Evaluate indicator choice (phenolphthalein) A student wishes to find the concentration of a $\pu{50 mL}$ solution of acetic acid $(K_\mathrm{a} = \pu{1.8e-5}).$ He performs a titration using a solution of $\pu{0.2 M}$ sodium hydroxide. The student selects phenolphthalein as his indicator choice which has a $K_\mathrm{a} = \pu{3.98e-10}.$ Justify the student’s choice of indicator.


My attempt: I found the $\mathrm{p}K_\mathrm{a}$ of the indicator to be $9.4$ and the fact that the titration was weak acid + strong base meant the $\mathrm{pH}$ at equivalence would be around that too, so the indicator was justified. In saying this, my justification isn’t very mathematical.
Easy marks for phenolphthalein usability are:

$99.9\%$ of acetic acid is titrated at about
$$\mathrm{pH} = \mathrm{p}K_\mathrm{a} + \log{1000} = 7.75$$

$100\%$ of acetic acid is titrated at
$$\mathrm{pH} = 7 + \frac12 \cdot (\mathrm{p}K_\mathrm{a} + \log c) = 9.385 + \frac12 \log c$$

Phenolphthalein color transition is $8.2-10.0.$