCl;34,35 solid triangles, DCl;34,35 and solid line, eq 4.

Figure 3.

Figure 3

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Limiting molar conductivity, log Λ°, of sodium hydroxide in H2O and D2O from 298 to 598 K: open circles, NaOH (this work); solid circles, NaOD (this work); open diamonds, NaOH;46 open triangles, NaOH;34,35 solid triangles, NaOD;34,35 and solid line, eq 4.

Figure 2.

Figure 2

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Limiting molar conductivity, log Λ°, of potassium hydroxide in H2O and D2O from 298 to 598 K: open circles, KOH (this work); solid circles, KOD (this work); open diamonds, KOH;46 open triangles, KOH;34,35 solid triangles, KOD;34,35 and solid line, eq 4.

Finally, limiting molar conductivities (Λ°) for each electrolyte were obtained by fitting the Fuoss–Hsia–Fernández–Prini extended form of the Fuoss–Onsager equation44 to the experimental molar conductivities measured at finite concentration. In the equation, the numerical values of all parameters are defined by theory, and the distance of the closest approach between ions of opposite charge is equal to the Bjerrum distance. Bianchi et al.45 have shown that this method yields the best results for 1–1 symmetrical electrolytes in the very dilute concentration range (0–0.01 mol dm–3). Because our solutions were so dilute (∼10–3 mol kg–1), ion association could be neglected. More information on the data analysis can be found in ref (33). Experimental conductivities (κ), molar conductivities (Λexp), and limiting conductivities (Λ°) for the solutions of HCl, DCl, NaOH, NaOD, KOH, and KOD in H2O and D2O are reported in Tables 1 and 2. The values for the physical properties of the solvents that were used to carry out our calculations are reported in the Supporting Information (Table S2).

3. Results

3.1. Comparison with Previous Studies

Ho et al. have reported limiting molar conductivities in light water for KOH46 and NaOH46 from 323 to 683 K at pressures up to 33 MPa, and for HCl47 from 373 to 683 K at pressures up to 31 MPa. Our experimental limiting conductivities agree with their results to within the combined experimental uncertainties up to T = 573 K, as shown in Figures 1–3. At T > 600 K, there are small systematic differences in the values of Λ°(NaOH) and Λ°(HCl) between our work and these studies, which were attributed to minor ion pairing formation in solution. The difference in the degree of ion pairing between NaOH and HCl [KA(NaOH) < KA(HCl)] supports this hypothesis. Because the goal of this study was to quantify the D2O isotope effect on ionic transport, no correction was done. Recently, Balashov et al.48 reported flow conductivity data for HCl from 373 to 673 K and pressures up to 28 MPa. Their results are consistent with our study, and with that of Ho and Palmer HCl47 up to 600 K, to within the combined experimental uncertainties.

The only high-temperature study with which to compare our results is that of Erickson et al.34,35 who measured the limiting conductivity of hydrochloric acid and sodium chloride in both H2O and D2O from 373 to 548 K at 20 MPa, using the same instrument as in the present study. As shown in Figure 1, the DCl results agree with ours to within the combined experimental uncertainties up to 548 K. The NaOD measurements from Erickson et al.34,35 are consistent with the present results above 423 K (Figure 3); however, the limiting conductivities reported at the two lowest temperatures (373 and 423 K) appear to be too low, suggesting an experimental problem with their solution.

3.2. Single-Ion Limiting Conductivities

Single-ion limiting conductivities for the H3O+, λ°(H3O+), D3O+, λ°(D3O+), hydroxide, λ°(OH–), and deuteroxide, λ°(OD–), ions were obtained by splitting the limiting molar conductivities Λ°(HCl), Λ°(DCl), Λ°(KOH), Λ°(KOD), Λ°(NaOH), and Λ°(NaOD) following Kohlrausch’s law

3.2. 3

where λ°(K+) and λ°(Cl–), measured at the exact same experimental conditions with the same apparatus, were taken from ref (33). The resulting single-ion limiting molar conductivities in H2O and D2O are reported in Tables 3 and 4 and plotted as a function of temperature in Figure 4. The solid lines in Figure 4 correspond to a fit of the empirical equation reported by Plumridge et al.33

3.2. 4

to the experimental values of λ°(H3O+), λ°(D3O+), λ°(OH–), and λ°(OD–). Here, ρw is the density of the solvent. The temperature of 228 K corresponds to the liquid–liquid critical point in supercooled H2O.49 The b term in the equation is designed to yield a discontinuity at the postulated critical temperature of liquid–liquid phase separation in supercooled water.49−52 We note that the critical temperature in supercooled D2O is thought to be T = (230 ± 5) K53,54 and is therefore consistent with eq 4 to within the margin of error. Numerical values for the fitted parameters a–d for each ion in both solvents are reported in Table 5. Standard relative uncertainties ur(λ°) were taken to be equal to the standard deviation of the relative conductivities, λ°exp/λ°fit.

Table 3. Single-Ion Limiting Conductivities (λ°) for Cl–, Na+, K+, H3O+, and OH– in H2O from T = 298 K to T = 598 K at p = 20 MPa.

T t°Cl (KCl)– λ°(Cl–) λ°(Na+) λ°(K+) λ°(H3O+) λ°(OH–)a λ°(OH–)b λE°(H3O+) λE°(OH–)a λE°(OH–)b
K   S·cm2 mol–1 S·cm2 mol–1 S·cm2 mol–1 S·cm2 mol–1 S·cm2 mol–1 S·cm2 mol–1 S·cm2 mol–1 S·cm2 mol–1 S·cm2 mol–1
298.2 0.5095 74.82 47.91 72.02 344.80   206.62 272.78   131.80
373.2 0.5195 205.28 146.25 189.88 633.12 437.99 467.82 443.24 232.72 262.54
423.3 0.5242 299.55 216.44 271.93 739.57   594.66 467.64   295.11
473.1 0.5278 390.65 289.25 349.46 799.46 679.60 668.32 450.00 288.95 277.67
522.9 0.5308 468.70 353.08 414.33 831.37 731.07 805.60 417.04 262.37 336.90
548.1 0.5321 489.92 370.28 430.83 815.20 746.03 806.49 384.37 256.11 316.57
572.8 0.5332 524.61 400.81 459.20 803.08 770.31 833.86 343.88 245.70 309.25
597.9 0.5343 559.47 433.17 487.60 768.92 775.87 864.49 281.32 216.40 305.02
urc   ±0.010 ±0.014 ±0.010 ±0.014 ±0.017 ±0.014 ±0.014 ±0.017 ±0.014
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a

Values derived from Λ°(NaOH).

b

Values derived from Λ°(KOH).

c

Standard relative uncertainty estimated from the values for other ions measured in this work (ur = δλ°/λ°).

Table 4. Single-Ion Limiting Conductivities (λ°) for Cl–, Na+, K+, D3O+, and OD– in D2O from T = 298 K to T = 598 K at p = 20 MPa.

T t°Cl (KCl)– λ°(Cl–) λ°(Na+) λ°(K+) λ°(D3O+) λ°(OD–)a λ°(OD–)b λE°(D3O+) λE°(OD–)a λE°(OD–)b
K   S·cm2 mol–1 S·cm2 mol–1 S·cm2 mol–1 S·cm2 mol–1 S·cm2 mol–1 S·cm2 mol–1 S·cm2 mol–1 S·cm2 mol–1 S·cm2 mol–1
298.2 0.5045 64.26 38.08 63.11 246.18   109.47 183.07   45.21
373.2 0.5145 177.89 131.01 167.89 485.75 294.82 287.97 317.86 116.93 110.08
423.3 0.5191 266.32 189.72 246.68 562.04   393.09 315.36   126.77
473.1 0.5228 343.82 269.04 313.81 624.60 535.82 480.26 310.79 192.00 136.44
522.9 0.5258 415.17 326.21 374.45 660.96 574.30 572.17 286.50 159.13 156.99
548.1 0.5271 437.17 344.72 392.25 670.99 578.19 572.57 278.74 141.02 135.40
572.8 0.5282 468.19 377.29 418.12 647.58 606.95 602.74 229.46 138.77 134.55
597.9 0.5293 500.26 415.83 444.84 608.62 609.28 621.98 163.78 109.02 121.72
urc   ±0.010 ±0.014 ±0.010 ±0.014 ±0.017 ±0.014 ±0.014 ±0.017 ±0.014
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a

Values derived from Λ°(NaOD).

b

Values derived from Λ°(KOD).

c

Standard relative uncertainty estimated from the values for other ions measured in this work (ur = δλ°/λ°).

Figure 4.

Figure 4

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Limiting molar conductivity of H3O+, D3O+, OH–, and OD– from 298 to 598 K: (a) open circles, H3O+; solid circles, D3O+; and (b) open diamonds, OH–; solid diamonds, OD–; solid line, eq 4.

Table 5. Fitted Parameters and Fit Standard Deviation for Single-Ion Limiting Conductivities for H3O+, D3O+, OH–, and OD– According to eq 4c.

Inline graphic
  fitted parameters (eq 4)
ion a b c d standard relative uncertainty (ur(λ°))
H3O+/H2O 0 17.1 –3950 582 0.020
D3O+/D2O 0 18.1 –4250 613 0.039
OH–/H2O 0 12.4 –2580 377 0.025
OD–/D2O 0 8.63 –2880 408 0.029
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c

Standard relative uncertainty ur(λ°) estimated from the standard deviation of the relative conductivities, λ°exp/λ°fit.

The recent paper by Balashov et al.48 reported a simplified version of the more complex model developed by Wood and his co-workers23,24 for the limiting conductivity of NaCl (aq). Their treatment relates ln (Λ°NaCl ηw) to temperature, solvent density, and solvent compressibility using six parameters obtained by regression. The Balashov model was not considered in this work as it has been designed for a limited temperature range (550–700 K).

4. Discussion

4.1. Temperature Dependence of λ°(D3O+), λ°(H3O+), λ°(OD–), and λ°(OH–)

Recently, our research group reported several experimental investigations as part of a definitive laboratory study aimed at achieving a better understanding of ionic diffusion under hydrothermal conditions in light25−27,37,54−56 and heavy waters.33−35 The effects of hydration on ionic diffusion were discussed using deviation plots, showing the difference between the experimental Walden products (λoηw) and predictions from the Stokes law

4.1. 5

This comparison provides a convenient, conceptually simple reference point at each temperature, corrected for the effects of viscosity, and is used in the discussion presented below.

Deviation plots of the limiting molar conductivities of D3O+, λ°(D3O+), H3O+, λ°(H3O+), deuteroxide, λ°(OD–), and hydroxide, λ°(OH–), are presented in Figure 5. Predictions from the Stokes law, λ°Stokes, using “slip” condition (f = 4) were calculated assuming that the crystallographic ionic radii of the bare H3O+ and D3O+ are the same [r(D3O+) = r(H3O+) = 1.40 Å], as are those for the bare hydroxide and deuteroxide [r(OD–) = r(OH–) = 1.33 Å]. The trends in Δλ/λ°Stokes for the two solvents are very similar. From ambient to hydrothermal conditions, the departure from the Stokes Law is much more important than that previously observed for other ions33 by several orders of magnitude. As an example, the deviation plots for chloride in D2O and H2O are shown in Figure 5 together with those for D3O+/H3O+ and OD–/OH–. Such results are to be expected because of the transport mechanism associated with proton hopping effects that operates in addition to simple ionic diffusion mechanisms in hydrogen-bonded solvents. This additional contribution to the ionic transport, which is commonly referred to as “anomalous”1 or “excess” conductivity4,5,8,18,19 λ°E, is typically estimated from the following assumptions8,11

4.1. 6
4.1. 7

where subscripts “exp” and “diff” stand for experimental and diffusion, respectively. In extending this assumption to elevated temperatures, we note that the recent study by Plumridge et al.33 determined that the Walden products of simple cations and anions in light and heavy waters differ by no more than 3% at temperatures above 400 K.

Figure 5.

Figure 5

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Relative deviation of the experimental conductivities of (a) chloride and (b) H3O+, D3O+, OH–, and OD– with respect to the Stokes law (using slip boundary conditions) from 298 to 598 K: open triangles, Cl–/H2O; solid triangles, Cl–/D2O; open circles, H3O+; solid circles, D3O+; open diamonds, OH–; solid diamonds, OD–; and solid line, eq 4.

The classic descriptions of proton hopping have been reviewed by Agmon9 and, more recently, by Marx.10 The mechanism presented by Tada et al.,8 which is typical of this approach, is shown in Figures 6 and 7 for the series of proton transfer steps responsible for the excess conductivities of H3O+ and OH– at 298.15 K. Briefly, the mechanism consists of the following steps. First, because of the repulsive effects between adjacent protons in the hydrogen-bonded liquid, the presence of the H3O+ ion causes the neighboring water molecule to rotate, exposing a lone pair and creating a hydrogen bond. The local electric field then drives the proton to transfer from the H3O+ to the neighboring water molecule. The newly formed H3O+ ion causes the next neighboring water molecule in the solvent to rotate, beginning the process anew. Hydroxide undergoes a similar hopping mechanism, in which protons migrate from an adjacent water molecule to a lone pair on the proton-deficient OH– ion. Proton hopping is less favored due to lower repulsive forces between the OH– ion and the neighboring water molecules8 and because the hydroxide ion has only one hydrogen vs three for the H3O+ ion. The D3O+ and OD– ions undergo the same type of hopping mechanism but, due to the greater mass of D vs H, the rotational step required to start the transfer is less energetically favorable.

Figure 6.

Figure 6

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Schematic representation of the reorientation of neighboring heavy-water molecules during the D3O+ hopping mechanism (redrawn from ref (8)).

Figure 7.

Figure 7

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Schematic representation of the reorientation of neighboring heavy-water molecules during the OD– hopping mechanism (redrawn from ref (8)).

The modern understanding is that proton hopping near ambient conditions includes contributions from both the classic Eigen mechanism described above and proton tunneling effects associated with the so-called Zundel species in which the proton is bound equally to two oxygen atoms in neighboring water molecules.10,57 The energy of activation of the two mechanisms is thought to be different. Ab initio molecular dynamics studies have yielded further insights into the details of these mechanisms.58 The two limiting states of the hydronium ion during the proton transfer process are thought to be the Eigen cation, H3O+, solvated by three water molecules to form the species (H2O)4H+ and the Zundel cation (H2O)2H+, which contains a delocalized proton between two water oxygens. The transfer involves a broad distribution of intermediate states, the “Zundel continuum” [refs (2, 61)]. The proton transfer of the hydroxide ion involves a process through which the proton “hole” (OH–) in the species (H2O)4OH– accepts three hydrogen bonds to form (H2O)3OH– and a Zundel anion (H2O)OH–.58

4.2. Deuterium Isotope Effects on Excess Conductivity and “Proton Hopping”

Only a few experimental studies have investigated the effect of temperature on the excess conductivity λE° under hydrothermal conditions.18,19,48,59 Franck et al.59 designed a static conductivity apparatus to measure this effect between 45 and 493 K at pressures up to 800 MPa using dilute solutions of KCl, HCl, and H2SO4. Ho et al.18,19 also used a static technique to measure the conductivity of dilute solutions of KOH, NaOH, KCl, and NaCl up to 873 K up to 300 MPa and derived λE° values for OH– and H3O+ using their results and previously published data by Noyes.60 Balashov et al.48 derived more accurate values for H3O+ based on very dilute flow conductivity measurements between 373 and 673 K up to 28 MPa.

Experimental values for the “excess” ionic conductivity, λE°, for H3O+ and OH– from the present study are tabulated in Table 3. The temperature dependence of λE° clearly shows distinct maxima for H3O+ at ∼423 K and for OH– at ∼500 K (Figure 8). The results are consistent with previous observations,18,19,40,59,60 with maxima at ∼423–448 K for H3O+ and ∼493 K for OH–. The maximum at ∼500 K for λE°(H3O+) compares well with the value of 481 K estimated by Sluyters and Sluyters–Rehbach.57 Experimental values for the “excess” ionic conductivity, λE°, of D3O+ and OD– are tabulated in Table 4 and plotted in Figure 8. The excess conductivity observed with OD–/OH– is less than for D3O+/H3O+. Moreover, there is no clear evidence that the temperatures of maximum λE° in D2O are different from those in H2O.

Figure 8.

Figure 8

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Excess limiting molar conductivity of H3O+, D3O+, OH–, and OD– from 298 to 598 K: (a) open circles, H3O+; solid circles, D3O+; and (b) open diamonds, OH–; solid diamonds, OD–; and solid line, eq 4.

The temperature dependence of λE° is due to two competing factors, which are responsible for the maxima observed. First, as temperature increases, the thermal energy in the system favors the rotation of solvent water molecules, which would increase the rate of proton hopping4,5,8 by both the classic Eigen mechanism and the Zundel proton tunneling mechanisms. Second, the coordination number of each solvent molecule decreases from approximately nw ≈ 4 near ambient conditions, dropping to nw ≈ 3 at 473 K and to nw ≈ 2 near the critical temperature and pressure of water (647.1 K and 22.1 MPa in H2O; 643.8 K and 21.7 MPa in D2O).61 These coordination numbers correspond to approximately tetrahedrally coordinated three-dimensional water networks, trigonally coordinated two-dimensional networks, and then one-dimensional chains and rings, which may break up the hydrogen-bonded “wires” required for proton hopping to take place.

In his discussion of excess conductivity effects under near-critical conditions, Balashov48 has speculated that the proton hopping from one H3O+ Eigen cation to form another through the Zundel intermediate cation is inhibited by the lower availability of neighboring hydrogen-bonded water molecules. We are aware of no other modern experimental studies that address the issue. To the best of our knowledge, the effects on proton transport mechanisms caused by the breakdown in tetrahedrally coordinated three-dimensional water “structure” in the ambient liquid into two-dimensional networks and one-dimensional chains or other structures under hydrothermal conditions have not yet been studied by ab initio simulation methods.

The temperature dependence of the experimental excess conductivities for the hydronium ion, plotted in Figure 8, shows a shallow maximum value at T ≈ 420 K for both D3O+ and H3O+, consistent with the progressive breakdown in the hydrogen-bonded networks of water, as suggested by Balashov.48 The excess conductivity of D3O+ is lower than that of H3O+ by about 30% over the entire temperature range, as must be the case if the rate-determining step involves the rotation of a D2O or H2O water molecule postulated by the classical mechanism shown in Figure 6.4,5,8

As shown in Figure 8, the excess conductivity for the hydroxide ion, OD–/OH–, reaches its maximum at a higher temperature than that of D3O+/H3O+, T ≈ 500 K. In addition, the value of λE° for OD– is much lower than that of OH–, ∼60% throughout the range, and the values for both ions decrease only slightly with increasing temperature above T > 500 K. The much larger deuterium isotope effect for the hydroxide ion undoubtedly arises because the proton hopping mechanism requires the rotation of a water molecule to orient one of its two hydrogens (see Figure 6), while the hydronium ion activation step requires the rotation of the H3O+ ion itself.6,8,58 The steep decrease in λE° for D3O+ and H3O+ above 423 K, compared to the almost temperature-independent values of λE° for OD– and OH– at these temperatures, may reflect the structures of the Eigen and Zundel transition-state species. We speculate that the breakdown in the coordination number of water solvent may destabilize the tetrahedral species (H2O)4H+ required to form the Zundel cation (H2O)2H+, while the Eigen species (H2O)3OH–, required to form the Zundel anion (H2O)OH–, may be much less destabilized. Finally, we note that the understanding of mechanisms derived from ab initio simulations at 25 °C is complicated by increasingly large electrostriction effects associated with long-range polarization of water dipoles and the sharp increase in the isothermal compressibility of liquid water from temperatures above ∼250 °C up to the critical point of water.62

4.3. Deuterium Isotope Effects: The Relative Contributions of Diffusion and Proton Hopping

As discussed above, the excess conductivities for D3O+ and OD– are smaller than those of H3O+ and OH– (Figures 8 and 9), suggesting that proton hopping in H2O is more efficient than in D2O near ambient conditions and that the effect persists up to near-critical conditions. As the number of coordinated solvent molecules decreases, it can be assumed that both excess conductivities, λE°(D3O+/H3O+) and λE°(OD–/OH–), would decrease relative to ionic conductivity by diffusion. Figure 9 presents plots of the ratio of excess conductivity to the total limiting conductivity, λE°/λ°, for D3O+, H3O+, OD–, and OH– as a function of temperature at p = 20 MPa based on the data in Tables 3 and 4. As expected, the results indicate that the contribution due to proton hopping decreases steadily as the critical temperature is approached. At ambient conditions, the excess conductivity for H3O+ makes up to 77% of the total conductivity, while the contributions for D3O+, OH–, and OD– are ∼70, ∼60, and ∼40%. These compare well with the work of Tada et al.4,8,5 who reported 298 K values of ∼80, ∼75, ∼60, and ∼50%. Once the temperature is increased to T > 598 K (Figure 9), the ratio of proton hopping to the total conductivity falls significantly, to values of ∼40, ∼35, ∼30, and ∼20% for H3O+, OH–, D3O+, and OD–, respectively.

Figure 9.

Figure 9

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Ratio of excess limiting conductivity to total limiting conductivity for (a) D3O+and H3O+ and (b) OD– and OH– with respect to the Stokes law (using slip boundary conditions) from 298 to 598 K: open diamonds, H3O+; solid diamonds, D3O+; open circles, OH–; solid circles, OD–; and solid line, eq 4.

Conclusions

This paper reports the first measurements of accurate electrical conductivities of aqueous acids and bases in D2O under hydrothermal conditions. The unique high-precision flow-through AC electrical conductance instrument designed by R.H. Wood and his co-workers was used to make quantitative measurements of the contribution of proton hopping to the limiting conductivities of D3O+, H3O+, OD–, and OH–. The flow capabilities of the instrument permitted impedance measurements for each solution to be made in sequence under nearly identical conditions, so that the systematic errors associated with high-temperature conductivity measurements cancel one another. This allowed deuterium isotope effects to be measured with the required precision. The “excess” conductivities of D3O+ and H3O+ show a well-defined maximum at T ≈ 420 K, which we attribute to the breakdown of a three-dimensional tetrahedral “structure” of liquid water into two-dimensional networks and chains at elevated temperatures. The deuterium isotope effect on the excess conductivities of the hydroxide ion is greater, and the maximum in excess conductivities is less pronounced, suggesting that the structures of the Eigen and Zundel intermediate anions are relatively more stable than those of the hydronium ion in the two-dimensional three-coordinate hydrogen-bonded water “structures” that exist under these conditions. These are the first results of their kind reported at temperatures above 318 K. Moreover, recent analyses have shown that the limiting conductivities for H3O+ and OH– from these measurements can be used with experimental conductivities for pure water to derive new values for the ionization constant of water, Kw,H2O, up to subcritical and supercritical conditions.41 A similar analysis for the ionization constant of D2O is more difficult because the degree of ionization is less, but may be possible at elevated temperatures where autoprotolysis is greater.

In addition to their significance to basic research, the results suggest that the corrosion and radiolysis processes, which proceed by proton hopping mechanisms, are significantly slower in heavy-water reactors relative to the light-water supercritical water (SCWR) and pressurized water (PWR) reactors and that the differences become less pronounced as temperatures approach near-critical conditions (Figure 9). The quantitative data diffusion constant data derived from these results may be useful for modeling radiolysis21 and corrosion reactions in the primary coolant circuits of light-water nuclear reactors (PWRs) and CANDU heavy-water nuclear reactors (HWPRs) since few other sources of transport property data exist under these very aggressive conditions.

Acknowledgments

The authors thank Ian Renaud and Casey Gielen of the electronics shop and machine shop in the College of Engineering and Physical Science at the University of Guelph for their very considerable expertise in maintaining and modifying the instrument and its data acquisition system. The measurements were carried out as part of Jeff Plumridge’s MSc Project at the University of Guelph. The research was supported by the Natural Science and Engineering Research Council of Canada (NSERC) Discovery Grant to PT and Project NNAPJ 424121-11, Ontario Power Generation Ltd. (OPG), and the University Network of Excellence in Nuclear Engineering (UNENE). Technical advice and encouragement were provided by Dr. Dave Guzonas, Atomic Energy of Canada Ltd.; Dr. Dave Evans, OPG; Prof. Greg Zimmerman, Bloomsburg University; Dr. Jenny Cox and Dr. Jacy Conrad, University of Guelph.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpcb.2c02929.

  • Experimental molar conductivities Λ (NaCl) measured before and after each set of runs for calibration purposes; and values used for density (ρ), viscosity (η), dielectric constant (εr), and pK of H2O and D2O from T = 298 K to T = 598 K at p = 20 MPa (PDF)

Author Present Address

@ Wiss, Janney, Elstner Associates, Inc. 330 Pfingsten Road, Northbrook, IL 60062, United States

The authors declare no competing financial interest.

Supplementary Material

jp2c02929_si_001.pdf (305.5KB, pdf)

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