Problem 177 - A Complete Resource Book in Chemistry for JEE Main

Question

An acid-base indicator has \(\mathrm{K}_{\mathrm{a}}=3.0 \times 10^{-5} .\) The acid form of the indicator is red and the basic form is blue. The \(\left[\mathrm{H}^{+}\right]\)required to change the indicator from \(75 \%\) red to \(75 \%\) blue is (a) \(8 \times 10^{-5} \mathrm{M}\) (b) \(9 \times 10^{-5} \mathrm{M}\) (c) \(1 \times 10^{-5} \mathrm{M}\) (d) \(3 \times 10^{-4} \mathrm{M}\)

Step-by-Step Solution

Answer

The required [H+] is \(1 \times 10^{-5} \text{ M}\). Option (c).

Step 1: Understand the Problem

We need to find the hydrogen ion concentration \([\mathrm{H}^+]\) that changes an acid-base indicator from being \(75\%\) red to \(75\%\) blue. Given the \(K_a\), we can link this to the Henderson-Hasselbalch equation to solve for \([\mathrm{H}^+]\).

Step 2: Setup the Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation for the acid-base indicator is \( \text{pH} = \text{pKa} + \log\left(\frac{[A^-]}{[HA]}\right) \). Here, \([A^-]\) is the concentration of the basic form (blue) and \([HA]\) is the concentration of the acid form (red).

Step 3: Determine the Ratio from Percentages

Since the solution changes from \(75\%\) red to \(75\%\) blue, at \(75\%\) blue, \([A^-] = 0.75\) and \([HA] = 0.25\). Thus the ratio \(\frac{[A^-]}{[HA]} = \frac{0.75}{0.25} = 3\).

Step 4: Find \\[ \text{pKa} \\]

We find \([\mathrm{pKa}]\) using the relation \( \text{pKa} = -\log(\text{Ka}) \). Here, \( \text{Ka} = 3.0 \times 10^{-5} \), so \( \text{pKa} = -\log(3.0 \times 10^{-5}) = 4.52 \).

Step 5: Calculate the Resulting \\[ ext{pH} \\]

Using the ratio from Step 3, substitute into the Henderson-Hasselbalch equation: \( \text{pH} = 4.52 + \log(3) \). Compute this to find \( \text{pH} = 4.52 + 0.477 = 4.997 \).

Step 6: Convert \\[ ext{pH} \\] to \\[ [\mathrm{H}^+] \\]

The relationship between pH and \[ [\mathrm{H}^+] \] is \( \text{pH} = -\log([\mathrm{H}^+]) \). Substitute and solve for \([\mathrm{H}^+]\): \( 4.997 = -\log([\mathrm{H}^+]) \) hence \([\mathrm{H}^+] = 10^{-4.997} \approx 1.0 \times 10^{-5} \).

Step 7: Select the Correct Answer

Compare the value \(1.0 \times 10^{-5} \text{ M}\) to the options and choose. The correct answer is (c) \(1 \times 10^{-5} \text{ M}\).

Key Concepts

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a fundamental formula in acid-base chemistry. It is used to calculate the pH of a buffer solution by relating it to the concentration of an acid and its conjugate base. The equation is expressed as: \[ \text{pH} = \text{pKa} + \log\left(\frac{[A^-]}{[HA]}\right) \]where

• \([A^-]\) represents the concentration of the base form of the indicator,
• \([HA]\) is the concentration of the acid form, and
• \(\text{pKa}\) is the negative logarithm of the acid dissociation constant \(K_a\).

This equation is particularly useful when dealing with buffer solutions or acid-base indicators because it connects the pH of the solution with the acid and base forms of a compound. By simplifying the relationship between the concentrations and the pH, it aids in determining the shift in equilibrium as the pH changes. Remember, the Henderson-Hasselbalch equation assumes that the concentrations of the buffer components are much greater than the concentrations of \(\mathrm{H}^+\) or \(\mathrm{OH}^-\), ensuring that the solution behaves predictably.

Acid-Base Indicator

Acid-base indicators are substances that change color depending on the pH of the solution in which they are present. This color change is due to the different structures that the molecule adopts at different pH levels.

• The acid form, often represented as \([HA]\), will display one color.
• The base form, \([A^-]\), will display another color.

Take, for example, the indicator from our exercise, which turns red in its acidic form and blue in its basic form. This transition occurs around the pKa of the indicator because that is the point where there is a roughly equal amount of the acid and base forms.The transition range, usually about ±1 pH unit around the pKa, is critical for selecting an appropriate indicator. If you're monitoring a reaction, you should choose an indicator whose transition range aligns with the pH range you're interested in. This is because it will assure you of a noticeable and distinct color change as the reaction progresses.

pKa and pH Relationship

The concept of pKa is pivotal in understanding how acids and bases behave in solution. pKa, the negative logarithm of the acid dissociation constant \(K_a\), reveals the strength of an acid.

• A lower pKa value indicates a stronger acid, which dissociates more in solution.
• A higher pKa signifies a weaker acid, as it holds onto its hydrogen ions more tightly.

The relationship between pKa and pH is essential for predicting the state of an acid-base system. When the pH of the solution is equal to the

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