What is the pH of a 0.500 M solution of trimethylamine (pK_b = 4.13)?

Question

Quizplus · Accepted Answer

Trimethylamine is a weak base, and its conjugate acid, trimethylammonium ion, is a weak acid. Using the relationship between pKa and pKb (pKw = pKa + pKb), we can find the Kb of trimethylamine: 
pKw = 14.00
pKa of trimethylammonium ion = 9.87 (since pKb of trimethylamine > 14, pKa of trimethylammonium ion < 14)
pKb of trimethylamine = pKw - pKa = 4.13
Using the relationship between Kb and Ka (Kw = Kb x Ka), we can find the Ka of trimethylammonium ion:
Ka = Kw / Kb = 1.0 x 10^-14 / 1.48 x 10^-5 = 6.76 x 10^-10
Then we can use the relationship between Ka and the concentrations of the acid and conjugate base (Ka = [H+][CH3N+H3] / [CH3N]) to find the pH:
Ka = (x)(x) / (0.500 - x)
where x is the concentration of H+ ions.
Solving for x gives: x = 1.18 x 10^-12 M
Therefore, pH = -log[H+] = 11.8

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