How to find the ionization constant of an unknown weak acid?


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> A $\pu{0.45 M}$ solution of a weak acid, $\ce{HX}$, has a $\pu{pH}$ of $4.5$. What is the ionization constant, $K\_a$, of the acid?
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* $\ce{[HX]}$ is already given as $\pu{0.45 M}$.
* $\ce{[H+]}$ is given by $10^{-4.5}$.


Since $\ce{pH}$ is $4.5$, $\ce{pOH} = 14 - 4.5 = 9.5$. Then we can get $\ce{[OH^-]}$ by computing $10^{-9.5}$. 


Then computing $K\_a$, I get 
$$\frac{10^{-4.5} \cdot 10^{-9.5}}{0.45} = 2.22 \times 10^{-14}.$$
But that is apparently not the right answer. What did I do wrong?




Look at the answer given here and modify it:
[How to set up equation for buffer reaction?](https://chemistry.stackexchange.com/a/95566/)


In your case the concentration of base $c\_B \approx 0$ and $[\ce{OH^-}] = K\_\mathrm{w}/[\ce{H^+}] \approx 0$, so that the equation reduces to $$K\_\mathrm{a} = \frac{[\ce{H^+}]^2}{c\_\mathrm{a}-[\ce{H^+}]} \approx \frac{[\ce{H^+}]^2}{c\_\mathrm{a}},$$ 
as $c\_\mathrm{a}$ is so large.


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Forget about $\ce{[OH^-]}$. This is $K\_\mathrm{a}$ we're talking about - the **acid dissociation constant**. Remember that the equation for $K\_\mathrm{a}$ is:


$$K\_\mathrm{a} = \frac{[\ce{H^{+}}] [\ce{A^{-}}]} {[\ce{HA}]}$$


As you can see, $\ce{[OH^{-}]}$ doesn't come into the equation. 


In terms of data, $\ce{[A^{-}]}$ hasn't been provided, and so we can assume that 
$$[\ce{H^+}][\ce{A^-}] \approx [\ce{H^{+}}]^2.$$
Also, the $\ce{[H^{+}]}$ you provided looks more like a $\ce{pH}$ conversion, so $[\ce{H^+}] = 10^{-4.5} = \pu{3.16x10^-5 mol dm^-3}$. Putting all of this together means that:


$$K\_\mathrm{a} \approx \frac{[\ce{H^{+}}]^2}{\ce{HA}} = \frac{(3.16\times10^{-5})^2}{0.45} = \pu{2.22x10^-9 mol dm^-3}.$$


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