Problem 51 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

An acid type indicator, HIn differs in colour from its conjugate base (In \(^{-}\)). The human eye is sensitive to colour differences only when the ratio \(\left[\mathrm{In}^{-}\right] /[\mathrm{HIn}]\) is greater than 10 or smaller than. \(0.1\). What should be the minimum change in the \(\mathrm{pH}\) of the solution to observe a complete colour change \(\left(K_{a}=1.0 \times 10^{-5}\right) ?\)

Step-by-Step Solution

Answer

The minimum change in pH for a complete color change is 2.

Step 1: Understanding the Problem

We need to find the minimum change in pH for a noticeable color change of the indicator. The color change is noticeable when \(\left[ \mathrm{In}^{-} \right] / \left[ \mathrm{HIn} \right]\) is greater than 10 or less than 0.1. We also have the acid dissociation constant \(K_a = 1.0 \times 10^{-5}\).

Step 2: Using the Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation relates the pH of a solution to the concentration of acid and its conjugate base:\[\text{pH} = \text{pKa} + \log \left( \frac{[\mathrm{In}^-]}{[\mathrm{HIn}]} \right)\]Given \(K_a\), find \(\text{pKa}\):\[\text{pKa} = -\log(K_a) = -\log(1.0 \times 10^{-5}) = 5\].

Step 3: Finding pH for [In⁻]/[HIn] = 10

Use the Henderson-Hasselbalch equation with \([\mathrm{In}^-]/[\mathrm{HIn}] = 10\):\[\text{pH}_1 = 5 + \log(10)\]\[\text{pH}_1 = 5 + 1 = 6\]

Step 4: Finding pH for [In⁻]/[HIn] = 0.1

Use the Henderson-Hasselbalch equation with \([\mathrm{In}^-]/[\mathrm{HIn}] = 0.1\):\[\text{pH}_2 = 5 + \log(0.1)\]\[\text{pH}_2 = 5 - 1 = 4\]

Step 5: Calculating Minimum Change in pH

The minimum change in pH to observe a complete color change is the difference between \(\text{pH}_1\) and \(\text{pH}_2\):\[\Delta \text{pH} = \text{pH}_1 - \text{pH}_2 = 6 - 4 = 2\]

Key Concepts

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a fundamental tool in chemistry. It helps us understand how acids and bases behave in a solution. This equation describes the relationship between the pH of a solution and the ratio of the concentration of an acid to its conjugate base. The formula used is:\[\text{pH} = \text{pKa} + \log \left( \frac{[\text{Conjugate Base}]}{[\text{Acid}]} \right)\]In this equation, \( \text{pKa} \) is the negative logarithm of the acid dissociation constant (\( K_a \)) of the acid. This constant reflects the strength of the acid, or how well it can donate a proton in a solution.
This equation is particularly useful for calculating the pH of buffer solutions, which are solutions that resist changes in pH when small amounts of acid or base are added.In the context of an acid-base indicator, the colors change when the ratio \( \left[ \mathrm{In}^{-} \right] / [\mathrm{HIn}] \) shifts. By using the Henderson-Hasselbalch equation, we can predict the pH at which these shifts produce visible color changes. Knowing \( K_a \), we derive \( \text{pKa} \), which is essential for further calculations.

pH Change

pH change is a crucial concept to grasp when studying acid-base chemistry. pH measures the acidity or basicity of a solution. It directly influences the color of indicators used to visualize pH levels.
Most indicators change color in a certain pH range, providing a visual cue to the pH level.In the exercise provided, we observed that the ratio of \( \left[ \mathrm{In}^{-} \right] / [\mathrm{HIn}] \) must exceed 10 or be less than 0.1 for a notable color change. This demands specific pH thresholds.
To calculate these, we used the Henderson-Hasselbalch equation to determine:

• \( \text{pH}_1 = 6 \) when \( [\mathrm{In}^-]/[\mathrm{HIn}] = 10 \), and
• \( \text{pH}_2 = 4 \) when \( [\mathrm{In}^-]/[\mathrm{HIn}] = 0.1 \).

The difference between these pH values, \( \Delta \text{pH} = 2 \), is the minimum change needed to notice a complete shift in indicator color.
This highlights how sensitive indicators can be to slight pH variations.

Conjugate Base

In acid-base chemistry, understanding the role of a conjugate base is fundamental. When an acid donates a proton, it transforms into its conjugate base. The conjugate base is the species remaining after the acid has given up a hydrogen ion. For instance, when the acid form of the indicator, \( \text{HIn} \), loses a proton, it becomes \( \text{In}^- \), which is its conjugate base.
This shift between acid and conjugate base is accompanied by noticeable color changes, depending on their concentrations.The relative concentrations of the acid and its conjugate base determine the pH of the solution, which can be calculated using the Henderson-Hasselbalch equation. The visible changes in color are due to the chemical balance between the acid and its conjugate base. These color shifts serve as practical tools for measuring pH in various chemical contexts.
Conjugate bases are therefore integral to understanding chemical equilibria and reaction dynamics, especially in buffer and indicator systems.