Problem 24 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

A weak acid \(\mathrm{H} X\) has the dissociation constant \(1 \times 10^{-5} \mathrm{M}\). It forms a salt \(\mathrm{Na} X\) on reaction with alkali. The percentage hydrolysis of \(0.1 \mathrm{M}\) solution of \(\mathrm{Na} X\) is (a) \(0.0001 \%\) (b) \(0.01 \%\) (c) \(0.1 \%\) (d) \(0.15 \%\)

Step-by-Step Solution

Answer

The percentage hydrolysis of \(\mathrm{Na} X\) is \(0.01\%\).

Step 1: Write the Hydrolysis Reaction

Focus on the hydrolysis reaction of the salt \( \mathrm{Na}X \), which can be written as: \( \mathrm{X}^- + \mathrm{H}_2\mathrm{O} \rightleftharpoons \mathrm{HX} + \mathrm{OH}^- \). This equation shows how the weak acid anion \( \mathrm{X}^- \) reacts with water to form the weak acid and a hydroxide ion.

Step 2: Define the Hydrolysis Constant

The hydrolysis constant \( K_h \) is given by the expression: \[ K_h = \frac{K_w}{K_a} \] where \( K_w \) is the ion-product constant of water \( (1 \times 10^{-14} \ \mathrm{M}^2) \) and \( K_a \) is the dissociation constant of the weak acid \( (1 \times 10^{-5} \ \mathrm{M}) \).

Step 3: Calculate the Hydrolysis Constant

Substitute the known values into the hydrolysis constant expression: \[ K_h = \frac{1 \times 10^{-14}}{1 \times 10^{-5}} = 1 \times 10^{-9} \ \mathrm{M} \].

Step 4: Calculate the Concentration of Hydroxide Ion

If \( h \) is the degree of hydrolysis, for initial concentration \( c = 0.1 \ \mathrm{M} \), the final concentration of \( \mathrm{OH}^- \) ions is \( ch = 0.1h \). From \( K_h = c h^2 \), substitute \( K_h \) and \( c \) to get: \[ 1 \times 10^{-9} = 0.1 h^2 \].

Step 5: Solve for Degree of Hydrolysis

Rearrange the equation from the previous step to obtain \( h^2 \): \[ h^2 = \frac{1 \times 10^{-9}}{0.1} = 1 \times 10^{-8} \]. Solve for \( h \) by taking the square root: \[ h = \sqrt{1 \times 10^{-8}} = 10^{-4} \].

Step 6: Calculate Percentage Hydrolysis

The percentage hydrolysis is obtained by multiplying \( h \) by 100: \[ \text{Percentage hydrolysis} = h \times 100 = 10^{-4} \times 100 = 0.01\% \].

Key Concepts

Weak Acid Dissociation

When weak acids dissolve in water, they do not completely dissociate into ions. This incomplete dissociation is what characterizes a weak acid. The dissociation constant, denoted by \(K_a\), is a measure of the extent of this dissociation. For a weak acid \( \ ext{HX} \), the dissociation can be represented as follows:

\[\text{HX} \rightleftharpoons \text{H}^+ + \text{X}^-\]
The dissociation constant \(K_a\) is given by the expression:

\[K_a = \frac{[\text{H}^+][\text{X}^-]}{[\text{HX}]}\]
A larger value of \(K_a\) indicates a stronger acid, meaning it dissociates more in solution. However, for weak acids, \(K_a\) is quite small, indicating that most of the acid remains undissociated. In the given problem, \(K_a\) is \(1 \times 10^{-5} \ \text{M}\), which is typical of a weak acid, as only a small portion ionizes in water.

Degree of Hydrolysis

Hydrolysis refers to the reaction of an ion with water to form a weak acid or base. When a salt derived from a weak acid, such as \( \text{NaX} \), undergoes hydrolysis, the anion \( \text{X}^- \) reacts with water to re-form the weak acid \( \text{HX} \) and produces hydroxide ions (\( \text{OH}^- \)).
The extent to which this hydrolysis occurs is known as the degree of hydrolysis, represented by \( h \). In the context of a solution, it represents the fraction of the initial salt concentration that is hydrolyzed.

• The degree of hydrolysis is obtained from the expression \( K_h = c h^2 \), where \( c \) is the initial concentration of the salt.
• The lower the \(K_a\) of the weak acid, the higher the degree of hydrolysis, as there are more ions available to react with water.

In our problem, for a \(0.1 \ \text{M}\) solution of \( \text{NaX} \), the calculated degree of hydrolysis \( h \) is \(10^{-4}\). This means that only a small fraction, specifically 0.01%, of the salt is hydrolyzed, which is typical for salts derived from weak acids.

Hydrolysis Constant

The hydrolysis constant \(K_h\) quantifies the extent of hydrolysis for a salt derived from a weak acid and a strong base. It is an important parameter as it helps predict the behavior of the salt in solution.

The expression for \(K_h\) is derived from the relationship between the equilibrium constants of hydrolysis:\[K_h = \frac{K_w}{K_a}\]
Where:

• \(K_w\) is the ion-product constant of water, typically \(1 \times 10^{-14} \ \text{M}^2\) at 25°C.
• \(K_a\) is the dissociation constant for the weak acid.

Applying these values to our exercise, the hydrolysis constant is calculated as \(1 \times 10^{-9}\ \text{M}\). This low value confirms that hydrolysis doesn't proceed extensively, as only a small amount of the salt ions will revert to the weak acid in water.