How is K_w defined, and what is its numerical value at 25 °C?

Verified step by step guidance

1. Understand that \( K_w \) is the ion product constant for water, which is a measure of the extent of water's self-ionization.
2. Recognize that the self-ionization of water can be represented by the chemical equation: \( 2H_2O(l) \rightleftharpoons H_3O^+(aq) + OH^-(aq) \).
3. The expression for \( K_w \) is given by the product of the concentrations of the hydronium ions \([H_3O^+]\) and hydroxide ions \([OH^-]\) in pure water: \( K_w = [H_3O^+][OH^-] \).
4. At 25 °C, the concentrations of \([H_3O^+]\) and \([OH^-]\) in pure water are both \(1.0 \times 10^{-7} \text{ M}\).
5. Thus, the numerical value of \( K_w \) at 25 °C is calculated by multiplying these concentrations: \( K_w = (1.0 \times 10^{-7})(1.0 \times 10^{-7}) \).

Key Concepts

Ion Product of Water (K_w)

K_w, or the ion product of water, is a constant that represents the equilibrium constant for the self-ionization of water. It quantifies the concentrations of hydrogen ions (H⁺) and hydroxide ions (OH⁻) in pure water at a given temperature. The self-ionization reaction can be expressed as 2 H₂O ⇌ H₃O⁺ + OH⁻.

Temperature Dependence of K_w

The value of K_w is temperature-dependent, meaning it changes with variations in temperature. At 25 °C, K_w has a specific numerical value, which is 1.0 x 10⁻¹⁴. This value is crucial for understanding the behavior of acids and bases in aqueous solutions, as it helps determine the pH and the concentrations of H⁺ and OH⁻ ions.

pH and pOH Relationship

The concepts of pH and pOH are directly related to K_w. pH is a measure of the hydrogen ion concentration, while pOH measures the hydroxide ion concentration. At 25 °C, the relationship pH + pOH = 14 holds true, allowing for the calculation of one from the other. This relationship is essential for understanding acid-base chemistry in aqueous solutions.