The equivalence point of an acid-base titration can be determined by choosing the appropriate colored indicator. This indicator is a weak acid $\ce{HA}$. The undissociated acid has a different color than it's ion $\ce{A-}$. The $\mathrm{p}K_\mathrm{a}$ value of a colored indicator is given to be $\mathrm{p}K_\mathrm{a}=6$. At what value of the $\mathrm{pH}$ will the color change? Please explain your answer. (Assume that both colors are equally well recognizable).
I've had this question on an exam and I can't seem to figure out what exactly the answer could be.
I would intuitively say that the color change happens at $\mathrm{pH}=6$. If the $\mathrm{pH}$ is lower than 6 then there are less $\ce{A-}$ ions and more $\ce{HA}$ (acid-indicator) in the solution. Hence the solution will take on the color of the undissociated acid-indicator. If the pH is higher than 6 then the $\ce{HA}$ starts to dissociate more to $\ce{H3O+}$ hence there being a higher $\ce{A-}$ concentration and the solution takes the color of the indicator's conjugate base. But I'm not sure if my thought process is right or how to 'prove' it's right.