Base Dissociation Constant

In chemistry, a base is a substance that can accept hydrogen ions (protons) or, more generally, donate a pair of valence electrons. The base dissociation constant, K_b, is a measure of basicity—the base's general strength. It is related to the acid dissociation constant, K_a, by the simple relationship pK_a + pK_b = 14, where pK_b and pK_a are the negative logarithms of K_b and K_a, respectively. The base dissociation constant can be expressed as follows:

$K_b = \dfrac{[\text{BH}^+][\text{OH}^-]}{\text{B}}$

where $\text{B}$ is the base, $\text{BH}^+$ is its conjugate acid, and $\text{OH}^-$ is hydroxide ions.

The Base Dissociation Constant

Historically, the equilibrium constant K_b for a base has been defined as the association constant for protonation of the base, B, to form the conjugate acid, HB⁺.

$B(aq) + H_2O(l) \leftrightharpoons HB^+(aq) + OH^-(aq)$

As with any equilibrium constant for a reversible reaction, the expression for K_b takes the following form:

$K_{b} = \frac{[OH^{-}][HB^{+}]}{[B]}$

K_b is related to K_a for the conjugate acid. Recall that in water, the concentration of the hydroxide ion, [OH⁻], is related to the concentration of the hydrogen ion by the autoionization constant of water:

$K_W=[H^+][OH^-]$

Rearranging, we have:

$[OH^{-}] = \frac{K_{w}}{[H^{+}]}$

Substituting this expression for [OH⁻] into the expression for K_b yields:

$K_{b} = \frac{K_{w}[HB^{+}]}{[B][H^{+}]} = \frac{K_{w}}{K_{a}}$

Therefore, for any base/conjugate acid pair, the following relationship always holds true:

$K_W=K_aK_b$

Taking the negative log of both sides yields the following useful equation:

$pK_a+pK_b=14$

In actuality, there is no need to define pK_b separately from pK_a, but it is done here because pK_b values are found in some of the older chemistry literature.

Calculating the pH of a Weak Base in Aqueous Solution

The pH of a weak base in aqueous solution depends on the strength of the base (given by K_b) and the concentration of the base (the molarity, or moles of the base per liter of solution). A convenient way to find the pH for a weak base in solution is to use an ICE table: ICE stands for "Initial," "Change," and"Equilibrium."

Before the reaction starts, the base, B, is present in its initial concentration [B]₀, and the concentration of the products is zero. As the reaction reaches equilibrium, the base concentration decreases by x amount; given the reaction's stoichiometry, the two products increase by x amount. At equilibrium, the base's concentration is [B]₀ – x, and the two products' concentration is x.

ICE diagram

An ICE diagram for a weak base in aqueous solution.

The K_b for the reaction is:

$K_{b} = \frac{[BH^+][OH^-]}{[B]}$

Filling in the values from the equilibrium line gives:

$K_{b} = \frac{x^2}{[B]_{0}-x}$

This quadratic equation can be solved for x. However, if the base is weak, then we can assume that x will be insignificant compared to [B]₀, and the approximation [B]₀– x ≈ [B]₀ can be used. The equation simplifies to:

$K_{b} = \frac{x^2}{[B]_{0}}$

Since x = [OH]^-, we can calculate pOH using the equation pOH = –log[OH]^-; we can find the pH using the equation 14 – pOH = pH.