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Water dissociation and pH

A remarkable property of pure water is that it dissociates to form hydrogen ions (H₃O⁺) and hydroxide (OH⁻) ions.

 

The ionic product, K_w

pH

Variation in K_w with temperature and pressure

Acidity, basicity, and the pK_a of water

Weak acids
Hydrogen ions
Hydroxide ions
Grotthuss mechanism
Ionic strength

 

The ionic product, K_w

Water dissociation (autoionization; self-ionization) occurs endothermically ^d due to electric field fluctuations between neighboring molecules. Dipole librations [191], resulting from thermal effects and favorable localized hydrogen bonding [567], together with nuclear quantum effects [2025, 4456], cause these fluctuations. The process may be facilitated by exciting the O-H stretch overtone vibration [393]. Once formed (at an average concentration of about 1 M H₂O-H⁺···OH⁻ at femtosecond timescales, as reported by Vasily Artemov's group [1984]),^h the ions may separate through the Grotthuss mechanism. Usually (>99.9%), these ion-clusters rapidly recombine (≈ 20 ps [2171]) with a frequency in the terahertz range to give the long-lived ions conventionally found. Thus the exact hydrogen ion concentrations depend on the time scale at which these are probed. Rarely (about once every eleven hours per molecule at 25 °C, or less than once a week at 0 °C), the localized hydrogen bonding arrangement breaks before allowing the separated ions to return [191]. The pair of ions (H⁺, OH⁻)^g hydrate independently and continue their separate existence^a for about 70 μs (this lifetimeis also dependent on the extent of hydrogen bonding, being shorter at lower temperatures). They tend to recombine when separated by only one or two water molecules.

 

H₂O   H⁺ + OH⁻
K_w = [H⁺] ˣ [OH⁻]

 

The kinetics of the process has been described using the equations, [4481]

 

The recombination (k_r2) is assumed to be controlled by diffusion ( 0.112 M⁻¹ ˣ ps⁻¹), k_d2 ≫k_d1., and k_r1 ≫k_r2 , (k_d1 =0.076 ˣ s⁻¹, k_r1 = 3.142 ˣ ps⁻¹. The dissociation rate (k_d2) was estimated at 2.04 ˣ 10⁻⁵ ˣ s⁻¹ . The paired ions were found to be ion pairs separated by two water molecules (0.58 nm).

 

pK_w is a dimensionless number (= 13.995, 25 °C [IAPWS]) as the included concentrations are relative to the molal standard states (e.g., 1 mol ˣ Kg⁻¹). ⁿ This low occurrence of the ions means that at neutrality (pH 7, at 25 °C) ^c, similarly charged ions are, on average, separated by vast distances (≈ 0.255 μm) in molecular terms and (for example) bacteria contain only a few tens of free hydrogen ions (~ 30 in an Escherichia coli). Contributing to this effect is the high dielectric constant (encouraging charge separation) and high concentration of H₂O (≈ 55.5 M; increasing the absolute amount dissociated). The mean lifetime of a hydroxonium ion (1 ps; about the same as that of a hydrogen bond) is such that the charge could be associated with over 10⁷ molecules of water before neutralization.

 

Although the extent of dissociation is tiny ([H⁺]/[H₂O] = 2.8  ˣ  10⁻⁹ at 37 °C), the dissociation and consequential changes in the tiny concentrations of hydrogen ions have absolute importance to living processes. Hydrogen and hydroxyl ions are produced already hydrated.

 

H₂O (liq)   H⁺(aq) + OH⁻ (aq)

 

where the subscripts, 'liq' and ‘aq’, indicate that the species are in or within the aqueous liquid phase.^l

 

The protons (H⁺, hydrons) initially hydrate as hydroxonium ions, H₃O⁺ (also called oxonium or hydronium ions) and do not exist as naked protons in the gas phase, or liquid, or solid water, where they interact extremely strongly with electron clouds. All three hydrogen atoms in the hydroxonium ion are held by strong covalent bonds and are equivalent (C_3v symmetry in a vacuum). The thermodynamic properties of the dissociation at 25 °C and 0.1 MPa are ΔU° = 59.5 kJ ˣ mol⁻¹, ΔV° = 22.13 cm³ ˣ mol⁻¹, ΔH° = 55.8 kJ ˣ mol⁻¹, ΔG° = 79.9 kJ ˣ mol⁻¹, ΔS° = -80.8 J ˣ K⁻¹ ˣ mol⁻¹ [1938]. The proton is never found unhydrated in an aqueous solution, and the hydroxonium ions, H₃O⁺, also have negligible independent existence in an aqueous environment [2134]. All the hydroxonium ion protons are predominantly hydrogen-bonded, causing further hydration to H₃O⁺(H₂O)_n, where n depends on the conditions such as temperature, solutes, pressure, and method of determination. [H⁺] is often written instead of [H₃O⁺].

 

To avoid the misleading presumption of the existence of bare protons similar to other bare cations (but see [2132] for an alternative view), the above equations are better written as:

 

2 H₂O(aq)   H₃O⁺(aq) + OH⁻(aq)

K_w = [H₃O⁺] ˣ [OH⁻]

 

 

Both ions are ionic kosmotropes, hydrating and creating order in forming stronger hydrogen bonds with surrounding water molecules than the water molecules form between themselves and creating short chains of hydrogen-bonded water molecules [2025]. Remarkably, the volume change in this reaction

 

2 H₂O_(l)   H₃O⁺_(aq) + OH⁻_(aq)  ΔV = -22.3 cm³ mol⁻¹

 

at 25 °C and infinite dilution, [1946], see Figure right,

due to the change in the hydration strength plus electrostriction, is about the same as one molecule of water (18.1 cm³ ˣ mol⁻¹); when one water molecule ionizes, its volume effectively disappears.

 

The concentrations of H₃O⁺ and OH⁻ are normally taken as the total concentrations of all the singly charged small clusters including these species. As other water molecules are required to promote the hydrolysis, the equations below includes the most important. None of these hydrated ions should be considered as 'fixed' structures, with further water molecules continually coming and going in aqueous solution with timescales of picoseconds. No bare protons occur, however.

 

4 H₂O   H₅O₂⁺ + H₃O₂⁻

10 H₂O   H₁₃O₆⁺ + H₇O₄⁻

 

The concentration of hydroxonium and hydroxide ions produced is equal to the square root of the dissociation constant (K_w).

 

A wide range of values for K_w has been calculated using Quantum Cluster Equilibrium (QCE) theory with various ab initio and density functional methods [3025].

 

The reverse reaction

                     H₃O⁺_(aq) + OH⁻_(aq)   2 H₂O_(aq)                      K_eq = 10¹⁴ ; 25 °C

 

greatly favors the neutralization reactions of acids and bases.

 

Aqueous OH⁻ does not ionize further as (O²⁻ + H₂O 2 OH⁻, K > 10²²). H₃O⁺ does not protonate further to the tetrahedral H₄O²⁺ ion as although H₄O²⁺ may be metastable in the gas phase, it is thermodynamically unstable to loss of a proton [2133], and would immediately dissociate in aqueous solution. [Back to Top ]

pH

The hydronium (oxonium) ion concentration (commonly called 'hydrogen ion concentration') is often given in terms of the pH, where

 

pH = Log₁₀(1/[H₃O⁺]) = -Log₁₀([H₃O⁺])

 

(that is, [H₃O⁺] = 10^−pH) ^f with the concentration of H₃O⁺ in mol ˣ L⁻¹. This equation is adequate for most solutions but may fail under higher solute concentrations.

 

More accurately and as determined using a pH meter,

 

pH = -Log₁₀(a_H) = -Log₁₀(m_H ˣ λ_H/m°)

 

where a_H, m_H, λ_H, and m° are the relative (molality based) activity, molality, molal activity coefficient^j, and standard molality (1 mol ˣ [kg solvent]⁻¹) of the hydrogen ions. This equation is correct under higher solute concentrations. Proof that the use of the equation pH = -Log₁₀([H⁺]) may give misleading results (and pH = -Log₁₀(a_H) is preferred) is easily shown as the pH of 0.1 M HCl decreases (i.e., it apparently becomes more acidic) when it is diluted with 5% M LiCl [1107]. Also, the -Log₁₀([H₃O⁺]) for average seawater at 25 °C is 8.19, but its -Log₁₀(a_H) is 8.32. As pH involves activities, the [H₃O⁺] calculated from pH = -Log₁₀([H⁺]) is often not the [H₃O⁺] concentration experimentally determined by titration. The hydrogen ion activity cannot be evaluated thermodynamically [3376] but may be determined kinetically.

 

Water can thus support acid-base equilibria over a range of about 16 pH units (see above right). At the low concentrations usually found, the molarity-based hydrogen ion concentration is close enough to the relative (molality based) activity for its use in most circumstances. The presence of salts and other solutes will generally reduce this activity, and the activity varies with temperature. The molal activity of hydrogen ions cannot be determined directly but may be determined using a glass electrode standardized to the response of standard buffer solutions of comparable ionic strength. ^o Glass electrode-determined pH values are error-prone, and calculated hydrogen ion concentrations should be treated with caution, particularly at the extremes of pH [1890]. For more information and a list of primary pH standards, see [813].

 

The pH scale was first introduced by Sørensen (as p_H·) in 1909 [1036] using colorimetric measurements and the hydrogen electrode, which generates an electrode potential proportional to pH. Today, we commonly use the glass electrode [3377] (see right), which also gives an electrode potential dependent on the activity of the H⁺and proportional to pH in dilute solution,

 

 

where F, R, and T are the Faraday constant, the gas constant, and absolute temperature, respectively. E refers to the potential (V) of the “unknown” solution being determined, and E_S and pH_S refer to the standard reference electrode solution. Once calibrated with a solution of Known pH

 

pH ∝ E

 

with a proportionality correction being applied for the temperature. The pH's are reasonably accurate where the pH meter is calibrated close to the pH at issue and with a solution of approximately the same ionic strength [3378].

 

The pH meter can be used to estimate hydrogen ion concentrations without concern for knowing activities [3376]. pH electrodes are now commonly used to determine the pH (see right).ⁱ The pH scale extends to negative numbers (for example, concentrated HCl has a pH of about -1.1) and to greater than 14 (for example, saturated NaOH has a pH of about 15.0) [1187]; although the glass electrode cannot be used at such extreme pH's. There is a review of the pH of natural water [1712].

 

Similarly to pH, pK_w is defined by

pK_w = Log₁₀(1/K_w) = -Log₁₀(K_w)

 

utilizing concentrations in mol ˣ L⁻¹.

 

Where the solution contains significant amounts of heavy water (D₂O), the measured pH will differ slightly from that expected from the sum of the amounts of the protonated species H₃O⁺ + H₂DO⁺ + HD₂O⁺ + D₃O⁺ [2953].

 

[Back to Top ]

Variation in K_w with T and P

K_w is very temperature dependent, increasing with temperature (that is, from 0.001 ˣ 10⁻¹⁴ mol² ˣ L⁻² at -35 °C (pH 8.5) [112], 0.112 ˣ 10⁻¹⁴ mol² ˣ L⁻² at 0 °C (pH 7.5), to 0.991 ˣ 10⁻¹⁴ mol² ˣ L⁻² at 25 °C (pH 7.0), to 9.311 ˣ 10⁻¹⁴ ˣ mol² ˣ L⁻² at 60 °C (pH 6.5) [87]), to 10⁻¹² ˣ mol² ˣ L⁻² at 300 °C (pH 6.0, ≈ 50 MPa) [456] in agreement with the high positive standard free energy. ^b There is a minimum at about 249 °C along the saturated pressure line for H₂O and at about 257 °C for D₂O (see right [1865]). The pK_w H₂O minimum is about 0.74 lower than that for D₂O [1865]. (see also conductivity maximum).

 

A theoretical treatment of this temperature dependence is available [763].

---

Footnotes

^a It follows that pure water droplets containing less than about 10⁸ water molecules (≈ 0.1 μm radius) would usually (i.e., on average) contain no ions in the absence of surface effects. [Back]

 

^b A bulk energy diagram for the dissociation in bulk water has been described [604]. [Back]

 

^c Note that acid-base neutrality only occurs when the concentration of hydrogen ions equals the concentration of hydroxyl ions (whatever the pH). This only occurs at pH 7 in pure water when at 25 °C. A solution is acidic when the hydrogen ion concentration is greater than the hydroxide ion concentration, whatever the pH. The pH of a neutral solution is numerically equal to half the pK_w of the solution. Therefore a pH of 7 at 0 °C indicates a slightly acid solution (neutrality is pH 7.5), whereas a pH of 7 at 50 °C indicates a slightly alkaline solution (neutrality is pH 6.6). [Back]

 

^d In a vacuum the reaction

H₂O (g)  H⁺ (g) + OH⁻ (g)

 

requires over three times more energy (1.66 MJ mol⁻¹; K_w = 1 ˣ 10⁻²⁸⁵ mol² ˣ L⁻² [3025]) than dissociation

 

H₂O (g)   H· (g) + ·OH (g)

 

(531 kJ ˣ mol⁻¹). In liquid water, the hydration of the ions (H⁺ ΔG° hydration -1112.5 kJ ˣ mol⁻¹, this includes H₃O⁺ ΔG° hydration -461.1 kJ ˣ mol⁻¹; OH⁻ ΔG° hydration -437.6 kJ ˣ mol⁻¹ [1067]) reduces the ΔG° of the reaction

 

2 H₂O (l)  H₃O⁺ (aq) + OH⁻ (aq)

 

to +99.78 kJ ˣ mol⁻¹ (These calculations assume that the standard state of the solvent water is taken as 1.0 M. If the standard state of the solvent water is the unit activity or mole fraction (= 1.0), the ΔG° is +79.907 kJ ˣ mol⁻¹ as given above). The dissociated radicals (H·, ·OH) are also somewhat stabilized in liquid water, as shown by the occasional dissociation of water [1066, see equations]. [Back]

 

^e Note that the 'acidity' of a material is commonly its pH and depends both on its pK_a and its concentration. The 'acidity' of drinks and foodstuffs, however, also depends on the response of our taste buds. [Back]

 

^f As Logarithms may only be taken of dimensionless numbers, all the concentrations (activities, partial pressures, etc.) in any Logarithmic expression are actually divided by unit values in the same units of that concentration (activity, partial pressure, etc.); thus, for example, here [H₃O⁺] (concentration of H₃O⁺ in mol ˣ L⁻¹) is actually [H₃O⁺]/(1.0 mol ˣ L⁻¹).

 

The p in pH originated as the arbitrary choice for the naming of the electrode solutions 'p' and 'q' by Sørensen [1036, 1891], but is now taken to mean the 'logarithm to the base 10 of the reciprocal of' (cologarithm) as in the function described above. [Back]

 

^g Strictly speaking, these equations should be expressed in terms of activities rather than concentrations; thus

 

K_w = [a_H+] ˣ [a_OH−]

 

The derivations are easier to follow as given, and usually the concentrations are so small that the difference is inconsequential. [Back]

 

^h Similar concentrations are found for ice [2171]. [Back]

 

ⁱ Different membrane glasses may be used dependent on the solution to be analyzed (e.g., high temperatures, strong alkali or HF).

 

The electrode reaction for the Ag/AgCl/saturated KCl reference electrode is

 

AgCl + e⁻ Ag⁰ + Cl⁻                E = +0.197 V (in saturated KCl)

[Back]

 

^j The activity coefficient of hydrogen ions depends on the acid concentration, temperature, ionic strength, the dielectric constant, the size of the ions, and the density of the medium. [Back]

 

^k Weak acids A weak acid is only partially dissociated, with both the undissociated acid and its dissociation product(s) being present, in solution, and in equilibrium with each other. The dissociation constant (K_a) may be determined by titration (see right).

 

If we consider the situation where the acid is one half dissociated, in other words where [A⁻] is equal to [HA], then the solution's pH equals the acid's pK_a. This means that an acid is half dissociated when the pH of the solution is numerically equal to the pK_a of the acid. Therefore acids with the lowest pK_a values are able to dissociate in solutions of low pH, i.e., even where the hydrogen ion concentration is high. Acids with higher pK_a values dissociate only in solutions of high (more alkaline) pH. The pK_a values of carboxylic acids are examined elsewhere as are those associated with CO₂. A compilation of weak acid pK_a values is available.

 

For any weak monoprotic acid (HA) of concentration c, the equilibrium is,

HA [H⁺] + [A⁻]

 

The concentration of hydrogen ions [H⁺] is given by the following four equations,

 

K_a= [H⁺] ˣ [A⁻]/[HA]

 

K_w = [H⁺] ˣ [OH⁻]

Balancing charges

[H⁺] = [OH⁻] + [A⁻]

Mass balance of the acid

c = [A⁻] + [HA]

Eliminating [A⁻], [HA] and [OH⁻] gives

                                                           [H⁺]³ + K_a ˣ [H⁺]² − (K_a ˣ c + K_w) ˣ [H⁺] − K_a ˣ K_w = 0                                     (A)

The pH is,

so long as c ≫ K_a and c ˣ K_a > K_w.

 

The pH, calculated using a numerical iterative method from the above exact equation (A), is compared with its approximation -0.5 ˣ Log₁₀(c ˣ K_a) for acetic acid (pK_a = 4.76, 25 °C) is shown on the right.

 

It can be seen that the approximation should not be used for concentrations below about 0.1 ˣ K_a. At very low acid concentrations (< 10⁻⁷ M), the pH cannot rise above pH 7.0.

 

 

 

 

 

 

^l Auto-ionization is not restricted to water but can occur with a few other substances. An example is liquid ammonia with ionization equilibrium

 

NH₃ + NH₃  NH₂⁻(in NH₃) + NH₄⁺(in NH₃)

 

K_NH3= [NH₄⁺][NH₂⁻]

 

The self-ionization constant (K_NH3) of ammonia (at ≈ 10⁻³⁰) is sixteen orders of magnitude smaller than that of water. [Back]

 

 

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