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Equilibrium constant

Animation of the self-ionization of water

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 ⇌ H3O+ + OH−

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

Keq=aH3O+⋅aOH−aH2O2{\displaystyle K_{\rm {eq}}={\frac {a_{\rm {H_{3}O^{+}}}\cdot a_{\rm {OH^{-}}}}{a_{\rm {H_{2}O}}^{2}}}}

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

Keq=aH+⋅aOH−aH2O{\displaystyle K_{\rm {eq}}={\frac {a_{\rm {H^{+}}}\cdot a_{\rm {OH^{-}}}}{a_{\rm {H_{2}O}}}}}

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.[1]

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:[2]

Keq≈aH3O+⋅aOH−{\displaystyle K_{\rm {eq}}\approx a_{\rm {H_{3}O^{+}}}\cdot a_{\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 Kw, may be given by:

Kw=[H3O+][OH−]{\displaystyle K_{\rm {w}}=[{\rm {H_{3}O^{+}}}][{\rm {OH^{-}}}]}

where [H3O+] is the molarity (≈ molar concentration)[3] of hydrogen 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 Kw{\displaystyle K_{\rm {w}}} depending on ionic strength and other factors (see below).[4]

At 25 °C and zero ionic strength, Kw is equal to 1.0×10−14. 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 (or nearly 1 molar). For many practical purposes, the molal (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,[3] or at elevated temperatures, like those prevailing in thermal power plants.

We can also define pKw≡{\displaystyle \equiv } −log10 Kw (which is approximately 14 at 25 °C). This is analogous to the notations pH and pKa for an acid dissociation constant, where the symbol p denotes a cologarithm. The logarithmic form of the equilibrium constant equation is pKw = pH + pOH.

Dependence on temperature, pressure and ionic strength

Temperature dependence of the water ionization constant at 25 MPa

Pressure dependence of the water ionization constant at 25 °C

Variation of pKw with ionic strength of NaCl solutions at 25 °C

The dependence of the water ionization on temperature and pressure has been investigated thoroughly.[5] The value of pKw 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.

pKw values for liquid water.[6]
Temperature

Pressure[7]
pKw0 °C

0.10 MPa

14.95
25 °C

0.10 MPa

13.99
50 °C

0.10 MPa

13.26
75 °C

0.10 MPa

12.70
100 °C

0.10 MPa

12.25
150 °C

0.47 MPa

11.64
200 °C

1.5 MPa

11.31
250 °C

4.0 MPa

11.20
300 °C

8.7 MPa

11.34
350 °C

17 MPa

11.92

With electrolyte solutions, the value of pKw is dependent on ionic strength of the electrolyte. Values for sodium chloride are typical for a 1:1 electrolyte. With 1:2 electrolytes, MX2, pKw decreases with increasing ionic strength.[8]

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

Comparison of pKw values for liquid water, superheated steam, and supercritical water.[1]
Temp.Pressure
350 °C
400 °C
450 °C
500 °C
600 °C
800 °C
0.1 MPa

47.961b
47.873b
47.638b
46.384b
40.785b17 MPa

11.920 (liquid)a

25 MPa

11.551 (liquid)c
16.566
18.135
18.758
19.425
20.113
100 MPa

10.600 (liquid)c
10.744
11.005
11.381
12.296
13.544
1000 MPa

8.311 (liquid)c
8.178
8.084
8.019
7.952
7.957
Notes to the table. The values are for supercritical fluid except those marked: a at saturation pressure corresponding to 350 °C. b superheated steam. ccompressed or subcooled liquid.

Isotope effects

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

D2O + D2O ⇌ 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:

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

Kw=[D3O+][OD−]{\displaystyle K_{\rm {w}}=[{\rm {D_{3}O^{+}}}][{\rm {OD^{-}}}]}

The following table compares the values of pKw for H2O and D2O.[9]

pKw values for pure water

T/°C

10
20
25
30
40
50
H2O

14.535
14.167
13.997
13.830
13.535
13.262
D2O

15.439
15.049
14.869
14.699
14.385
14.103
Ionization equilibria in water–heavy water mixtures

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

Mechanism

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

NN0=e−ΔE‡kT{\displaystyle {\frac {N}{N_{0}}}=e^{-{\frac {\Delta E^{\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.[10] Random fluctuations in molecular motions occasionally (about once every 10 hours per water molecule[11]) 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.[12][13][14]

The inverse recombination reaction

H3O+ + OH− → 2 H2O

is among the fastest chemical reactions known, with a reaction rate constant of 1.3×1011 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.[15]

Relationship with the neutral point of water

Water molecules dissociate into equal amounts of H3O+ and OH−, so their concentrations are equal to 1.00×10−7 mol dm−3 at 25 °C. 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

1/2pKw.

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.

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

References

^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the “Gold Book”) (1997). Online corrected version:  (2006–) “autoprotolysis constant”. doi:10.1351/goldbook.A00532

^ a b Stumm, Werner; Morgan, James (1996). Aquatic Chemistry. Chemical Equilibria and Rates in Natural Waters (3rd ed.). John Wiley & Sons, Inc. ISBN 9780471511847.

^ Harned, H. S.; Owen, B. B. (1958). The Physical Chemistry of Electrolytic Solutions (3rd ed.). New York: Reinhold. pp. 635.

^ International Association for the Properties of Water and Steam (IAPWS)

^ Bandura, Andrei V.; Lvov, Serguei N. (2006). “The Ionization Constant of Water over Wide Ranges of Temperature and Density” (PDF). Journal of Physical and Chemical Reference Data. 35 (1): 15–30. Bibcode:2006JPCRD..35…15B. doi:10.1063/1.1928231.

^ 0.1 MPa for T < 100 °C. Saturation pressure for T > 100 °C.

^ Harned, H. S.; Owen, B. B. (1958). The Physical Chemistry of Electrolytic Solutions (3rd ed.). New York: Reinhold. pp. 634–649, 752–754.

^ Lide, D. R., ed. (1990). CRC Handbook of Chemistry and Physics (70th ed.). Boca Raton (FL):CRC Press.

^ Geissler, P. L.; Dellago, C.; Chandler, D.; Hutter, J.; Parrinello, M. (2001). “Autoionization in liquid water”. Science. 291 (5511): 2121–2124. Bibcode:2001Sci…291.2121G. CiteSeerX 10.1.1.6.4964. doi:10.1126/science.1056991. PMID 11251111.

^ Eigen, M.; De Maeyer, L. (1955). “Untersuchungen über die Kinetik der Neutralisation I” [Investigations on the kinetics of neutralization I]. Z. Elektrochem. 59: 986.

^ Stillinger, F. H. (1975). Theory and Molecular Models for Water. Adv. Chem. Phys. Advances in Chemical Physics. 31. pp. 1–101. doi:10.1002/9780470143834.ch1. ISBN 9780470143834.

^ Rapaport, D. C. (1983). “Hydrogen bonds in water”. Mol. Phys. 50 (5): 1151–1162. Bibcode:1983MolPh..50.1151R. doi:10.1080/00268978300102931.

^ Chen, S.-H.; Teixeira, J. (1986). Structure and Dynamics of Low-Temperature Water as Studied by Scattering Techniques. Adv. Chem. Phys. Advances in Chemical Physics. 64. pp. 1–45. doi:10.1002/9780470142882.ch1. ISBN 9780470142882.

^ Tinoco, I.; Sauer, K.; Wang, J. C. (1995). Physical Chemistry: Principles and Applications in Biological Sciences (3rd ed.). Prentice-Hall. p. 386.

General Chemistry – Autoionization of WatervteChemical equilibriaConcepts
Acid dissociation constant
Binding constant
Binding selectivity
Buffer solution
Chemical equilibrium
Chemical stability
Chelation
Determination of equilibrium constants
Dissociation constant
Distribution coefficient
Dynamic equilibrium
Equilibrium chemistry
Equilibrium constant
Equilibrium unfolding
Equilibrium stage
Hammett acidity function
Henry’s law
Liquid–liquid extraction
Macrocycle effect
Phase diagram
Predominance diagram
Phase rule
Phase separation
Reaction quotient
Self-ionization of water
Solubility equilibrium
Stability constants of complexes
Thermodynamic equilibrium
Thermodynamic activity
Vapor–liquid equilibrium