Question

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

Short answer

The indicator changes color at a pH of approximately 5.7.

Step-by-step solution

Understand the Equilibrium of the Indicator

The equilibrium reaction for the indicator can be described as \[ \text{HIn (green)} \rightleftharpoons \text{In}^- \text{ (red)} + \text{H}^+ \]. \( K_a \) provides the equilibrium constant for this reaction, which helps us determine the \( pH \) at which the indicator changes color.

Write the Expression for the Equilibrium Constant

For the reaction \( \text{HIn} \rightleftharpoons \text{In}^- + \text{H}^+ \), the expression for the equilibrium constant \( K_a \) is given by \[ K_a = \frac{[\text{In}^-][\text{H}^+]}{[\text{HIn}]} \]. Given \( K_a = 2.0 \times 10^{-6} \), we can use this to find the \( pH \) at which the color change occurs.

Determine pH at Color Change

The color change occurs most noticeably when \([\text{In}^-] = [\text{HIn}]\), meaning the concentrations of the acidic (green) and basic (red) forms of the indicator are equal. Under this condition, the expression for the equilibrium constant becomes \( K_a = [\text{H}^+] \). Substituting \( K_a = 2.0 \times 10^{-6} \), we find \([\text{H}^+] = 2.0 \times 10^{-6} \).

Calculate the pH

With \([\text{H}^+] = 2.0 \times 10^{-6} \), we can find the \( pH \) using the formula \[ pH = -\log([\text{H}^+]) \]. Calculating, we have \[ pH = -\log(2.0 \times 10^{-6}) \approx 5.7 \].

Key concepts

Equilibrium Constant

The equilibrium constant, represented as \( K_a \), is a crucial factor when studying acid-base indicators. It measures the extent to which an acid dissociates into its ions in a solution. When talking about indicators, \( K_a \) helps us understand at what point in a titration the color will shift, revealing the endpoint of the reaction.
When an indicator is added to a solution, it undergoes a chemical reaction:

• \( ext{HIn (green)} \rightleftharpoons ext{In}^- ext{ (red)} + ext{H}^+ \)

Here, \( ext{HIn} \) is the acidic form (green), and \( ext{In}^- \) is the basic form (red). The equilibrium constant for this reaction, \( K_a \), quantifies the balance in concentrations between these forms. A large \( K_a \) value would mean the system favors the production of ions, whereas a small \( K_a \) indicates it remains largely undissociated.
In the given exercise, \( K_a = 2.0 \times 10^{-6} \) means that the indicator dissociates only slightly. Its \( K_a \) is crucial for determining the \( pH \) at which the color change happens.

pH Calculation

Calculating \( pH \) is essential for understanding when an acid-base indicator will change color in a titration. \( pH \) is determined by the concentration of hydrogen ions, \( [ ext{H}^+] \), in the solution.
The exercise specifies that the color change is most distinct when the concentrations of \( ext{HIn} \) and \( ext{In}^- \) are equal. This condition implies:

• \( [ ext{In}^-] = [ ext{HIn}] \)
• Thus, \( K_a = [ ext{H}^+] \)

Substituting the provided \( K_a \) (\( 2.0 \times 10^{-6} \)), we find that \( [ ext{H}^+] = 2.0 \times 10^{-6} \).
To find the \( pH \), use the formula:

• \( pH = -\log([ ext{H}^+]) \)

Applying this formula gives:

• \( pH = -\log(2.0 \times 10^{-6}) \approx 5.7 \)

This value indicates that at \( pH = 5.7 \), the indicator will transition between its two color forms, signifying its role in showing the endpoint of the titration.

Color Change

An acid-base indicator exhibits a color change due to the equilibrium between its acidic and basic forms. In this context, the transition signals the extent of a titration. The indicator starts as a green \( ext{HIn} \) and turns red as \( ext{In}^- \) takes over.
The light shift occurs when the concentration of \( ext{HIn} \) equals that of \( ext{In}^- \), specifically at \( pH = 5.7 \) for our example case.
Some key notes on color change with indicators:

• The speed of color change can vary based on the indicator's specific \( K_a \) value and the buffering capacity of the solution.
• The range of \( pH \) over which the indicator changes color is called the "transition range." It encompasses the \( pH \) at which half the indicator is \( ext{HIn} \) and half \( ext{In}^- \).

In practical use, different indicators have unique transition points, making them suitable for detecting particular \( pH \) endpoints. Choosing the right indicator depends on the chemical nature of the titration involved and the desired accuracy of results.