Auto-Ionization of Water and pH Calculations

Auto-Ionization of Water

Concept of Auto-Ionization

Auto-ionization occurs when water molecules react with one another in an aqueous solution, producing hydronium and hydroxide ions. This process is fundamental to understanding acid-base chemistry in aqueous solutions.

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

• Alternatively: $H_2O(l) \rightleftharpoons H^+(aq) + OH^-(aq)$

Ionization Constant of Water ($K_w$)

The equilibrium constant for the auto-ionization of water is called the ionization constant of water, $K_w$. It is defined as:

• $K_w = [H^+][OH^-]$

• At 25°C, $K_w = 1.0 \times 10^{-14}$

Relationship between pH and pOH:

• $pH + pOH = 14$

Temperature Dependence of $K_w$

The value of $K_w$ changes with temperature. The following table summarizes $K_w$ values at different temperatures:

Temperature (°C)	$K_w$
0	0.114 × 10-14
25	1.00 × 10-14
50	5.48 × 10-14
75	19.5 × 10-14
100	51.3 × 10-14

Calculating pH and Solution Classification

pH Calculation

The pH of a solution is a measure of its acidity or basicity and is calculated using the concentration of hydronium ions:

• $pH = -\log[H^+]$

• $pOH = -\log[OH^-]$

• $pH + pOH = 14$ (at 25°C)

Example Calculation

Example: A particular aqueous solution at 25°C contains $3.7 \times 10^{-4}$ M of hydronium ions. Calculate the pH and identify whether the solution is acidic, basic, or neutral.

• $pH = -\log(3.7 \times 10^{-4}) = 3.43$

• Since $pH < 7$, the solution is acidic.

Practice Problems

• Practice 1: If a solution contains $7.42 \times 10^{-4}$ M of hydroxide ions, calculate the pH.

  • $pOH = -\log(7.42 \times 10^{-4}) = 3.13$

  • $pH = 14 - 3.13 = 10.87$

  • Since $pH > 7$, the solution is basic.

• Practice 2: Calculate the $K_w$ for pure water given the pH = 7.34.

  • $[H^+] = 10^{-7.34} = 4.57 \times 10^{-8}$ M

  • Since pure water: $[H^+] = [OH^-]$

  • $K_w = [H^+][OH^-] = (4.57 \times 10^{-8})^2 = 2.09 \times 10^{-15}$

Classification of Solutions

• Acidic: $pH < 7$

• Neutral: $pH = 7$

• Basic: $pH > 7$

Additional info: The auto-ionization of water is a key concept in acid-base equilibria and is foundational for understanding pH, pOH, and the behavior of aqueous solutions in general chemistry.