The Ionization Constant of Water over Wide Ranges of Temperature and Density Andrei V. Bandura Department of Quantum Chemistry, St. Petersburg State University, 26 University Prospect, Petrodvoretz, St. Petersburg 198504, Russia Serguei N. Lvova… The Energy Institute and Department of Energy & Geo-Environmental Engineering, The Pennsylvania State University, 207 Hosler Building, University Park, Pennsylvania 16802 ͑Received 26 April 2004; revised manuscript received 1 March 2005; accepted 5 April 2005; published online 8 December 2005͒ A semitheoretical approach for the ionization constant of water, KW , is used to fit the available experimental data over wide ranges of density and temperature. Statistical ther- modynamics is employed to formulate a number of contributions to the standard state chemical potential of the ionic hydration process. A sorption model is developed for calculating the inner-shell term, which accounts for the ion–water interactions in the immediate ion vicinity. A new analytical expression is derived using the Bragg–Williams approximation that reproduces the dependence of a mean ion solvation number on the solvent chemical potential. The proposed model was found to be correct at the zero- density limit. The final formulation has a simple analytical form, includes seven adjust- able parameters, and provides good fitting of the collected KW data, within experimental uncertainties, for a temperature range of 0 – 800 °C and densities of 0 – 1.2 g cmϪ3.©2006 American Institute of Physics. ͓DOI: 10.1063/1.1928231͔ Key words: Bragg–Williams approximation; high temperatures and low densities; ionization constant of water; statistical thermodynamics solvation model. Contents 3. References on the experimentally obtained KW data used for estimating the empirical parameters Nomenclature.............................. 16 given in Tables 1 and 2...................... 24 List of Symbols............................ 16 4. Negative logarithm ͑base 10͒ of the ionization 1. Introduction................................ 16 constant of water, KW , calculated using Model 2. Theory and Computations. ................... 17 II ͓Eq. ͑39͔͒............................... 25 2.1. Thermodynamic Consideration............. 17 2.2. Model................................. 17 2.2.1. Cavity Formation Energy............ 18 List of Figures 2.2.2. Solvation Contributions............. 18 1. Contributions to the residual part of the standard 2.2.3. Electrostatic Polarization Term. ...... 20 chemical potential change, ⌬␮R, as a function 2.3. Simplified Analytical Expression........... 20 of density at 400 ° ......................... 20 2.4. Fitting Procedure........................ 21 2. Temperature–density distribution of the available 2.5. Choice of Experimental Data.............. 22 experimental KW data: ...................... 24 2.5.1. Conductivity...................... 22 3. Deviation of calculated lg KW values from 2.5.2. Potentiometry..................... 23 experimental data in supercritical and low-density 2.5.3. Calorimetry....................... 23 regions at 0.08рDр1.00 g cmϪ3 ............. 26 3. Results and Discussion....................... 24 4. Deviation of calculated lg KW values from the 4. Acknowledgments.......................... 27 experimental data in high-density region at 1.0 5. Appendix.................................. 27 рDр1.2 g cmϪ3 ........................... 26 6. References................................. 29 5. Temperature dependence of lg KW at the saturated vapor pressure for Tу200°C......... 26 List of Tables 6. Temperature dependence of lg KW at a constant 1. Empirical parameters calculated using Model I. 22 density ................................... 26 ͑ 2. Empirical parameters calculated using Model II. 22 7. Density dependence of lg KW at 400 °C upper curves͒ and 800 °C ͑lower curves͒ ............. 27 Ϫ ͒ 3 a Corresponding author; electronic mail: [email protected] 8. Dependence of lg KW from lg(D/gcm ) at 400 © 2006 American Institute of Physics. and800°C................................ 27 Õ Õ Õ Õ Õ 0047-2689 2006 35„1… 15 16 $40.0015 J. Phys. Chem. Ref. Data, Vol. 35, No. 1, 2006 16 A. V. BANDURA AND S. N. LVOV 9. Bragg–Williams sorption isotherm, ␪ϭ␪(z), R residual part of the chemical calculated using different values of ␻/RTϭ5 potential ͑left curves͒ and ␻/RTϭ30 ͑right curves͒ ...... 28 S solvation complex 10. Excess chemical potential ⌬␮˜ of the sorbent W water particle in the Bragg–Williams approximation X chemical component calculated using different values of ␻/RTϭ5 ͑left curve͒ and ␻/RTϭ30 ͑right curve͒ ............ 28 1. Introduction Nomenclature The standard-state thermodynamic properties of aqueous Abbreviations ions and molecules over wide ranges of temperature and den- BMCSL Boublik–Mansoori–Carnahan–Starling–Leland sity are required for modeling most of the physicochemical M&F Marshall and Franck processes in aqueous media related to a number of important MSA mean spherical approximation applications. The ionization constant of water (KW)isa benchmark property in aqueous solution chemistry and KW List of Symbols has been experimentally obtained over wide ranges of tem- Symbol Physical quantity Unit perature and pressure ͑see a brief review of published papers Ϫ below͒. In 1981, Marshall and Franck ͑M&F͒ proposed an D mass density g cm 3 empirical equation for pK , which was an empirical ap- f fugacity MPa W proach based on the available experimental measurements G Gibbs energy ͑Marshall and Franck 1981͒. The M&F equation has a simple K ionization constant analytical form and well represents the experimental data for k Boltzmann’s constant densities above 0.4 g cmϪ3. While the M&F equation, tech- l. u. decimal logarithmic unit Ϫ3 Ϫ nically, allows an extrapolation below 0.4 g cm , there is M molar mass g mol 1 some concern ͑Chen et al. 1994a͒ that the M&F formulation n ion coordination number may not give accurate values of K at low densities. P pressure MPa W Ϫ Ϫ Several attempts have been made for predicting the stan- R gas constant (8.31451 J mol 1 K 1) dard chemical potentials of aqueous electrolytes at condi- r particle radius Å tions other than ambient and most of the approaches have T absolute temperature K employed the well-known Born ͑1920͒ equation for estimat- V volume ing the continuum dielectric polarization by an ion. An im- v molecular free-volume parameter provement of this approximation led to the so-called semi- z sorption isotherm parameter continuum models where the solvent in the region ␧ interparticle interaction neighboring the ion is treated as a medium with discrete parameter; relative permittivity molecules, while the outside of this region is considered as a ␨ particle diameter ratio dielectric continuum ͑see, for example, Ferna´ndez-Prini ␩ reduced density et al. 1992͒. The semicontinuum models have been success- ␪ occupation fraction in solvation ful in a number of respects but cannot be used alone for complex calculating the ionization constant of water over a wide ␭ absolute activity range of densities. In 1982, Pitzer made an attempt to calcu- ␮ chemical potential late the ionization constant of water at high temperatures and ⌶ grand partition function low densities ͑Pitzer 1982͒. He regarded water vapor as a ␳ number particle density near-perfect gas mixture of hydrates of different composi- ␴ parameter for interaction between tions and used a reliable set of mass spectrometry data ob- solvation complex and bulk tained by Kebarle ͑1977͒ and Lau et al. ͑1982͒. It was found ␻ water–water interaction energy in that K values predicted by Pitzer decrease much faster than solvation complex W that obtaned from the M&F equation at densities below Subscripts and superscripts 0.4 g cmϪ3, and the difference between M&F’s and Pitzer’s ° standard state ͑pure fluid for water calculations can be up to 10 orders of magnitude. Tanger and and 1 mol kgϪ1 solution exhibiting Pitzer ͑1989a, b͒ extended the Pitzer’s approach to a region infinitely diluted of higher temperatures and densities to permit predictions up properties for ionic species͒ to 1000 °C and 500 MPa. The revised semicontinuum model B boundary between solvation complex of Tanger and Pitzer is a more realistic representation of both and bulk the inner-shell and outer-shell contributions to the hydration C cavity process. These authors concluded that their semicontinuum E electrostatic polarization model is more reliable than the M&F equation at pressures G ideal gas state and temperatures where the density of water is less than I ion 0.4 g cmϪ3. J. Phys. Chem. Ref. Data, Vol. 35, No. 1, 2006 IONIZATION CONSTANT OF WATER 17 ͑ ͒ ␮G ␳ Klots 1984 derived an equation for calculating the pH The ideal gas chemical potential X(T, X), can be calcu- values of steam along the coexistence curve with liquid lated using the well-known statistical thermodynamic ap- phase using a rather different approach than that of Tanger proach ͑Hill 1956͒ and Pitzer ͑1989a, b͒. Klots considered a distribution of ion– ␮G͑T,␳ ͒ϭϪRT ln͑q /⌳3 ␳ ͒ water clusters in the framework of a liquid-drop model. The X X X X X author concluded that any extrapolation using the M&F ϭ␮°G͑ ͒ϩ ͑ G ͒ ͑ ͒ X T RT ln xXP /P0 , 3 equation might overestimate the ionic content of vapor at ͑ ͒ ⌳ low temperatures and densities. where qX is the intrinsic vibrational partition function, X ͑ ͒ is the kinetic ͑translational and rotational͒ partition function, The paper of Tawa and Pratt 1995 should also be men- G tioned here as an example of the continuum approach. The xX is the mole fraction of the component X, and P is the ␮°G authors have applied a dielectric solvation model to predict total pressure for the gas-phase reference system. X (T)in ͑ ͒ the equilibrium ionization of liquid water over wide ranges Eq. 3 is the gas-phase standard chemical potential, which of density and temperature. In addition, the model includes can be presented as: an approximate description of the polarizability of dissoci- ␮°G͑ ͒ϭϪ ͑ ⌳3 ͒ ͑ ͒ X T RT ln kTqX / XP0 , 4 ated water molecules. It was found that the calculated pK W ͑ ͒ values are extremely sensitive to the water molecule radius, where P0 is the standard pressure 0.1 MPa .