# Self-ionisation of Water and Kw

This is part of the HSC Chemistry course under the topic Quantitative Analysis.

### HSC Chemistry Syllabus

• calculate and apply the dissociation constant (Ka) and pK(pKa = –log10 (Ka)) to determine the difference between strong and weak acids (ACSCH098)

### What is Self-ionisation of Water?

Self-ionisation or auto-ionisation of water is a reversible reaction that occurs between two molecules of water. In this reaction, a proton is transferred from one water molecule to another, forming a hydroxide and hydronium ion.

$$2H_2O(l) \rightleftharpoons H_3O^+(aq) + OH^-(aq)$$

This reversible reaction occurs in pure water and all aqueous solutions e.g. acid or base solutions. In a closed chemical system, this reaction can reach an equilibrium.

The equilibrium constant of this reaction is represented by K_w = 1.0 \times 10^{-14} at 25 ºC.

$$K_w = [H_3O^+][OH^-] = 1.0 \times 10^{-14}$$

This equation can be used to calculate the concentration of either ions in water or an aqueous solution when the concentration of the other is given.

### Example 1

The hydrogen ion concentration of a hydrofluoric acid solution is 0.0700 mol/L at 25ºC. What is the hydroxide concentration?

Solution:

$$[H_3O^+][OH^-] = 1.0 \times 10^{-14}$$

$$[OH^-] = \frac{1.0 \times 10^{-14}}{0.0700}$$

$$[OH^-] = 1.43 \times 10^{-13} \text{ mol/L (3 s.f.)}$$

### Effect of Temperature on Water's Self-ionisation

The self-ionisation of water is an endothermic reaction (\Delta H > 0) as the energy absorbed during bond breaking is greater than the energy released during bond formation.

Since K_w is an equilibrium constant, its value is affected by temperature.

• At a temperature > 25ºC, the equilibrium position shifts to the product side to produce more ions, thereby increasing the value of K_w.
• At a temperature < 25ºC, the equilibrium position shifts to the reactant side, thereby reducing the equilibrium concentration of hydroxide and hydronium ions and decreasing the value of K_w.

At 25ºC, the extremely small value of K_w suggests that the equilibrium concentrations of H_3O^+ and OH^- ions are small. In pure water, the production of equal concentrations of these ions makes it neutral. When these ion concentrations change with temperature (as explained above), they remain equal and thus pure water remains neutral at all temperatures.

### Relationship Between Kw, Ka and Kb

Self-ionisation of water does not only occur in pure water, it occurs in all aqueous solutions. For example, the above equilibrium can occur in a solution of acid or base.

This is what you need to remember and understand: K_w is the product of K_a of a weak acid and K_b of its conjugate base.

$$K_w = K_a \times K_b$$

Derivation (do not memorise): For a generic acid-base system,

$$\text{Acid}: \hspace{0.5cm} HA_{(aq)} + H_2O_{(l)} \leftrightharpoons A^-_{(aq)} + H_3O^+_{(aq)} \hspace{0.5cm} K_a = \frac{[A^-][H_3O^+]}{[HA]}$$

$$\text{Conjugate base}: \hspace{0.5cm} A^-_{(aq)} + H_2O_{(l)} \leftrightharpoons HA_{(aq)} + OH^-_{(aq)} \hspace{0.5cm} K_b = \frac{[HA][OH^-]}{[A^-]}$$

Let's multiply the expressions for K_a and K_b:

$$K_a \times K_b = \frac{[{A^-}][H_3O^+]}{[HA]} \times \frac{[HA][OH^-]}{[A^-]}$$

$$K_a \times K_b = [H_3O^+_{(aq)}][OH^-_{(aq)}] = 1.0 \times 10^{-14}$$

This means at 25℃ (298 K):

$$K_a = \frac{10^{-14}}{K_b} \hspace{3cm} K_b = \frac{10^{-14}}{K_a}$$

These equations suggest that K_a of a weak acid's ionisation is inversely proportional to the K_b of its conjugate base's ionisation. The stronger an acid is, the weaker its conjugate base is; the weaker an acid is, the stronger its conjugate base is. This relationship is also true for a weak base and its conjugate acid.

Since these equilibrium constant values also indicate the strength of the acid and base respectively, we can formulate the following relationship:

 Weak acid Conjugate weak base High Ka & low pKa High strength Low strength Low Ka & high pKa Low strength High strength

 Weak base Conjugate weak acid High Kb & low pKb High strength Low strength Low Kb & high pKb Low strength High strength

Note the concept of pK_a is discussed separately here