Question

The ionization constant \(K_{\mathrm{a}}\) of an indicator HIn is \(1.0 \times\) \(10^{-6}\). The color of the nonionized form is red and that of the ionized form is yellow. What is the color of this indicator in a solution whose \(\mathrm{pH}\) is \(4.00 ?\)

Short answer

With the given pH of 4 and Ka value, calculate the ratio of the non-ionized to ionized form of the indicator; depending on which is more abundant, the observed color of the indicator would be red for the non-ionized form and yellow for the ionized form.

Step-by-step solution

Understand ionization equilibrium

The indicator \(HIn\) is subject to the following ionization equilibrium in aqueous medium: \[HIn \leftrightarrow H^{+} + In^{-}\]The color changes from red (non-ionized form, \(HIn\)) to yellow (ionized form, \(In^{-}\)) when \(HIn\) loses a proton H+ to become \(In^{-}\). It is important to note that pH affects this equilibrium, as pH is a measure of the concentration of \(H^{+}\) ions.

Calculate the hydrogen ion concentration

Based on the given pH value, the \(H^{+}\) ion concentration can be calculated using the formula: \[[H^{+}] = 10^{-\mathrm{pH}}\]The pH of the solution is 4. Therefore, the concentration of the hydrogen ion \([H^{+}]\) is \(10^{-4} M\).

Calculate the ionized to non-ionized form ratio

The ionization constant (Ka) represents the equilibrium between the ionized and non-ionized forms, which is given by \[K_{a} = \frac{[H^{+}][In^{-}]}{[HIn]}\]We know that \([H^{+}]\) = \([In^{-}]\) because for each \(HIn\) that ionizes, one \(H^{+}\) and one \(In^{-}\) are produced. So we replace in the Ka expression \([In^{-}]\) by \([H^{+}]\) which is equal to \(10^{-4}\), and we get\[K_{a} = ([H^{+}])^{2}/[HIn]\]From this it is possible to find the relation \([HIn]\)/\([H^{+}]\) = \(([H^{+}])/K_{a}\). As the value of Ka is given \((1.0 \times 10^{-6})\), the ratio of \([HIn]\) to \([H^{+}]\) in the solution can be calculated.

Determine the dominant form and color of the indicator

If the ratio obtained in the previous step is greater than 1, it means that the concentration of \(HIn\) (red) is higher than \(In^{-}\) (yellow), so the indicator will be red. If the ratio is less than 1, \(In^{-}\) (yellow) is more abundant, so the indicator will be yellow.

Key concepts

Ionization Constant

To understand the behavior of acid-base indicators in solutions, we must first learn about the ionization constant. The ionization constant, typically denoted as \(K_{a}\), is a measure of the degree to which a chemical species can donate protons (\(H^{+}\)) in an aqueous solution. It is a crucial concept in chemistry because it helps predict the strength of acids and bases.
When an indicator such as HIn is placed in water, it establishes an equilibrium between the ionized and non-ionized forms:

• Non-ionized form \(HIn\): typically has a distinct color, in this case, red.
• Ionized form \(In^{-}\): has a different color, here, yellow.

This equilibrium can be expressed mathematically as: \[ K_{a} = \frac{[H^{+}][In^{-}]}{[HIn]} \]In essence, the \(K_{a}\) value tells us how much of the indicator molecule will be in its ionized form compared to its non-ionized form. A low \(K_{a}\) indicates that the substance is a weak acid, meaning it does not ionize significantly in solution. Understanding this helps us predict the color-change behavior of an indicator under different conditions of pH.

pH Calculation

Determining the pH of a solution is a fundamental aspect of chemistry, especially when dealing with acid-base equilibria. pH is a measurement of hydrogen ion concentration \([H^+]\) within a solution and is crucial for calculating the degree to which a substance will ionize.
The pH is calculated using the formula:

• \([H^{+}] = 10^{-\text{pH}}\)
• For example, a pH of 4 would have a hydrogen ion concentration of \(10^{-4} M\).

This calculation allows us to determine the concentration of hydrogen ions in the solution, which is essential when analyzing how an indicator like HIn will behave. A lower pH value typically indicates a higher concentration of hydrogen ions, influencing the ionization equilibrium of the indicator.
Thus, knowing the pH allows one to determine whether the solution will favor the ionized or non-ionized form, and subsequently, predict the resulting color change of an indicator.

Acid-Base Indicator

Acid-base indicators are fascinating substances that change color when they encounter different pH levels. They are typically weak acids or bases that exhibit different colors in their ionized and non-ionized forms. This color change is due to structural changes in the molecule as it ionizes or returns to its original form.
In the context of this exercise:

• The indicator HIn appears red in its non-ionized state and yellow when ionized.
• The ionization equilibrium determines the dominant form of the indicator based on the pH of the solution.

At a given pH, if the concentration of \([HIn]\) is higher than \([In^{-}]\), the solution will appear red, indicating a predominance of the non-ionized form. Conversely, if \([In^{-}]\) is greater, the solution will appear yellow, showing the ionized form is dominant.
This principle allows chemists to use indicators to visually estimate the pH of a solution within a certain range, making these substances invaluable for both educational purposes and practical applications in labs.