Question

If a salt of strong acid and weak base hydrolyses appreciably \((\alpha=0.1)\), which of the following formula is to be used to calculate degree of hydrolysis ' \(\alpha\) '? (a) \(\alpha=\sqrt{\frac{K_{w}}{K_{a} \cdot a}}\) (b) \(\alpha=\sqrt{\frac{K_{w}}{K_{b} \cdot a}}\) (c) \(\alpha=\sqrt{\frac{K_{w}}{K_{a} \cdot K_{b}}}\) (d) None of these

Short answer

Option (b) \(\alpha =\sqrt{\frac{K_{w}}{K_{b} \cdot a}}\) is the correct formula for calculating the degree of hydrolysis \(\alpha\) for a salt derived from a strong acid and a weak base.

Step-by-step solution

Identify the nature of the salt

Recognize that the salt is formed from a strong acid and a weak base. Hydrolysis of such salts results in the solution being acidic because the weak base's anion reacts with water to produce the weak base and hydronium ions.

Recognize the correct hydrolysis equation

For salts of strong acids and weak bases, the degree of hydrolysis, represented by \(\alpha\), is given by the equation \(\alpha = \sqrt{\frac{K_{w}}{K_{b} \cdot a}}\), where \(K_{w}\) is the ionic product of water, \(K_{b}\) is the dissociation constant of the weak base, and \(a\) is the concentration of the salt solution.

Choose the correct formula

The correct option that represents the degree of hydrolysis for a salt made of strong acid and weak base is option (b) \(\alpha =\sqrt{\frac{K_{w}}{K_{b} \cdot a}}\).

Key concepts

Ionic Product of Water

The ionic product of water, denoted by \(K_w\), is a constant value that represents the equilibrium concentration of hydrogen ions (\(H^+\)) and hydroxide ions (\(OH^-\)) in pure water at a specific temperature, usually 25°C. This constant is crucial in understanding acid-base reactions and the hydrolysis of salts.

Here's a simple equation that expresses this equilibrium for water: \[H_2O \rightleftharpoons H^+ + OH^-\] \(K_w = [H^+][OH^-]\) is typically valued at \(1 \times 10^{-14}\) at 25°C. This value implies that in pure water, the concentration of hydrogen and hydroxide ions will both be \(1 \times 10^{-7} M\), leading to a neutral pH of 7. When dealing with salt hydrolysis, \(K_w\) helps to determine the extent to which water molecules dissociate, influencing the pH of the solution.

Dissociation Constant

The dissociation constant \(K_a\) or \(K_b\) is a measure of the strength of an acid or a base, respectively. It is a quantitative representation of the degree to which an acid donates protons \(H^+\) or a base accepts protons. The strength of the acid or base dictates how easily it dissociates in water - a higher dissociation constant indicates a stronger acid or base.

In our exercise, \(K_b\), the dissociation constant for a weak base, is particularly significant. \(K_b\) is defined by the equilibrium expression for the base when dissolved in water: \[B + H_2O \rightleftharpoons HB^+ + OH^-\] \(K_b = \frac{[HB^+][OH^-]}{[B]}\) A lower value of \(K_b\) signifies that the base is weaker and less likely to accept protons from water, which is essential when calculating the degree of hydrolysis in salt solutions.

Acid-Base Hydrolysis

Acid-base hydrolysis refers to the chemical reaction where a salt (the product of an acid and base reaction) reacts with water to form its corresponding acid or base. Depending on the strengths of the original acid and base from which the salt was formed, the solution can become acidic, basic, or remain neutral.

In the exercise, we're dealing with a salt from a strong acid and a weak base. This salt will undergo hydrolysis to produce the weak base and hydronium ions (\(H_3O^+\)). The hydrolysis reaction can be represented as: \[A^- + H_2O \rightleftharpoons HA + OH^-\] where \(A^-\) indicates the anion from the weak base. Since the base is weak, the reaction shifts left, favoring the formation of \(A^-\) and \(H_3O^+\), leading to an acidic solution.

The degree of hydrolysis (\(\alpha\)) is vital for understanding the extent of this reaction. It is calculated under the assumption that the concentration of hydronium ions is directly proportional to the degree of hydrolysis, which, for a salt of a strong acid and a weak base, is given by: \[\alpha = \(\sqrt{\frac{K_{w}}{K_{b} \cdot a}}\)\] Understanding these concepts helps in predicting the pH of the solution and the behavior of salts in various chemical processes.