The dissociation of water into H₃O⁺ and OH^– ions depends on temperature. At 0 °C the [H₃O⁺] = 3.38 x 10^–8 M, at 25 °C the [H₃O⁺] = 1.00 x 10^–7 M, and at 50 °C the [H₃O⁺] = 2.34 x 10^–7 M.
b. What is the value of K_w at 0 °C and 50 °C?

Verified step by step guidance

1. Step 1: Recall the expression for the ionization constant of water, K_w, which is defined as the product of the concentrations of hydronium ions ([H₃O⁺]) and hydroxide ions ([OH⁻]). Mathematically, K_w = [H₃O⁺] × [OH⁻].
2. Step 2: Understand that water is neutral, meaning the concentrations of [H₃O⁺] and [OH⁻] are equal at any given temperature. Therefore, [OH⁻] = [H₃O⁺].
3. Step 3: Substitute the given values of [H₃O⁺] at 0 °C and 50 °C into the formula for K_w. For 0 °C, [H₃O⁺] = 3.38 × 10⁻⁸ M, so K_w = (3.38 × 10⁻⁸ M) × (3.38 × 10⁻⁸ M). For 50 °C, [H₃O⁺] = 2.34 × 10⁻⁷ M, so K_w = (2.34 × 10⁻⁷ M) × (2.34 × 10⁻⁷ M).
4. Step 4: Perform the multiplication for each temperature to find the value of K_w. For 0 °C, multiply (3.38 × 10⁻⁸) by itself. For 50 °C, multiply (2.34 × 10⁻⁷) by itself.
5. Step 5: Express the final values of K_w for both temperatures in scientific notation, ensuring the units are in M² (molar squared) since K_w is a product of two molar concentrations.

Key Concepts

Dissociation of Water

The dissociation of water refers to the process where water (H₂O) separates into hydronium ions (H₃O⁺) and hydroxide ions (OH⁻). This equilibrium is crucial for understanding acid-base chemistry and is influenced by temperature, as the concentrations of these ions change with thermal conditions.

Ion Product of Water (K_w)

The ion product of water, denoted as K_w, is the equilibrium constant for the dissociation of water. It is defined as the product of the concentrations of H₃O⁺ and OH⁻ ions in pure water. K_w varies with temperature, and its value at 25 °C is 1.0 x 10⁻¹⁴, serving as a reference point for calculating ion concentrations at different temperatures.

Temperature Dependence of K_w

K_w is temperature-dependent, meaning its value changes with temperature variations. As temperature increases, the dissociation of water becomes more favorable, leading to higher concentrations of H₃O⁺ and OH⁻ ions, thus increasing K_w. Understanding this relationship is essential for calculating K_w at different temperatures, such as 0 °C and 50 °C.