Final answer:
The pH of a 1.32 M solution of diethylamine hydrochloride, a salt derived from a weak base and a strong acid, is calculated using the base's Kb value to determine the concentration of produced H3O+ ions. The resulting pH is approximately 0.88.
Step-by-step explanation:
The question involves calculating the pH of a 1.32 M aqueous solution of diethylamine hydrochloride ((C₂H₅)₂NH₂Cl). Diethylamine hydrochloride is the hydrochloride salt form of diethylamine, which is a weak base with a given Kb. Since the salt is derived from a weak base and a strong acid (HCl), it will hydrolyze in water to form the weak base and HCl. The resulting solution will be slightly acidic.
The hydrolysis of diethylamine hydrochloride can be represented by the equation:
(C₂H₅)₂NH₂Cl (aq) → (C₂H₅)₂NH (aq) + HCl (aq)
Because diethylamine is a weak base, we can use its Kb to calculate its conjugate acid's Ka using the relation Kw = Ka × Kb. The concentration of the resultant hydrochloride acid will be equal to the initial concentration of diethylamine hydrochloride since it is a one-to-one reaction from the salt to the acid. The pH of the solution can then be found by calculating the concentration of the H₃O⁺ ions produced.
To calculate Ka, use the equation:
Ka = Kw / Kb
Using the given Kb of diethylamine (1.30×10⁻³) and the Kw of water at 25°C (1.00×10⁻¹⁴), we can find Ka:
Ka = 1.00×10⁻¹⁴ / 1.30×10⁻³ = 7.69×10⁻¹²
The concentration of the H₃O⁺ ions (H₃O⁺ = [diethylamine hydrochloride] for one-to-one reaction) is 1.32 M, and the pH can be calculated as the negative logarithm of the H₃O⁺ ion concentration using the equation pH = -log[H₃O⁺]. Treating the formed hydrochloric acid as a strong acid which dissociates completely, we get:
pH = -log[1.32] ≈ 0.88