I have been working on this lab forever and have not been able to get a $K_\text{a}$ anywhere close to the accepted value. I’m sure I’m missing something really simple.
We titrated $\ce{CH3COOH}$ with $\ce{NaOH}$.
I used $10.00\ \mathrm{mL}$ of $\ce{CH3COOH}$. Volume of $\ce{NaOH}$ at the equivalence point was $10.14\ \mathrm{mL}$, and the concentration of $\ce{NaOH}$ was $0.1530\ \mathrm{mol \cdot L^{-1}}$.
First I found the concentration of $\ce{CH_3COOH}$ by using $c_\text{a} \cdot V_\text{a} = c_\text{b} \cdot V_\text{b}$, so $c_\text{a} \cdot 10.00\ \mathrm{mL} = 0.1530\ \mathrm{mol \cdot L^{-1}} \cdot 10.14\ \mathrm{mL}$
$c_\text{a} = 0.1551\ \mathrm{mol \cdot L^{-1}}$
Then I used the equation: $\ce{CH3COOH + H2O <=> H3O+ + CH3COO-}$ and the pH (at the midpoint of $5.07\ \mathrm{mL}\ \ce{NaOH}$) which was $4.87$
$$K_\text{a}= \frac{[\ce{H3O+}] [\ce{CH3COO-}]}{[\ce{CH3COOH}] [\ce{H3O+}]}=[\ce{CH3COO-}]= 10^{-4.87}= 1.36 \times 10^5 $$ $$ K_\text{a}= \frac{(1.36 \times 10^5)^2}{ 0.1551} = 1.19 \times 10^{-9} $$ Any help on what I am doing wrong would be greatly appreciated!