Answer
$K_a = 2.4 \times 10^{-6} $
Work Step by Step
1. Draw the ICE table for this equilibrium:
$$\begin{vmatrix}
Compound& [ HA ]& [ A^- ]& [ H_3O^+ ]\\
Initial& 0.115 & 0 & 0 \\
Change& -x& +x& +x\\
Equilibrium& 0.115 -x& 0 +x& 0 +x\\
\end{vmatrix}$$
2. Write the expression for $K_a$, and substitute the concentrations:
- The exponent of each concentration is equal to its balance coefficient.
$$K_a = \frac{[Products]}{[Reactants]} = \frac{[ A^- ][ H^+ ]}{[ HA ]}$$
$$K_a = \frac{(x)(x)}{[ HA ]_{initial} - x}$$
3. Using the pH, find the $H_3O^+$ concentration:
$$[H_3O^+] = 10^{-pH} = 10^{-3.29} = 5.1 \times 10^{-4} \space M$$
$[H_3O^+] = x = 5.1 \times 10^{-4} $
4. Substitute the value of x and calculate the $K_a$:
$$K_a = \frac{( 5.1 \times 10^{-4} )^2}{ 0.115 - 5.1 \times 10^{-4} }$$
$K_a = 2.4 \times 10^{-6} $
