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

The color of the indicator is red in a solution with pH 4.00.

Step-by-step solution

Write the Ionization Equation

For the weak acid indicator HIn, the ionization can be represented as:\[\text{HIn} \rightleftharpoons \text{H}^+ + \text{In}^-\]where \(\text{HIn}\) is the nonionized (red) form and \(\text{In}^-\) is the ionized (yellow) form.

Express the Ionization Constant

The ionization constant \(K_a\) for the indicator HIn is given by:\[K_a = \frac{[\text{H}^+][\text{In}^-]}{[\text{HIn}]}\]Given that \(K_a = 1.0 \times 10^{-6}\).

Calculate the [H⁺] in the Solution

Since the pH of the solution is 4.00, use the relation:\[pH = -\log[\text{H}^+]\]\[4.00 = -\log[\text{H}^+]\]Solving for \([\text{H}^+]\):\[[\text{H}^+] = 10^{-4} \text{ M}\]

Determine the Dominant Form of the Indicator

We need to find the ratio of \([\text{In}^-]\) to \([\text{HIn}]\), using \(\[ \text{H}^+\] \):\[ \frac{[\text{In}^-]}{[\text{HIn}]} = \frac{K_a}{[\text{H}^+]} = \frac{1.0 \times 10^{-6}}{10^{-4}} = 0.01\]This indicates that the concentration of \([\text{HIn}]\) is 100 times that of \([\text{In}^-]\), so the non-ionized (red) form is dominant.

Conclude the Color of the Solution

Since the non-ionized (red) form is present in higher concentration compared to the ionized (yellow) form, the indicator will appear red in the solution.

Key concepts

Ionization Constant

The ionization constant, often denoted as \(K_a\) for acids, is an essential concept in understanding how indicators work in chemistry. This constant provides insight into the degree of ionization of a weak acid in solution. In the context of acid-base indicators, the ionization constant tells us how readily the indicator dissociates into its ionized and non-ionized forms. For our specific indicator HIn, the given \(K_a\) is \(1.0 \times 10^{-6}\), indicating a relatively low level of ionization. This means that the equilibria strongly favor the non-ionized form (in this case, red) over the ionized form (yellow), unless perturbed by a pH change. Understanding \(K_a\) helps in predicting the behavior of indicators in different pH solutions, crucial for determining the color change point.

pH Calculations

pH calculations are crucial for determining the color of an acid-base indicator in a solution. The pH of a solution is a measure of its acidity or basicity, and it affects the equilibrium of the indicator between its ionized and non-ionized forms.

• The pH can be calculated using the formula: \(pH = -\log[\text{H}^+]\).
• In our situation, a solution with a pH of 4.00 corresponds to a hydronium ion concentration of \([\text{H}^+] = 10^{-4} \text{ M}\).

Knowing the pH allows us to calculate the proportions of each form of the indicator present. It tells us how much of each form we have at equilibrium, which directly informs us about the color the solution will display. In the exercise provided, the calculation helps us to see that at pH 4, the majority form is non-ionized, resulting in the red color.

Color Change in Indicators

The color of an acid-base indicator arises from its chemical structure. These structures often change with differing pH, leading to changes in color. A common feature of many indicators is their ability to exist in two different forms: a non-ionized form and an ionized form.

• At low pH, indicators tend to be in their non-ionized form, while high pH favors the ionized form.
• For the indicator HIn, red represents the non-ionized form, and yellow represents the ionized form.

In the context of this exercise, the indicator exhibits a reddish color at pH 4 because the concentration of the non-ionized form (red) is significantly higher than that of the ionized form (yellow). When \(\text{HIn}\) outweighs \(\text{In}^-\) by a factor of 100, the red color is dominant. This demonstrates the sensitivity of the indicator to changes in the pH environment, making it a useful tool for detecting changes in acidity or basicity in different solutions.