# Determining the Equilibrium Constant

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

"An equilibrium constant expression describes the relationship among the concentrations (or partial pressures) of the substances present in a system at equilibrium" (General Chemistry).

For a given chemical reaction:

$aA + bB \rightleftharpoons cC + dD$

the equilibrium constant expression is as follows:

$K_c = \frac{[C]^c[D]^d}{[A]^a[B]^b}$

To calculate the equilibrium constant (also known as the dissociation constant), the concentrations of each species in the reaction at equilibrium must be measured. Consider the general acid dissociation equation:

$HA + H_2O \rightleftharpoons A^- + H_3O^+$

Where HA is the acid, H2O is water, A- is the conjugate base of the acid, and H3O+ is the hydronium ion, a protonated water molecule.

Any equilibrium problem given provides some needed information, including a variety of numbers and/or units. Either Ka or Kb, the dissociation constants for the acid or the base in the equation, respectively, will be provided. For this example, Ka = 2.2 x 10-4 and the initial concentration of HA, the acid in solution is 2 mol L-1.

The equilibrium concentrations are calculated using an ICE Table. An ICE Table is a way to keep all your information organized throughout the process. ICE stands for 'Initial', 'Concentration', and 'Equilibrium'.

$HA + H_2O \rightleftharpoons A^- + H_3O^+$

HA + H2O A- + H3O+
I 2 mols/L n/a 0 0
C -x n/a +x +x
E 2-x n/a x x

In this case,

$K_a = \frac{[A^-][H_3O^+]}{[HA]}$

H2O is ignored in the ICE Table and when calculating the dissociation constant because it is a pure liquid. Any hypothetical solids would also be ignored.

Substituting the values from the "E" row of the table,

$\begin{eqnarray} K_a &=& \frac{(x)(x)}{2-x} \\ &=& \frac{x^2}{2-x} \end{eqnarray}$

A simple way to solve this equation is to first estimate the value for x. Assume x is very small, so the quantity 2-x is essentially 2. Keep in mind that the value for Ka is known. The equation simplifies to the following:

$x^2 = 2K_a$

The calculated value for x is very close to 0. It can be plugged into the 2-x term in the original equation to find a more accurate value for x. A few iterations of this process will converge on the correct concentration of H3O+ and, equivalently, A- at equilibrium in this specific case. In this example, the concentration of H3O+ and A- is 0.02086 or 2.09 x 10-2 mol L-1

The relationship between Ka and K b is the following:

$K_W = K_a \times K_b$

Kw is equal to 10-14. If the problem provides a value for Ka but the reaction involves a base, Kb can be calculated by dividing Kw by Ka. The same holds true if Kb is given for an acid dissociation problem.