Problem 115 - A Complete Resource Book in Chemistry for JEE Main

Question

If equilibrium constant for the reaction, \(\mathrm{XO}^{-}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(1) \rightleftharpoons \mathrm{HXO}(\mathrm{aq})+\mathrm{OH}^{-}(\mathrm{aq})\) is \(0.36 \times 10^{-6}\) then find the value of dissociation constant \(\left(\mathrm{K}_{2}\right)\) for \(\mathrm{HXO} ?\) (a) \(0.36 \times 10^{-8}\) (b) \(2.8 \times 10^{-8}\) (c) \(2.8 \times 10^{-10}\) (d) \(0.36 \times 10^{-6}\)

Step-by-Step Solution

Answer

The dissociation constant \( K_2 \) for \( \mathrm{HXO} \) is approximately \( 2.8 \times 10^{-8} \).

Step 1: Identify the given equilibrium constant

In the reaction \( \mathrm{XO}^{-} + \mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{HXO} + \mathrm{OH}^{-} \), we are given the equilibrium constant \( K = 0.36 \times 10^{-6} \). This constant is for the equilibrium involving the base \( \mathrm{XO}^{-} \).

Step 2: Understand the relationship between equilibrium constants

The equilibrium constant for the base reaction \( K_b = K_{eq} = 0.36 \times 10^{-6} \). The relationship between \( K_a \) of the acid \( \mathrm{HXO} \) and \( K_b \) of the base \( \mathrm{XO}^{-} \) is given by \( K_w = K_a \cdot K_b \), where \( K_w \) is the ion-product constant of water \( 1.0 \times 10^{-14} \).

Step 3: Calculate the dissociation constant \( K_2 \)

To find \( K_2 \) for \( \mathrm{HXO} \), use the equation \( K_2 = \frac{K_w}{K_b} \). Substituting the known values: \[ K_2 = \frac{1.0 \times 10^{-14}}{0.36 \times 10^{-6}} = 2.778 \times 10^{-8} \].

Step 4: Choose the closest answer

From the calculation in step 3, \( K_2 \approx 2.78 \times 10^{-8} \). The closest option to this value is (b) \( 2.8 \times 10^{-8} \).

Key Concepts

Dissociation Constant

The dissociation constant, often referred to in formulas as \( K_a \), is a crucial concept in acid-base chemistry. It measures the strength of an acid in solution.
When an acid dissolves in water, it disassociates into its constituent ions. The dissociation constant quantifies this process, providing insight into how completely an acid dissociates into its ions.

• For strong acids, \( K_a \) is large because they dissociate almost entirely.
• For weak acids, \( K_a \) is smaller, indicating partial dissociation.

Understanding \( K_a \) is important because it allows chemists to predict the behavior of acids in various chemical environments. In our exercise, the dissociation constant \( K_2 \) for \( \mathrm{HXO} \) aids in identifying its potential as an acid in the given reaction.

Base Ionization Constant

The base ionization constant, or \( K_b \), is similarly important but applies to bases instead of acids. It measures the extent to which a base dissociates in water to form hydroxide ions \( \text{OH}^- \).
This constant plays a vital role in understanding the basicity of a compound and thus its ability to accept protons.

• Bases with a high \( K_b \) are considered strong bases because they robustly form hydroxide ions.
• Bases with lower \( K_b \) values are identified as weak bases.
In this exercise, the base \( \mathrm{XO}^- \) has an ionization constant \( K_b \) equal to \( 0.36 \times 10^{-6} \). This is essential for determining the acidic nature of the complementary acid \( \mathrm{HXO} \) by using its relationship with \( K_a \) and the water ion-product constant \( K_w \).

Water Ion-Product Constant

The water ion-product constant, \( K_w \), is a foundational constant in the study of aqueous solutions. It represents the autoionization of water, which is an equilibrium process where water partially ionizes into hydrogen and hydroxide ions.

• \( K_w \) at 25°C is approximately \( 1.0 \times 10^{-14} \).

This constant is crucial in connecting \( K_a \) and \( K_b \) for any conjugate acid-base pair via the equation \( K_w = K_a \cdot K_b \). Understanding this relationship is significant, especially when evaluating the equilibrium behavior of acids and bases.
In our problem, \( K_w \) is used to calculate the dissociation constant \( K_2 \) for \( \mathrm{HXO} \) with the base ionization constant \( K_b \) of \( \mathrm{XO}^- \), offering insights into the strength and behavior of these chemical species.