Problem 158 - A Complete Resource Book in Chemistry for JEE Main

Question

An acid base indicator has \(\mathrm{K}_{\mathrm{a}}=3 \times 10^{-5} .\) The acid form of the indicator is red and the basic form is blue. By how much must the \(\mathrm{pH}\) change in order to change the indicator from \(75 \%\) red to \(75 \%\) blue \((\log 3=0.4770)\) (a) \(0.95\) (b) \(2.3\) (c) \(0.75\) (d) 5

Step-by-Step Solution

Answer

The pH must change by 0.95.

Step 1: Identify the Form of the Question

The problem involves moving from 75% red (acidic) to 75% blue (basic), indicating a change in the pH that requires us to use equilibrium concepts of acid and base indicators.

Step 2: Determine Initial Ionization State

At the beginning, the indicator is 75% in its acidic form and 25% in its basic form, meaning that the ratio \( \frac{\text{[base]}}{\text{[acid]}} = \frac{25}{75} = \frac{1}{3} \). From Henderson–Hasselbalch equation, we know the pH to be \( pK_a + \log{\frac{1}{3}} \).

Step 3: Calculate Initial pH

Use the formula: \[ pH = pK_a + \log{\left( \frac{\text{[base]}}{\text{[acid]}} \right)} \]With \(K_a = 3 \times 10^{-5}\), \( pK_a = -\log{(3 \times 10^{-5})} = 5 - 0.4770 \), giving us:\[pH = 4.523 + \log{\left( \frac{1}{3} \right)} = 4.523 - 0.4770 = 4.046\]

Step 4: Determine Final Ionization State

When 75% of the indicator is in the basic (blue) form, 25% is in the acid form, giving a ratio \( \frac{\text{[base]}}{\text{[acid]}} = \frac{3}{1} \). The new pH is given by \( pK_a + \log{3} \).

Step 5: Calculate Final pH

For 75% blue, the pH can be calculated as:\[ pH = 4.523 + \log{3} = 4.523 + 0.4770 = 5.0 \].

Step 6: Calculate Change in pH

The change in pH required is:\[ \Delta \text{pH} = 5.0 - 4.046 = 0.954 \approx 0.95 \].

Key Concepts

pH Calculation

Understanding how to calculate pH is crucial in chemistry, especially when dealing with acid-base reactions. The pH scale measures the acidity or basicity of a solution. It is an important concept for predicting how different substances will interact within a solution. The pH is calculated using the concentration of hydrogen ions (^+]) in a solution. The formula for calculating pH is:
\[ pH = -\log [H^+] \]
In the context of this exercise, we are tasked with determining the pH change required to alter the color of an acid-base indicator. This process involves using calculated pH values from the Henderson-Hasselbalch equation, which helps to understand the pH at which the indicator changes color.

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a key tool for chemists. It provides a way to estimate the pH of a buffer solution. This equation relates the pH to the concentration of an acid and its conjugate base. The formula is:
\[ pH = pK_a + \log\left(\frac{\text{[base]}}{\text{[acid]}}\right) \]
In the exercise, we use the Henderson-Hasselbalch equation to calculate the initial and final pH of the indicator solution. Initially, the concentrated form of the indicator is red (acidic), and it turns blue (basic) as the pH lowers. By applying this equation, students learn how a small change in the pH influences the color of the indicator, here shifting from 75% red to 75% blue.

Equilibrium Concepts

Equilibrium concepts highlight how chemical reactions reach a state where the forward and reverse reactions occur at the same rate. This balance is crucial for understanding acid-base indicators, where the equilibrium between different forms determines the color shown.
In this scenario, the equilibrium shifts from favoring the acid form of the indicator when it is red, to favoring the base form when it is blue. As students solve the exercise, they see how altering the pH can shift equilibrium and change the concentration ratio between the base and acid forms. This understanding of equilibrium is vital because it helps in predicting how long it will take for a reaction to reach equilibrium and how different conditions can affect the outcome.

Ka and pKa Values

The K_a and pK_a values are fundamental in the study of acids and bases. These values help to describe the strength of an acid in a solution. K_a, the acid dissociation constant, quantifies the extent of dissociation of an acid in water.
For weak acids, it is common to use pK_a, which is the negative logarithm of the K_a value:
\[ pK_a = -\log K_a \]
In the exercise, the given K_a value of the indicator helps to determine its pK_a, an essential step in calculating the pH change. The pK_a value represents the pH at which half of the indicator exists in its acid form and half in its base form. This point, often referred to as the halfway point of the transition, is critical because indicators change color around this pH, demonstrating the acidity level's impact on the chemical's behavior.