Reactivity 3.1.5 – The Ion Product Constant of Water

Autoionization of Water

• Water is a unique molecule because it can undergo a reaction with itself — this is called autoionization or self-ionization.
• In this equilibrium, one water molecule donates a proton (acting as a Brønsted–Lowry acid), and another accepts the proton (acting as a base):

  \( \text{H}_2\text{O}(l) + \text{H}_2\text{O}(l) \rightleftharpoons \text{H}_3\text{O}^+(aq) + \text{OH}^-(aq) \)

  or more simply:
  \( \text{H}_2\text{O}(l) \rightleftharpoons \text{H}^+(aq) + \text{OH}^-(aq) \)

• Although the concentration of ions is very small, this equilibrium is essential in understanding all acid–base behavior in aqueous solutions.   

What is \( K_w \)?

• \( K_w \) is the ionic product of water. It is the equilibrium constant for the autoionization reaction of water.
• Since water is a pure liquid, its concentration is constant and not included in the equilibrium expression.
• Thus, the expression for the equilibrium becomes:

  \( K_w = [\text{H}^+][\text{OH}^-] \)

• This equation shows that the concentration of hydrogen ions and hydroxide ions are inversely related: when one increases, the other must decrease (since the product must stay constant at a fixed temperature).

Standard Value of \( K_w \)

• At a temperature of 25°C (298 K), which is the standard temperature used in chemistry unless otherwise stated, the value of \( K_w \) is:

  \( K_w = 1.0 \times 10^{-14} \, \text{mol}^2\,\text{dm}^{-6} \)

• This means that in pure water:

  \( [\text{H}^+] = [\text{OH}^-] = \sqrt{K_w} = \sqrt{1.0 \times 10^{-14}} = 1.0 \times 10^{-7} \, \text{mol·dm}^{-3} \)

• This very small value indicates that water ionizes only very slightly — most water molecules remain undissociated.

Inverse Relationship Between \( [\text{H}^+] \) and \( [\text{OH}^-] \)

• In any aqueous solution, the product of the concentrations of H⁺ and OH⁻ must always equal \( K_w \) at that temperature.
• This means if the concentration of H⁺ increases (more acidic), the concentration of OH⁻ must decrease to maintain the value of \( K_w \).
• Similarly, if OH⁻ increases (more basic), H⁺ must decrease.
• This is a fundamental principle in determining whether a solution is acidic, neutral, or basic.

Temperature Dependence of \( K_w \)

• The autoionization of water is an endothermic reaction (it absorbs heat).
• According to Le Châtelier’s Principle, increasing the temperature favors the forward reaction (more ionization), so \( [\text{H}^+] \) and \( [\text{OH}^-] \) both increase.
• This means that \( K_w \) increases as temperature increases.
• For example, at 50°C, \( K_w \approx 5.5 \times 10^{-14} \), which is higher than at 25°C.
• Important: Since both ion concentrations increase equally, the solution is still neutral (H⁺ = OH⁻), but the pH of neutral water at higher temperatures is less than 7.
• So, pH = 7 is only neutral at 25°C — this is important in higher-level pH calculations.

Significance of \( K_w \) in Chemistry

• \( K_w \) forms the basis for all pH calculations and allows you to move between [H⁺] and [OH⁻] in any problem.
• It is critical in calculating unknown concentrations in acid–base titrations, buffer problems, and equilibrium conditions.
• It also helps you recognize whether a solution is acidic, basic, or neutral by comparing [H⁺] and [OH⁻].

Using \( [\text{H}^+] \) and \( [\text{OH}^-] \) to Determine the Nature of a Solution

• All aqueous solutions obey the relationship:
  \( K_w = [\text{H}^+][\text{OH}^-] \)
• This value is constant at a given temperature (e.g., \( 1.0 \times 10^{-14} \) at 25°C).
• You can compare the relative concentrations of \( [\text{H}^+] \) and \( [\text{OH}^-] \) to determine whether a solution is:
  • Acidic → \( [\text{H}^+] > [\text{OH}^-] \)
  • Neutral → \( [\text{H}^+] = [\text{OH}^-] \)
  • Basic → \( [\text{H}^+] < [\text{OH}^-] \)

Relationship to pH and pOH

• Since \( \text{pH} = -\log_{10}[\text{H}^+] \) and \( \text{pOH} = -\log_{10}[\text{OH}^-] \), we can also classify solutions using:
  • Acidic: pH < 7 (so pOH > 7)
  • Neutral: pH = 7 (so pOH = 7)
  • Basic: pH > 7 (so pOH < 7)
• The relationship between pH and pOH is:
  \( \text{pH} + \text{pOH} = 14 \) (at 25°C)

Important Clarifications for IB Chemistry:

• These values are only valid at 25°C (298 K). At other temperatures, the neutral point may not be pH = 7.
• \( K_w \) increases with temperature → the pH of neutral water becomes less than 7 at higher temperatures.
• Always report concentrations in mol·dm⁻³ and pH values to 2 decimal places when doing calculations.