pKb Value (Base Constant)

The pKb value is the negative base-10 logarithm of the base constant Kb: $pKb = -lg Kb$. The base constant describes the equilibrium of the protolysis of a base with water, B + H₂O ⇌ BH⁺ + OH⁻, and reads $Kb =\frac{ [BH^{+}]\cdot{}[OH^{-}]}{[B]}$. Example ammonia: $Kb =\frac{ 1.78\times{}10^{-5} mol}{L}$ gives $pKb = -lg(1.78\times{}10^{-5}) = 4.75$. The smaller the pKb, the stronger the base. Often, however, one does not know Kb itself but the pKa of the conjugate acid; then one uses the relation $pKb = 14 - pKa$, which holds at 25 °C for conjugate pairs. This gives the pKb of a base directly from the tabulated pKa of its corresponding acid.

The relation follows from the ion product of water. For a conjugate acid-base pair $\frac{HA}{A^{-}}$ the acid constant Ka of the acid and the base constant Kb of the conjugate base multiply exactly to the ion product of water: $Ka\cdot{}Kb = Kw$. At 25 °C, $Kw = 10^{-14}$. Taking the logarithm of this equation and multiplying by minus one turns the product into a sum: $pKa + pKb = pKw = 14$. Intuitively this means: the stronger an acid, the weaker its conjugate base, and vice versa. It is important that the 14 holds only at 25 °C, because Kw is temperature-dependent, and only for a genuinely conjugate pair, not for arbitrary acids and bases.

For a weak base you first calculate the pOH and from it the pH. The approximation is $pOH = \frac{1}{2}\cdot{}(pKb - lg c_{0})$, where c₀ is the initial concentration of the base. It holds when c₀ is much larger than Kb, so the protolysis proceeds only to a small extent. Then you use $pH = 14 - pOH$. Example: $\frac{0.1 mol}{L}$ ammonia with $pKb = 4.75$ gives $pOH = \frac{1}{2}\cdot{}(4.75 - lg 0.1) = \frac{1}{2}\cdot{}(4.75 + 1) = 2.88$ and thus $pH = 14 - 2.88 = 11.12$. A common mistake is to leave the pOH as the final answer; almost always the pH is asked for. For strong bases this approximation does not hold; there you calculate $pOH = -lg c$ directly.

The pKa measures the strength of an acid, the pKb the strength of a base. The pKa is the negative logarithm of the acid constant Ka, which describes the equilibrium HA + H₂O ⇌ H₃O⁺ + A⁻; a small pKa means a strong acid. The pKb is the negative logarithm of the base constant Kb, which describes the equilibrium B + H₂O ⇌ BH⁺ + OH⁻; a small pKb means a strong base. Both quantities are linked via the conjugate pair: $pKa + pKb = 14$ at 25 °C. In practice tables usually list only the pKa; the pKb of the corresponding base is calculated from it. Both share the same logarithmic logic: one unit of difference corresponds to a factor of ten in the constant.

Ammonia NH₃ is a weak base because it protolyses only incompletely in water. In the equilibrium reaction NH₃ + H₂O ⇌ NH₄⁺ + OH⁻ the equilibrium lies predominantly on the left side; only a small fraction of the ammonia molecules actually take up a proton. This is expressed in the small base constant $Kb =\frac{ 1.78\times{}10^{-5} mol}{L}$ and the pKb of 4.75. For comparison: a strong base like sodium hydroxide dissociates completely. Nevertheless an ammonia solution reacts distinctly basic, because even the small amount of hydroxide ions formed raises the pH above seven; a 0.1-molar solution reaches about pH 11. Weak thus refers to the degree of protolysis, not to the solution being only slightly basic.