Problem 185 - A Complete Resource Book in Chemistry for JEE Main

Question

The percentage hydrolysis of \(\mathrm{NaCN}\) in \(\left(\frac{\mathrm{N}}{80}\right)\) aqueous solution [Dissociation constant of \(\mathrm{HCN}\) is \(1.3 \times 10^{-9}\) and \(\left.\mathrm{K}_{w}=1.0 \times 10^{-14}\right]\) is (a) \(8.2\) (b) \(9.6\) (c) \(5.26\) (d) \(2.48\)

Step-by-Step Solution

Answer

The percentage hydrolysis of \(\mathrm{NaCN}\) is \(2.48\%\). Therefore, option (d) is correct.

Step 1: Determine Concentration of NaCN

Since the solution's normality is given as \(\frac{\mathrm{N}}{80}\), we know this equals \(\frac{1}{80}\) normal. As \(\mathrm{NaCN}\) fully dissociates, this is equal to the molarity of \(\mathrm{NaCN}\). Thus, the concentration \([\mathrm{NaCN}]\) is \(0.0125\, \text{M}\) (since \(N_1=1\), \(\text{Molarity} = \text{Normality}\)).

Step 2: Understand Hydrolysis of NaCN

\(\mathrm{NaCN}\) dissociates completely in water, forming \(\mathrm{CN}^-\). Here, \(\mathrm{CN}^-\) undergoes hydrolysis: \[ \mathrm{CN}^- + \mathrm{H}_2\mathrm{O} \rightleftharpoons \mathrm{HCN} + \mathrm{OH}^- \] This equilibrium is controlled by the hydrolysis constant \(K_h\).

Step 3: Calculate Hydrolysis Constant (Kh)

Hydrolysis constant \(K_h\) is given by \(K_w / K_a\). Here, \[ K_a = 1.3 \times 10^{-9} \] and \[ K_w = 1.0 \times 10^{-14} \]. Thus, \[ K_h = \frac{1.0 \times 10^{-14}}{1.3 \times 10^{-9}} = 7.69 \times 10^{-6} \].

Step 4: Relate Hydrolysis and Initial Concentration

The degree of hydrolysis \(h\) can be calculated as: \[ h = \sqrt{\frac{K_h}{[\mathrm{NaCN}]}} \] Substitute the values: \[ h = \sqrt{\frac{7.69 \times 10^{-6}}{0.0125}} = \sqrt{6.152 \times 10^{-4}} \approx 0.0248 \]

Step 5: Calculate Percentage Hydrolysis

Percentage hydrolysis is obtained by multiplying the degree of hydrolysis \(h\) by 100: \[ \text{Percentage hydrolysis} = 0.0248 \times 100 = 2.48\% \].

Key Concepts

Dissociation Constant

The dissociation constant, often symbolized as \( K_a \), is a measure of the strength of an acid in solution. It is defined as the equilibrium constant for the dissociation of an acid into its ions:
\[\mathrm{HA} \rightleftharpoons \mathrm{H}^+ + \mathrm{A}^-\]
Here, \( \mathrm{HA} \) represents the undissociated acid, \( \mathrm{H}^+ \) is the hydrogen ion, and \( \mathrm{A}^- \) is the conjugate base.

• A larger \( K_a \) value indicates a stronger acid, which dissociates more completely in water.
• In weak acids, \( K_a \) is generally a small value, showing that few molecules dissociate.

Understanding \( K_a \) helps predict how acidic a solution is and plays a crucial role in calculating the hydrolysis reactions of salts derived from weak acids.

Hydrolysis Constant

The hydrolysis constant \( K_h \) is the equilibrium constant for the hydrolysis reaction. Hydrolysis occurs when an ion from a salt reacts with water:
\[\mathrm{CN}^- + \mathrm{H}_2\mathrm{O} \rightleftharpoons \mathrm{HCN} + \mathrm{OH}^-\]
This specific reaction involves the cyanide ion from \( \mathrm{NaCN} \), a salt of weak acid \( \mathrm{HCN} \).

• To find \( K_h \), we use the formula: \( K_h = \frac{K_w}{K_a} \), where \( K_w \) is the ionization constant of water.
• Here, \( K_w = 1.0 \times 10^{-14} \) and \( K_a = 1.3 \times 10^{-9} \), giving \( K_h = 7.69 \times 10^{-6} \).

The hydrolysis constant helps understand the extent of hydrolysis and the formation of an acidic or basic solution in water.

Percentage Hydrolysis

Percentage hydrolysis reflects the fraction of a solute that undergoes hydrolysis, expressed as a percentage of the total initial concentration. To calculate it:

• First determine the degree of hydrolysis \( h \) using: \( h = \sqrt{\frac{K_h}{[\mathrm{NaCN}]}} \).
• For \( \mathrm{NaCN} \), \( h = \sqrt{\frac{7.69 \times 10^{-6}}{0.0125}} \).
• This results in \( h \approx 0.0248 \).
• Finally, multiply \( h \) by 100 to convert this value to a percentage, yielding the percentage hydrolysis of \( 2.48\% \).

This value shows how much of the substance has reacted with water, providing insights into the solution's nature and acidity.

Degree of Hydrolysis

The degree of hydrolysis \( (h) \) is a crucial factor in determining how much of a solute carries out hydrolysis. It provides a ratio that shows the amount hydrolyzed compared to the initial concentration.

• Calculated as \( h = \sqrt{\frac{K_h}{[\mathrm{solute}]}} \).
• In our context, \( \mathrm{NaCN} \) concentration is \( 0.0125\, \text{M} \).
• The degree of hydrolysis exposes the extent to which the ions are reacting with water in solution.

Knowing the degree of hydrolysis helps chemists understand and predict the properties of a solution, such as pH and ionic strength.