The \(K_{\mathrm{a}}\) of a certain indicator is \(2.0 \times 10^{-6} .\) The color of HIn is green and that of In \(^{-}\) is red. A few drops of the indicator are added to a HCl solution, which is then titrated against a \(\mathrm{NaOH}\) solution. At what \(\mathrm{pH}\) will the indicator change color?

Acid-base titrations are fundamental procedures in chemistry to determine the concentration of an unknown acid or base solution. During the process, a titrant (a solution of known concentration) is added progressively to the titrand (the solution with an unknown concentration) until the chemical reaction between the two is complete. This point of completion is known as the equivalence point and is often indicated by a sharp change in pH, which can be detected by a pH indicator.

The pH indicator is a special dye that changes color at a specific pH level, thus signaling that the stoichiometric amount of titrant has been added to neutralize the titrand. In the classroom or laboratory, understanding how to perform a titration and interpret its results is essential for analyzing the composition of solutions and their acid or base properties.

The pKa is a crucial value in acid-base chemistry, representing the acid dissociation constant (Ka) on a logarithmic scale. Understanding the relationship between Ka and pKa helps students analyze the strength of an acid and its tendency to donate a proton. To calculate the pKa, use the formula:

\[pKa = -\text{log}(Ka)\]
For instance, if the Ka of an indicator is given as \(2.0 \times 10^{-6}\), the pKa can be calculated by inserting this value into the formula. The formula, a logarithmic transformation, provides ease in comparing acid strengths since pKa values on this scale are more manageable numbers.

The pH is a scale used to specify how acidic or basic a water-based solution is. It's calculated as the negative logarithm of the hydrogen ion concentration, using the formula:

\[pH = -\text{log}([H^+])\]
In the context of a titration, the pH will change as the acid and base react. The pH at which an indicator changes color is particularly valuable because it helps determine the endpoint of the titration, which is slightly different from the equivalence point. The pH at this stage may not necessarily be neutral (pH 7); its value depends on the particular qualities of the acid and base being titrated, as well as the indicator used.

A chemical indicator is a substance that undergoes a distinct observable change, usually in color, at a specific pH range during an acid-base titration. It is this property that makes indicators an invaluable tool in titrations to signal the endpoint of the reaction.

Different indicators have different pKa values and, thus, change color at different pH levels. For instance, the problem describes an indicator with a green color in its acidic form (HIn) and red in its basic form (In\(^-\)). Based on the calculations of pKa from the provided Ka value, the pH at which this indicator changes color can be precisely determined. This precise point of color change, where the concentration of HIn equals In\(^-\) , will occur at the pKa value of the indicator.