self-ionization of water

The self-ionization of water was first proposed in 1884 by Svante Arrhenius as part of the theory of ionic dissociation which he proposed to explain the conductivity of electrolytes including water. Arrhenius wrote the self-ionization as H2O <=> H+ + OH-. At that time, nothing was yet known of atomic structure or subatomic particles, so he had no reason to consider the formation of an H+ ion from a hydrogen atom on electrolysis as any less likely than, say, the formation of a Na+ ion from a sodium atom.

In 1923 Johannes Nicolaus Brønsted and Martin Lowry proposed that the self-ionization of water actually involves two water molecules: H2O + H2O <=> H3O+ + OH-. By this time the electron and the nucleus had been discovered and Rutherford had shown that a nucleus is very much smaller than an atom. This would include a bare ion H+ which would correspond to a proton with zero electrons. Brønsted and Lowry proposed that this ion does not exist free in solution, but always attaches itself to a water (or other solvent) molecule to form the hydronium ion H3O+ (or other protonated solvent).

Later spectroscopic evidence has shown that many protons are actually hydrated by more than one water molecule. The most descriptive notation for the hydrated ion is H+(aq), where aq (for aqueous) indicates an indefinite or variable number of water molecules. However the notations H+ and H3O+ are still also used extensively because of their historical importance. This article mostly represents the hydrated proton as H3O+, corresponding to hydration by a single water molecule.

File:Autoionizacion-agua.gif

Chemically pure water has an electrical conductivity of 0.055 μS/cm. According to the theories of Svante Arrhenius, this must be due to the presence of ions. The ions are produced by the water self-ionization reaction, which applies to pure water and any aqueous solution:

: H₂O + H₂O {{eqm}} H₃O⁺ + OH⁻

Expressed with chemical activities {{mvar|a}}, instead of concentrations, the thermodynamic equilibrium constant for the water ionization reaction is:

:$K\_\{\backslash rm\; eq\}\; =\; \backslash frac\{a\_\{\backslash rm\{H\_3O^+\}\}\; \backslash cdot\; a\_\{\backslash rm\{OH^-\}\}\}\{a\_\{\backslash rm\{H\_2O\}\}^2\}$

which is numerically equal to the more traditional thermodynamic equilibrium constant written as:

:$K\_\{\backslash rm\; eq\}\; =\; \backslash frac\{a\_\{\backslash rm\{H^+\}\}\; \backslash cdot\; a\_\{\backslash rm\{OH^-\}\}\}\{a\_\{\backslash rm\{H\_2O\}\}\}$

under the assumption that the sum of the chemical potentials of H⁺ and H₃O⁺ is formally equal to twice the chemical potential of H₂O at the same temperature and pressure.{{cite web|url=http://www.iapws.org/relguide/Ionization.pdf |title=Release on the Ionization Constant of H₂O |publisher=The International Association for the Properties of Water and Steam |location=Lucerne|date=August 2007}}

Because most acid–base solutions are typically very dilute, the activity of water is generally approximated as being equal to unity, which allows the ionic product of water to be expressed as:{{GoldBookRef|file=A00532|title=autoprotolysis constant}}

:$K\_\{\backslash rm\; eq\}\; \backslash approx\; a\_\{\backslash rm\{H\_3O^+\}\}\; \backslash cdot\; a\_\{\backslash rm\{OH^-\}\}$

In dilute aqueous solutions, the activities of solutes (dissolved species such as ions) are approximately equal to their concentrations. Thus, the ionization constant, dissociation constant, self-ionization constant, water ion-product constant or ionic product of water, symbolized by K_w, may be given by:

:$K\_\{\backslash rm\; w\}=[\{\backslash rm\{H\_3O^+\}\}][\{\backslash rm\{OH^-\}\}]$

where [H₃O⁺] is the molarity (molar concentration){{cite book |first1=Werner |last1=Stumm |first2=James |last2=Morgan |title=Aquatic Chemistry. Chemical Equilibria and Rates in Natural Waters |edition=3rd |publisher=John Wiley & Sons, Inc. |year=1996 |isbn=9780471511847}} of hydrogen cation or hydronium ion, and [OH⁻] is the concentration of hydroxide ion. When the equilibrium constant is written as a product of concentrations (as opposed to activities) it is necessary to make corrections to the value of $K\_\{\backslash rm\; w\}$ depending on ionic strength and other factors (see below).{{cite book|last1=Harned|first1=H. S. |last2=Owen |first2=B. B. |title=The Physical Chemistry of Electrolytic Solutions|url=https://archive.org/details/physicalchemistr0003harn|url-access=registration|edition=3rd|year=1958|publisher=Reinhold|location=New York|pages=[https://archive.org/details/physicalchemistr0003harn/page/635 635]}}

At 24.87 °C and zero ionic strength, K_w is equal to {{val|1.0|e=-14}}. Note that as with all equilibrium constants, the result is dimensionless because the concentration is in fact a concentration relative to the standard state, which for H⁺ and OH⁻ are both defined to be 1 molal (= 1 mol/kg) when molality is used or 1 molar (= 1 mol/L) when molar concentration is used. For many practical purposes, the molality (mol solute/kg water) and molar (mol solute/L solution) concentrations can be considered as nearly equal at ambient temperature and pressure if the solution density remains close to one (i.e., sufficiently diluted solutions and negligible effect of temperature changes). The main advantage of the molal concentration unit (mol/kg water) is to result in stable and robust concentration values which are independent of the solution density and volume changes (density depending on the water salinity (ionic strength), temperature and pressure); therefore, molality is the preferred unit used in thermodynamic calculations or in precise or less-usual conditions, e.g., for seawater with a density significantly different from that of pure water, or at elevated temperatures, like those prevailing in thermal power plants.

We can also define pK_w $\backslash equiv$ −log₁₀ K_w (which is approximately 14 at 25 °C). This is analogous to the notations pH and pK_a for an acid dissociation constant, where the symbol p denotes a cologarithm. The logarithmic form of the equilibrium constant equation is pK_w = pH + pOH.

The dependence of the water ionization on temperature and pressure has been investigated thoroughly.[http://www.iapws.org/ International Association for the Properties of Water and Steam (IAPWS)] The value of pK_w decreases as temperature increases from the melting point of ice to a minimum at c. 250 °C, after which it increases up to the critical point of water c. 374 °C. It decreases with increasing pressure

With electrolyte solutions, the value of pK_w is dependent on ionic strength of the electrolyte. Values for sodium chloride are typical for a 1:1 electrolyte. With 1:2 electrolytes, MX₂, pK_w decreases with increasing ionic strength.{{cite book|last1=Harned|first1=H. S. |last2=Owen |first2=B. B. |title=The Physical Chemistry of Electrolytic Solutions|url=https://archive.org/details/physicalchemistr0003harn|url-access=registration|edition=3rd|year=1958|publisher=Reinhold|location=New York|pages=[https://archive.org/details/physicalchemistr0003harn/page/634 634]–649, 752–754}}

The value of K_w is usually of interest in the liquid phase. Example values for superheated steam (gas) and supercritical water fluid are given in the table.

:

:Notes to the table. The values are for supercritical fluid except those marked: ^a at saturation pressure corresponding to 350 °C. ^b superheated steam. ^c compressed or subcooled liquid.

Heavy water, D₂O, self-ionizes less than normal water, H₂O;

:D₂O + D₂O {{eqm}} D₃O⁺ + OD⁻

This is due to the equilibrium isotope effect, a quantum mechanical effect attributed to oxygen forming a slightly stronger bond to deuterium because the larger mass of deuterium results in a lower zero-point energy.

Expressed with activities a, instead of concentrations, the thermodynamic equilibrium constant for the heavy water ionization reaction is:

:$K\_\{\backslash rm\; eq\}\; =\; \backslash frac\{a\_\{\backslash rm\{D\_3O^+\}\}\; \backslash cdot\; a\_\{\backslash rm\{OD^-\}\}\}\{a\_\{\backslash rm\{D\_2O\}\}^2\}$

Assuming the activity of the D₂O to be 1, and assuming that the activities of the D₃O⁺ and OD⁻ are closely approximated by their concentrations

:$K\_\{\backslash rm\; w\}=[\{\backslash rm\{D\_3O^+\}\}][\{\backslash rm\{OD^-\}\}]$

The following table compares the values of pK_w for H₂O and D₂O.{{cite book |editor-last=Lide|editor-first= D. R.|title=CRC Handbook of Chemistry and Physics |url=https://archive.org/details/handbookofchemis00crcp|url-access=registration|edition=70th |publisher=Boca Raton (FL):CRC Press |year=1990|isbn= 978-0-8493-0471-2}}

:

In water–heavy water mixtures equilibria several species are involved: H₂O, HDO, D₂O, H₃O⁺, D₃O⁺, H₂DO⁺, HD₂O⁺, HO⁻, DO⁻.

The rate of reaction for the ionization reaction

:2 H₂O → H₃O⁺ + OH⁻

depends on the activation energy, ΔE^‡. According to the Boltzmann distribution the proportion of water molecules that have sufficient energy, due to thermal population, is given by

:$\backslash frac\{N\}\{N\_0\}\; =\; e^\{-\backslash frac\{\backslash Delta\; E^\backslash ddagger\}\{kT\}\}$

where k is the Boltzmann constant. Thus some dissociation can occur because sufficient thermal energy is available. The following sequence of events has been proposed on the basis of electric field fluctuations in liquid water.{{cite journal |last1= Geissler|first1= P. L.|last2= Dellago|first2= C.|author2-link=Christoph Dellago|last3=Chandler|first3= D.|last4=Hutter|first4= J.|last5=Parrinello|first5= M. |year = 2001 |title = Autoionization in liquid water |journal = Science |volume = 291 |pages = 2121–2124 |doi = 10.1126/science.1056991 |pmid = 11251111 |issue = 5511|bibcode = 2001Sci...291.2121G |citeseerx= 10.1.1.6.4964}} Random fluctuations in molecular motions occasionally (about once every 10 hours per water molecule{{cite journal |last1= Eigen|first1= M. |last2=De Maeyer|first2= L. |year = 1955|trans-title=Investigations on the kinetics of neutralization I |title = Untersuchungen über die Kinetik der Neutralisation I |journal = Z. Elektrochem. |volume = 59 |pages = 986}}) produce an electric field strong enough to break an oxygen–hydrogen bond, resulting in a hydroxide (OH⁻) and hydronium ion (H₃O⁺); the hydrogen nucleus of the hydronium ion travels along water molecules by the Grotthuss mechanism and a change in the hydrogen bond network in the solvent isolates the two ions, which are stabilized by solvation. Within 1 picosecond, however, a second reorganization of the hydrogen bond network allows rapid proton transfer down the electric potential difference and subsequent recombination of the ions. This timescale is consistent with the time it takes for hydrogen bonds to reorientate themselves in water.{{cite book |last= Stillinger|first= F. H. |title= Advances in Chemical Physics |year = 1975 |journal = Adv. Chem. Phys. |volume = 31 |pages = 1–101 |doi = 10.1002/9780470143834.ch1 |chapter = Theory and Molecular Models for Water|isbn= 9780470143834 }}{{cite journal |last= Rapaport|first= D. C. |year = 1983 |journal = Mol. Phys. |volume = 50 |pages = 1151–1162 |doi = 10.1080/00268978300102931 |title = Hydrogen bonds in water |issue = 5|bibcode = 1983MolPh..50.1151R }}{{cite book |last1= Chen|first1= S.-H. |last2=Teixeira|first2= J. |year = 1986 |journal = Adv. Chem. Phys. |volume = 64 |pages = 1–45 |doi = 10.1002/9780470142882.ch1 |title = Structure and Dynamics of Low-Temperature Water as Studied by Scattering Techniques|series= Advances in Chemical Physics |isbn= 9780470142882 }}

The inverse recombination reaction

:H₃O⁺ + OH⁻ → 2 H₂O

is among the fastest chemical reactions known, with a reaction rate constant of {{val|1.3|e=11|u=M⁻¹ s⁻¹}} at room temperature. Such a rapid rate is characteristic of a diffusion-controlled reaction, in which the rate is limited by the speed of molecular diffusion.{{cite book|last1=Tinoco|first1= I.|last2= Sauer |first2=K. |last3=Wang |first3=J. C. |title=Physical Chemistry: Principles and Applications in Biological Sciences|url=https://archive.org/details/solutionsmanualp00tino|url-access=registration|edition=3rd |publisher=Prentice-Hall |date=1995| page=386|isbn= 978-0-13-435850-5}}

Water molecules dissociate into equal amounts of H₃O⁺ and OH⁻, so their concentrations are almost exactly {{val|1.00|e=-7|u=mol dm⁻³}} at 25 °C and 0.1 MPa. A solution in which the H₃O⁺ and OH⁻ concentrations equal each other is considered a neutral solution. In general, the pH of the neutral point is numerically equal to {{sfrac|2}}pK_w.

Pure water is neutral, but most water samples contain impurities. If an impurity is an acid or base, this will affect the concentrations of hydronium ion and hydroxide ion. Water samples that are exposed to air will absorb some carbon dioxide to form carbonic acid (H₂CO₃) and the concentration of H₃O⁺ will increase due to the reaction H₂CO₃ + H₂O = HCO₃⁻ + H₃O⁺. The concentration of OH⁻ will decrease in such a way that the product [H₃O⁺][OH⁻] remains constant for fixed temperature and pressure. Thus these water samples will be slightly acidic. If a pH of exactly 7.0 is required, it must be maintained with an appropriate buffer solution.

{{Portal|Water}}

• Acid–base reaction
• Chemical equilibrium
• Molecular autoionization (of various solvents)
• Standard hydrogen electrode

• [http://www.vias.org/genchem/acidbase_equ_12591_04.html General Chemistry] – Autoionization of Water

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