The acid-base indicator bromcresol green is a weak acid. The yellow acid and blue base forms of the indicator are present in equal concentrations in a solution when the \(\mathrm{pH}\) is \(4.68 .\) What is the \(\mathrm{p} K_{a}\) for bromcresol green?

Acid-base indicators are compounds that change color in response to changes in pH, making them valuable tools in chemistry for determining the acidity or basicity of a solution. Indicators are themselves weak acids or bases which exist in two forms: one with a color that shows in acidic environments and another with a different color that shows in basic conditions.

For instance, bromcresol green, the indicator mentioned in the exercise, is yellow when it forms its acid (HA) and blue when it forms its conjugate base (A-). When the pH of the solution equals the pKa of the indicator, both forms are present in equal concentrations, which results in a distinct intermediate color. This is the principle used to deduce the pKa value for an acid-base indicator.

The pKa of a substance is a quantitative measurement that reflects the strength of an acid in solution. It is derived from the acid dissociation constant (Ka), which measures the ability of an acid to donate a proton (H+). The pKa is simply the negative base-10 logarithm of this constant: \( pKa = -\text{log}_{10}(Ka) \)

A lower pKa value indicates a stronger acid because it suggests a greater tendency to lose the proton. As seen in our example with bromcresol green, when the concentrations of an acid and its conjugate base are equal, the pH of the solution is equal to the pKa. This relationship provides an essential way to understand how different chemical species will behave under varying pH conditions.

To calculate the pH of a solution, which is a measure of its acidity or basicity, we often use the Henderson-Hasselbalch equation as a tool for solutions involving weak acids and their salts (which form a conjugate base). The equation \( pH = pKa + \text{log} \frac{[A^-]}{[HA]} \) helps to determine the pH knowing the concentration of the acid and its conjugate base. In solutions where these concentrations are equal, as in a buffer system at its pKa, the log term becomes zero, and the pH is equal to the pKa. The exercise provided demonstrates a clear situation where this application is useful for determining the pKa of a weak acid; by knowing the pH when the acid and base forms are in equilibrium, the pKa value is readily obtained.

Weak acids are characterized by their partial dissociation in solution, which contrasts with strong acids that dissociate completely. This dissociation behavior can be quantified by the acid dissociation constant (Ka) and consequently the pKa. A key property of weak acids is that they establish an equilibrium between the un-dissociated acid (HA) and its dissociate form (A- and H+).

Bromcresol green functions as a typical weak acid in the given textbook exercise. It demonstrates a fundamental behavior of weak acids—a balance between their acid and base forms at a specific pH level, which is the acid's pKa. It also exemplifies how weak acids create buffer solutions that can resist changes in pH upon the addition of small amounts of acids or bases, due to the presence of both the acid and its conjugate base.