Question

The hydrolysis constant \(K_{h}\) of a salt of sodium hydroxidc and weak acid (IIX) if the \(K_{a}\) of the acid is \(5 \times 10^{-6}\) is (1) \(2 \times 10^{-8}\) (2) \(5 \times 10^{-6}\) (3) \(2.5 \times 10^{-7}\) (4) \(5 \times 10^{-9}\)

Short answer

The hydrolysis constant is \(5 \times 10^{-6}\). Therefore, the correct option is (2).

Step-by-step solution

- Identify the relationship between constants

Hydrolysis constant (\(K_{h}\)) is related to the ionization constant (\(K_{a}\)) of the acid and the ionization constant (\(K_{b}\)) of its conjugate base.

- Use the formula for hydrolysis constant

The general relationship is given by \[ K_{h} = \frac{K_{w}}{K_{b}} \]where \(K_{w}\) is the ion-product constant of water.

- Relate ionization constants

Since the salt is derived from a weak acid (HA) and a strong base (NaOH), the conjugate base (A\textsuperscript{-}) will hydrolyze. Thus,\(K_b\) is related to \(K_a\) by \(K_b = \frac{K_{w}}{K_a}\)

- Substitute and simplify

Substitute \(K_b\) from Step 3 into the equation from Step 2: \[ K_{h} = \frac{K_{w}}{K_{b}} = \frac{K_{w}}{\frac{K_{w}}{K_{a}}} = K_{a} \]Given that \(K_{a} = 5 \times 10^{-6}\)

- Calculate hydrolysis constant

Thus, \(K_{h} = 5 \times 10^{-6}\)

- Identify the correct option

From the given options, (2) matches our calculated value.

Key concepts

salt hydrolysis

Salt hydrolysis refers to the reaction of a salt with water to produce an acidic or basic solution. When a salt forms from a reaction between a strong base and a weak acid, its conjugate base (the part that comes from the weak acid) will tend to hydrolyze in water.
This means it will react with water to form hydroxide ions and create a basic solution.
Conversely, if a salt forms from a strong acid and a weak base, the conjugate acid will hydrolyze and produce hydronium ions, leading to an acidic solution.
Salt hydrolysis can be summarized like this:

• If the salt comes from a strong acid and a weak base, it yields an acidic solution.
• If the salt comes from a strong base and a weak acid, it yields a basic solution.
• If the salt comes from a strong acid and a strong base, it does not hydrolyze and creates a neutral solution.

Understanding salt hydrolysis is important because it helps predict whether a solution of salt in water will be acidic, basic, or neutral.

ion-product constant

The ion-product constant, denoted as \(K_{w}\), is a fundamental property of water. It is defined as the product of the concentrations of hydrogen ions (\(H^{+}\)) and hydroxide ions (\(OH^{-}\)) in water.
Mathematically, it is expressed as: \[K_{w} = [H^{+}][OH^{-}]\] At 25°C, the value of \(K_{w}\) is \(1 \times 10^{-14}\).
This constant is crucial in the context of hydrolysis and equilibrium reactions because it allows us to relate the strength of an acid to the strength of its conjugate base. When dealing with salt hydrolysis, knowing \(K_{w}\) helps in calculating other constants like the hydrolysis constant (\(K_{h}\)).
For example, if we know the ionization constant of an acid, \(K_{a}\), we can use \(K_{w}\) to find the ionization constant of its conjugate base, \(K_{b}\), using the relation: \[K_{w} = K_{a} \times K_{b}\] This relation is particularly helpful in problems involving acid-base equilibrium and hydrolysis.

ionization constant

The ionization constant encompasses two types: the ionization constant of an acid, \(K_{a}\), and the ionization constant of a base, \(K_{b}\). These constants are measures of the strength of an acid or a base.
For an acid, \(K_{a}\) is the equilibrium constant for the ionization (dissociation) of the acid in water. It is defined as: \[K_{a} = \frac{[H^{+}][A^{-}]}{[HA]}\] where \[ [H^{+}] \] is the concentration of hydrogen ions, \[ [A^{-}] \] is the concentration of the conjugate base, and \[ [HA] \] is the concentration of the undissociated acid.
Similarly, for a base, \(K_{b}\) is the equilibrium constant for the ionization of the base in water: \[K_{b} = \frac{[BH^{+}][OH^{-}]}{[B]}\] These ionization constants help in determining the pH and pOH of a solution and are key in understanding acid-base balance in chemical reactions.
If you know the \(K_{a}\) of an acid and you have \(K_{w}\), you can find \(K_{b}\) of the conjugate base using the relationship \[ K_{b} = \frac{K_{w}}{K_{a}} \] This is critical in problems like the hydrolysis of salts, where you might need to determine the resulting pH of the solution.