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 concentration of \\( [\mathrm{H}^+] \\\) required is \\( 1.0 \times 10^{-5} \mathrm{M} \\\), which is option (c).

Step 1: Understand the Problem

The problem involves an acid-base indicator that changes color depending on the pH of the solution. We want to find the \( [\mathrm{H}^+] \) concentration required to change the indicator from 75% red (acid form) to 75% blue (base form).

Step 2: Define the Relationship

The problem describes an equilibrium scenario where the acid form (HIn) and the base form (In-) of the indicator are in equilibrium: \( \mathrm{HIn} \rightleftharpoons \mathrm{H^+} + \mathrm{In^-} \). The equilibrium constant is given as \( \mathrm{K}_a = \dfrac{[\mathrm{H}^+][\mathrm{In^-}]}{[\mathrm{HIn}]} \).

Step 3: Set Ratios for Color Change

Initially, the indicator is 75% red, so \( [\mathrm{HIn}] = 0.75 \) and \( [\mathrm{In^-}] = 0.25 \). We want to find when it's 75% blue, meaning \( [\mathrm{In^-}] = 0.75 \) and \( [\mathrm{HIn}] = 0.25 \). The concentration ratio is \( \dfrac{[\mathrm{In^-}]}{[\mathrm{HIn}]} = \dfrac{0.75}{0.25} = 3 \).

Step 4: Use the Equilibrium Expression

Rearranging the \( \mathrm{K}_a \) expression gives \( [\mathrm{H}^+] = \dfrac{\mathrm{K}_a \cdot [\mathrm{HIn}]}{[\mathrm{In^-}]} = \dfrac{3.0 \times 10^{-5} \times 0.25}{0.75} \).

Step 5: Perform the Calculation

Calculate \( [\mathrm{H}^+] \) as follows: \( \dfrac{3.0 \times 10^{-5} \times 0.25}{0.75} = \dfrac{0.75 \times 10^{-5}}{0.75} = 1.0 \times 10^{-5} \mathrm{M} \).

Step 6: Select the Correct Answer

The calculated \( [\mathrm{H}^+] = 1.0 \times 10^{-5} \mathrm{M} \) matches option (c).

Key Concepts

Equilibrium Constant (Ka)

In the context of acid-base indicators, the equilibrium constant, denoted as \( K_a \), is fundamental in understanding how these indicators switch colors at different pH levels. The \( K_a \) offers insight into the stability of the ions formed in equilibrium. It also facilitates in determining which direction the reaction will favor (towards reactants or products) depending on the solution's pH.
For an indicator, \( K_a \) is expressed in the equilibrium reaction:

• \( \text{HIn} \rightleftharpoons \text{H}^+ + \text{In}^- \)

Here, \( \text{HIn} \) represents the acid form of the indicator, and \( \text{In}^- \) represents the base form. The \( K_a \) value is calculated using:

• \( K_a = \dfrac{[\text{H}^+][\text{In}^-]}{[\text{HIn}]} \)

Using this equilibrium expression allows chemists to understand at which pH level a significant color change will be observed. A higher \( K_a \) value indicates a stronger tendency for the indicator to dissociate into its ions, while a lower \( K_a \) implies weaker dissociation.
This aspect is crucial for determining the \( [\text{H}^+] \) required to shift equilibrium from one color form to another as shown in the exercise. Thus, \( K_a \) is not merely a number; it is a guide to predicting the behavior of indicators as they encounter solutions of varying acidity.

Chemical Equilibrium

Chemical equilibrium is a dynamic state where the rate of the forward reaction equals that of the reverse reaction, maintaining constant concentrations of products and reactants over time. For acid-base indicators, this equilibrium determines the color observed based on the relative concentrations of the acid and base forms.
The equation governing this balance for an acid-base indicator is:

• \( \text{HIn} \rightleftharpoons \text{H}^+ + \text{In}^- \)

In this context, if we have 75% of the indicator in its acid (red) form and 25% in its base (blue) form, equilibrium is shifted towards the acid form. As the solution becomes more basic, \( \text{H}^+ \) ions are consumed, shifting the equilibrium towards the base form, hence changing the color.
To predict such color changes accurately, the knowledge of equilibrium ratios, like \( \dfrac{[\text{In}^-]}{[\text{HIn}]} \), is necessary. Changing the concentration of \( [\text{H}^+] \) adjusts this ratio and prompts the observable shift from red to blue or vice versa.

This exercise portrays how equilibrium dictates the perceptible change due to the delicate balance between \( \text{HIn} \) and \( \text{In}^- \) and how the indicator responds to alterations in \( [\text{H}^+] \).

Colorimetric Analysis

Colorimetric analysis is a technique that involves quantifying the concentration of chemical substances by measuring the intensity of their color. For acid-base indicators, colorimetric analysis is pivotal as it visually manifests the changes in chemical equilibrium.
These indicators exhibit distinct colors depending on their dissociation state in the solution, governed by the concentration of \( [\text{H}^+] \). For instance, a color shift from red to blue occurs with a change in \( [\text{H}^+] \), indicating a transition from acidic to more basic conditions.
In practical applications, this process becomes a simple yet powerful tool for determining pH levels or the acidity/basicity of a solution:

• If an indicator is 75% red and transitions to 75% blue, a clear shift is recognized by the observer.
• This method is non-invasive, quick, and cost-effective.
• It relies solely on visually picking up the change without sophisticated equipment.

Thus, colorimetric analysis entwines with both the concepts of \( K_a \) and chemical equilibrium to provide a practical means to visually and accurately understand chemical conditions in various settings.