Question

Phenolphthalein is the common indicator for the titration of a strong acid with a strong base. (a) If the \(\mathrm{p} K_{\mathrm{a}}\) of phenolphthalein is \(9.10,\) what is the ratio of the nonionized form of the indicator (colorless) to the ionized form (reddish pink) at pH \(8.00 ?\) (b) If 2 drops of \(0.060 M\) phenolphthalein are used in a titration involving a \(50.0-\mathrm{mL}\) volume, what is the concentration of the ionized form at \(\mathrm{pH} 8.00 ?\) (Assume that 1 drop \(=\) \(0.050 \mathrm{~mL}\).)

Short answer

(a) The ratio \([\text{colorless}]/[\text{reddish pink}]\) at pH 8.00 is about 12.7. (b) Ionized form concentration at pH 8.00 is 0.0000089 M.

Step-by-step solution

Understand the Henderson-Hasselbalch Equation

For part (a), we will use the Henderson-Hasselbalch equation, which relates pH, pKa, and the ratio of base to acid forms of an indicator: \[ \text{pH} = \text{pKa} + \log \left( \frac{\text{[A}^-\text{]}}{\text{[HA]}} \right) \] Here, \( \text{[A}^-\text{]} \) represents the concentration of the ionized form (reddish pink), and \( \text{[HA]} \) represents the concentration of the nonionized form (colorless).

Rearrange the Equation for the Concentration Ratio

Rearrange the equation to solve for the ratio \( \frac{\text{[A}^-\text{]}}{\text{[HA]}} \): \[ \frac{\text{[A}^-\text{]}}{\text{[HA]}} = 10^{(\text{pH} - \text{pKa})} \] Substitute in \( \text{pH} = 8.00 \) and \( \text{pKa} = 9.10 \).

Calculate the Concentration Ratio

Using the rearranged equation from Step 2: \[ \frac{\text{[A}^-\text{]}}{\text{[HA]}} = 10^{(8.00 - 9.10)} = 10^{-1.10} \approx 0.079 \] This means the ratio of the ionized form to nonionized form at pH 8.00 is approximately 0.079.

Find Total Volume and Concentration for Part (b)

For part (b), first determine the total volume of solution used in the titration: \( 50.0 \text{ mL} + 2 \times 0.050 \text{ mL} = 50.1 \text{ mL} \). Convert this to liters: \( 50.1 \text{ mL} = 0.0501 \text{ L} \). Find the concentration of the phenolphthalein solution: \( 0.060 \text{ M} \times \frac{0.0001 \text{ L}}{0.0501 \text{ L}} \approx 0.00012 \text{ M} \).

Calculate Ionized Form Concentration

Now, use the ratio from Step 3 to find the ionized concentration: \[ \text{[A}^-\text{]} = \frac{0.079}{1 + 0.079} \times 0.00012 \text{ M} \approx 0.0000089 \text{ M} \]. This is the concentration of the ionized form (reddish pink) at pH 8.00.

Key concepts

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a fundamental tool used in acid-base chemistry to relate the pH of a solution to the pKa and the ratio of the concentrations of the ionized and nonionized forms of an acid. It is written as: \[ \text{pH} = \text{pKa} + \log \left( \frac{\text{[A}^-\text{]}}{\text{[HA]}} \right) \]Here's how it works:

• pH: Measures the acidity or basicity of the solution.
• pKa: The negative logarithm of the acid dissociation constant, a measure of the acid strength.
• \([A^-]\): Represents the concentration of the ionized form, which is often colored.
• \([HA]\): Represents the concentration of the nonionized form, typically colorless.

In practice, you rearrange the equation to find the ratio \(\frac{\text{[A}^-\text{]}}{\text{[HA]}}\), allowing you to determine the proportion between the two states at a given pH. This becomes crucial in tasks like determining the color of an indicator in a titration.

Phenolphthalein Indicator

Phenolphthalein is a popular indicator used in titrations, especially for detecting end points in strong acid-strong base reactions. It has a clear color change around a specific pH range:

• Nonionized Form: Colorless, associated with a low pH environment.
• Ionized Form: Reddish pink, typically seen in a more basic environment.

This indicator's transition occurs around a pH of 8.00 to 10.00. This color change is thanks to the shifting balance between its nonionized and ionized forms as described by the Henderson-Hasselbalch equation. For example, when the pH of the solution is 8.00, we can calculate the ratio of the colorless form to the colored form using this equation, which helps you visually approximate the concentration of hydrogen ions during a titration.

pKa and pH Relationship

The concepts of pKa and pH are intertwined in understanding how acids and bases interact in solutions. Here's what you need to know:

• pKa: Represents the strength of an acid, indicating how easily it donates protons. A lower pKa means a stronger acid.
• pH: A scale used to specify the acidity or basicity of an aqueous solution. Lower pH values correspond to more acidic solutions.

The relationship between pKa and pH is critical in determining the state of an acid in a solution. When the pH equals the pKa, the concentrations of ionized and nonionized forms are equal.
As the pH deviates from the pKa, this balance shifts:

• pH < pKa: More nonionized form \([HA]\).
• pH > pKa: More ionized form \([A^-]\).

This relationship helps explain the color change in indicators like phenolphthalein. Understanding this relationship allows you to predict how changes in pH will affect the balance between the ionized and nonionized forms, and consequently, the visual appearance of the solution.