Problem 91 - A Complete Resource Book in Chemistry for JEE Main

Question

The dissociation constant of a weak acid is \(4.9 \times 10^{-8}\), its percentage ionization at \(0.1 \mathrm{M}\) is (a) \(0.07 \%\) (b) \(0.007 \%\) (c) \(0.7 \%\) (d) \(0.0007 \%\)

Step-by-Step Solution

Answer

The percentage ionization is 0.07%, which corresponds to option (a).

Step 1: Write the dissociation equation

For a weak acid, represented as \( HA \), the dissociation equation in water is \( HA \rightarrow H^+ + A^- \).

Step 2: Define the expression for the acid dissociation constant

The dissociation constant \( K_a \) for the weak acid \( HA \) is given by the expression \( K_a = \frac{[H^+][A^-]}{[HA]} \).

Step 3: Assume initial concentrations and changes

Assume the initial concentration of \( HA \) is \( 0.1 \text{ M} \). Let \( x \) be the concentration of \( H^+ \) produced at equilibrium, then \([H^+] = [A^-] = x\) and \([HA] = 0.1 - x \).

Step 4: Simplify the expression using the small \( x \) approximation

Because \( HA \) is a weak acid and only partially dissociates, we can approximate \( x \ll 0.1 \), so \( [HA] \approx 0.1 \).

Step 5: Solve for \( x \)

Substitute the approximations into the dissociation constant expression: \( K_a = 4.9 \times 10^{-8} = \frac{x^2}{0.1} \). This simplifies to \( x^2 = 4.9 \times 10^{-9} \), which gives \( x = \sqrt{4.9 \times 10^{-9}} = 7.0 \times 10^{-5} \).

Step 6: Find percentage ionization

Percentage ionization is given by \( \frac{[H^+]}{[HA]_{initial}} \times 100\% = \frac{7.0 \times 10^{-5}}{0.1} \times 100\% \). Calculate this to get \( 0.07\% \).

Step 7: Select the correct answer

From the calculated percentage ionization, the correct option is \( a) 0.07\% \).

Key Concepts

Weak Acid Ionization

A weak acid does not fully dissociate into ions when dissolved in water. This is a key characteristic differentiating weak acids from strong acids, which dissociate completely. When we consider a weak acid, symbolized as \(HA\), its dissociation in water can be represented by the equation \(HA \rightarrow H^+ + A^-\). Here, \(HA\) denotes the weak acid, \(H^+\) the hydrogen ion, and \(A^-\) the conjugate base.

• *Partial Ionization*: Unlike strong acids that release all their hydrogen ions, weak acids release only a fraction, leading to an equilibrium between the undissociated acid and the ions.
• *Equilibrium State*: A balance is achieved in the solution between the ionized and non-ionized forms of the acid, offering a steady state known as acid base equilibrium.

Understanding the partial nature of weak acid ionization is essential for grasping how these acids behave in solution and affect the pH.

Percentage Ionization

Percentage ionization helps to quantify how much of the acid has dissociated at equilibrium, giving a numerical value to understand its ionization extent. It is a critical metric in the study of weak acids.
The formula for percentage ionization is:\[ \text{Percentage Ionization} = \left( \frac{[H^+]}{[HA]_{\text{initial}}} \right) \times 100\% \]Where:

• *[H+] represents the concentration of ionized hydrogen ions (in moles per liter) present in the solution.
• *[HA]_{\text{initial}} is the initial concentration of the weak acid.

Higher percentage ionization means a greater portion of the acid has dissociated, showing more acid molecules releasing their hydrogen ions.

Acid-Base Equilibrium

In chemical systems, equilibrium provides a snapshot of how reactions progress over time. For weak acids, acid-base equilibrium illustrates the dynamic balance between the dissociated and undissociated form.

• *Dynamic Equilibrium*: In a weak acid solution, the forward reaction where \(HA\) dissociates into \(H^+\) and \(A^-\) is continuously occurring alongside the reverse reaction where \(H^+\) and \(A^-\) recombine to form \(HA\).
• *Le Chatelier's Principle*: Any change in conditions (such as concentration or temperature) shifts the equilibrium to oppose that change, influencing ionization.

This equilibrium concept is vital for predicting how a weak acid's behavior changes in different chemical environments and is central to calculative approaches used to find concentrations and predict reaction outcomes.

Dissociation Equation

To understand the nature of acids in solution, writing out the dissociation equation is foundational. A dissociation equation displays the process by which an acid separates into ions when in water.
For a generic weak acid denoted as \(HA\), the dissociation equation is written as:\[ HA \rightarrow H^+ + A^- \]This formula captures the exchange happening at a molecular level:

• *Acid Symbol (HA)*: Represents the molecule of the weak acid.
• *Dissociation Arrow*: Indicates the direction of dissociation, showing the substances in reactants changing into the products.
• *Products (H+ and A-)*: Depict the ions formed as a result of the dissociation.

Drafting the dissociation equation is the preliminary step in many calculations regarding acid strength, ionization, or equilibrium in reactions. It lays the groundwork for deeper analyses into the acid behavior and its interactions in solutions.