Answer
See explanation below.
Work Step by Step
1
To see a distinct color in a mixture of two colors, we need one of the colors to have about 10 times the intensity of the other.
The concentration ratio is given by [HIn] / [In-]
For one of these to be the case, the concentration
ratio [HIn] / [In-] has to be either greater than 10 or less than 1/10.
i.e. [HIn] / [In-] > 10 OR [HIn] / [In-] < 1/10
pH = pKa + log([HIn] / [In-])
Thus, when [HIn] / [In-] = 10,
pH = pKa + 1
And, when [HIn] / [In-] = 1/10
pH = pKa - 1
In short, pH = pKa +/- 1, thus giving a pH range of about 2 units.