The indicator dinitrophenol is an acid with a \(K_{\mathrm{a}}\) of \(1.1 \times 10^{-4} .\) In a \(1.0 \times 10^{-4}-M\) solution, it is colorless in acid and yellow in base. Calculate the pH range over which it goes from \(10 \%\) ionized (colorless) to \(90 \%\) ionized (yellow).

Acid-base indicators are substances that change color with pH, helping us visualize the acid or basic nature of a solution. An indicator like dinitrophenol is useful in experiments to signal a specific pH range where it changes from colorless to yellow.

For students, grasping the science behind these indicators is critical. It's not just about memorizing color changes; it's understanding that these changes signal the degree of ionization of the indicator itself. Indicators are themselves weak acids (or bases), which dissociate subtly in solution. The percentage of ionization—an indicator's ratio of ionized to unionized form—varies with pH, leading to the observable color change.

The Henderson-Hasselbalch equation is a simplified version of the acid dissociation constant expression, often used for buffer solutions. It relates the pH of a solution to the pKa (which is the acid dissociation constant expressed in logarithmic form) and the ratio of the concentrations of the deprotonated (A-) and protonated (HA) form of the acid.

This is particularly important when trying to find the pH at certain degrees of ionization, as seen in this exercise. The equation is written as: \[\begin{equation}pH = pKa + \text{log}\bigg(\frac{[A^-]}{[HA]}\bigg)\text{\end{equation}\]}Understanding this equation allows you to compute not just the pH at any given ionization percentage, but also to predict how a buffer system will respond to additions of acid or base.

Ionization of acids is a fundamental concept in chemistry where an acid releases hydrogen ions (\[\begin{equation}\text{H}^+\text{\end{equation}\]}) into solution. When dealing with weak acids like dinitrophenol, not all of the acid will ionize, which is why we refer to the 'degree of ionization.' This is essential for students to understand as it directly impacts the pH of the solution.

When the ionization percentages are 10% and 90%, it shows that at specific pH levels, the acid exists predominantly in one form over the other. This is the foundational concept when calculating pH ranges for such indicators, highlighting the dynamic nature of acid dissociation in different pH conditions.

pKa is a crucial term in acid-base chemistry that represents the strength of an acid; it is the negative logarithm of the acid dissociation constant (\[\begin{equation}Ka\text{\end{equation}\]}). The relationship between pKa and Ka is logarithmic, meaning that as Ka increases (stronger acid), pKa decreases. Understanding this inverse relationship allows students to predict the behavior of an acid in solution.

For example, if an acid has a high pKa value, it means that it is a weaker acid (since its Ka value is low), and it will not dissociate much in solution. This is key to determining the pH range within which an indicator will operate and ensures that students can approach problems like these systematically, without simplifying to mere memorization tasks.