# Base Dissociation Constant

#### Learning Objective

- Calculate the Kw (water dissociation constant) using the following equation: Kw = [H+] x [OH−] and manpulate the formula to determine [OH−] = Kw/[H+] or [H+]=Kw/[OH-]

#### Key Points

- The base dissociation constant K
_{bE}measures a base's basicity, or strength. - K
_{b}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. - K
_{b}and K_{a }are also related through the ion constant for water, K_{w}, by the relationship$K_W=K_a\times K_b$.

#### Term

- conjugate acidthe species created when a base accepts a proton

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.*

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.

Show Sources

Boundless vets and curates high-quality, openly licensed content from around the Internet. This particular resource used the following sources:

**"Acid dissociation constant."**

Wikipedia

CC BY-SA 3.0.

**"General Chemistry/Chemical Equilibria/Acid-Base Equilibrium."**

Wikibooks

CC BY-SA 3.0.