Problem 253 - A Complete Resource Book in Chemistry for JEE Main

Question

Calculate \(\mathrm{pH}\) at which an acid indicator HIn with concentration \(0.1 \mathrm{M}\) changes its colour \(\left(\mathrm{K}_{\mathrm{a}}\right.\) for \(\mathrm{HIn}=\) \(\left.1 \times 10^{-5}\right)\)

Step-by-Step Solution

Answer

The pH at which the indicator changes color is 5.

Step 1: Understand the Problem

We need to find the pH at which an acid indicator (HIn) changes color. The given concentration of the indicator is 0.1 M, and its acid dissociation constant, \(K_a\), is \(1 \times 10^{-5}\).

Step 2: Use the Ionization Equation

The dissociation of the indicator HIn can be expressed as: \[ \text{HIn} \rightleftharpoons \text{H}^+ + \text{In}^- \] The equilibrium expression can be written as: \[ K_a = \frac{[\text{H}^+][\text{In}^-]}{[\text{HIn}]} \]

Step 3: Make Assumptions for Calculation

For indicators, the color change typically occurs around the equilibrium where \([\text{HIn}] \approx [\text{In}^-]\). Therefore: \[ K_a \approx [\text{H}^+] \]

Step 4: Calculate the pH

Since \( K_a \approx [\text{H}^+] \), we have: \[ [\text{H}^+] = 1 \times 10^{-5} \] The pH is calculated as: \[ \text{pH} = -\log_{10}([\text{H}^+]) \] Substitute \([\text{H}^+]\): \[ \text{pH} = -\log_{10}(1 \times 10^{-5}) = 5 \]

Step 5: Conclusion

Thus, the pH where the indicator changes color is 5.

Key Concepts

Acid Dissociation Constant

The Acid Dissociation Constant, often represented as \(K_a\), is a crucial component in understanding acid strength and behavior in solutions. It quantifies the tendency of the acid to donate a proton \(H^+\) in an aqueous solution. In the context of the exercise, the acid indicator \(\text{HIn}\) has a \(K_a\) of \(1 \times 10^{-5}\). This value suggests that \(\text{HIn}\) is a weak acid because its \(K_a\) is much smaller than 1.
\(K_a\) is calculated using the concentrations of the species involved in the acid's equilibrium with the expression:

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

This equilibrium expression will make it possible to predict the change in color of the indicator based on the \([\text{H}^+]\) concentration in the solution.

Equilibrium Expression

Chemical equilibrium is when the rate of the forward reaction equals the rate of the reverse reaction. This means the concentrations of reactants and products remain constant over time. For an acid such as \(\text{HIn}\), the equilibrium expression is crucial to determining when the indicator changes color. This expression is given by:

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

This formula expresses the balance between the proton \([\text{H}^+]\) concentration and the conjugate base \([\text{In}^-]\) with the remaining undissociated acid \(\text{HIn}\). At equilibrium, the concentrations of these components will not change unless external conditions (like pH or concentration) shift, which would demand a new equilibrium to be established. Understanding and utilizing the equilibrium expression is vital to predict the conditions under which the observed color change will occur.

Ionization Equation

The ionization equation illustrates the dissociation of an acid into its ions. For the indicator \(\text{HIn}\), this dissociation is represented by:

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

This equation shows the reversible reaction where \(\text{HIn}\) splits into a hydrogen ion \(\text{H}^+\) and its conjugate base \(\text{In}^-\). In the realm of pH calculations, knowing the ionization equation helps to determine the concentration of \(\text{H}^+\) ions, which is directly related to pH via the formula \(\text{pH} = -\log_{10}(\text{[H}^+])\).
In this example, it is understood for color indicators that the color change happens when the concentrations of \(\text{HIn}\) and \(\text{In}^-\) are roughly equal. Hence, \([\text{HIn}] \approx [\text{In}^-]\), simplifying the calculation to \(K_a \approx [\text{H}^+]\). This allows us to find at which \(\text{H}^+\) concentration the color change occurs.

Color Change of Indicators

Indicators, such as \(\text{HIn}\), are compounds that change color depending on the pH of the solution they are in. This color change is due to the ionization equilibrium of the indicator itself. As the \([\text{H}^+]\) concentration changes, so does the balance between \(\text{HIn}\) and \(\text{In}^-\). Most indicators have a distinct pH range over which they change color, often around \(\pm1\) pH unit around the pKa value of the indicator.
The color change occurs because each form of the indicator (the non-ionized and ionized) has a different color. When \([\text{HIn}]\) is nearly equal to \([\text{In}^-]\), the indicator shows its transition color which is often a blend of the two forms. For \(\text{HIn}\), its color change signifies the pH at which \(\text{H}^+\) concentration equals \(K_a\), hence, resulting in the noticeable shift in color signaling the change of pH in the environment.