Problem 278 - A Complete Resource Book in Chemistry for JEE Main

Question

The \(\mathrm{pH}\) of a \(0.1\) molar solution of the acid \(\mathrm{HQ}\) is 3 . The value of the ionization contstant, Ka of this acid is [2012] (a) \(1 \times 10^{-7}\) (b) \(3 \times 10^{-1}\) (c) \(1 \times 10^{-3}\) (d) \(1 \times 10^{-5}\)

Step-by-Step Solution

Answer

The ionization constant, \(K_a\), is \(1 \times 10^{-5}\).

Step 1: Understanding pH Formula

The pH of a solution is given by the formula \( \text{pH} = -\log_{10}[H^+] \), where \([H^+]\) is the concentration of hydrogen ions in the solution.

Step 2: Calculate Hydrogen Ion Concentration

Given that the pH is 3, we can calculate the concentration of hydrogen ions \([H^+]\) by rearranging the pH formula: \([H^+] = 10^{-\text{pH}} = 10^{-3} \ mol/L\).

Step 3: Write the Ionization Equation

The ionization of the acid HQ in water can be represented as \( \mathrm{HQ} \rightleftharpoons \mathrm{H^+} + \mathrm{Q^-} \).

Step 4: Apply the Expression for Ka

The ionization constant \( K_a \) for the acid is given by \( K_a = \frac{[H^+][Q^-]}{[HQ]} \). We assume full ionization does not occur as pH is not zero.

Step 5: Substitute Concentrations into Ka Expression

With \([H^+] = 10^{-3}\) and assuming \([Q^-] = [H^+]\), we substitute these values into the \( K_a \) expression. The starting concentration of HQ is 0.1 M, so \( [HQ] = 0.1 - 10^{-3} \approx 0.1\).

Step 6: Solve for Ka

Substitute the values into the expression: \( K_a = \frac{(10^{-3})(10^{-3})}{0.1} = \frac{10^{-6}}{0.1} = 10^{-5} \)

Key Concepts

pH calculation

Understanding how to calculate pH is essential when dealing with solutions of acid and base. The pH scale ranges from 0 to 14 and provides an indication of the acidity or basicity of a solution.
It is defined by the formula:

• \( ext{pH} = - ext{log}_{10}[H^+] \)

Where \([H^+]\) is the concentration of hydrogen ions in the solution. If the pH is low (under 7), the solution is acidic, and if it's high (over 7), it is basic—or alkaline. A pH of exactly 7 indicates a neutral solution.
When given the pH value, such as a pH of 3, you can determine the corresponding hydrogen ion concentration by rearranging the formula to solve for \([H^+]\). The formula becomes:

• \([H^+] = 10^{- ext{pH}}\)

So, for a pH of 3, the hydrogen ion concentration is \(10^{-3}\) mol/L.

Hydrogen ion concentration

The hydrogen ion concentration tells us how many hydrogen ions are present in a solution. It is a critical factor in determining the pH of the solution.
As mentioned, for a solution with pH 3, the concentration of hydrogen ions is \(10^{-3}\) mol/L.

• The more hydrogen ions present, the more acidic a solution is, which means a lower pH.
• Conversely, fewer hydrogen ions result in a higher pH, indicating a more alkaline solution.

Understanding this concentration helps us in various calculations, such as determining the strength of acids and bases. This concentration is derived directly from the pH using the formula for pH calculation, making it an integral part of acid-base chemistry.

Ionization equation

An ionization equation provides a detailed representation of how a compound dissociates in a solution. For the acid \(HQ\), the ionization equation reflects the process it undergoes when dissolved in water:

• \(\mathrm{HQ} \rightleftharpoons \mathrm{H^+} + \mathrm{Q^-}\)

This equation shows that \(HQ\) splits into hydrogen ions \((H^+)\) and conjugate base ions \(Q^-\). The double arrow in the ionization equation indicates that the reaction is reversible, reaching an equilibrium where the rate of the forward reaction equals the rate of the backward reaction.
In this case, the solution's pH indicates that not all \(HQ\) molecules fully ionize, which is common for weak acids. Observing the ionization equation helps understand the equilibrium conditions and the degree to which dissociation occurs.

Acid-Base equilibrium

Acid-base equilibrium is a state in which the concentration of acids and bases in a solution remain constant over time. It is a balance achieved when the rates of the forward and backward ionization processes are equal.
The equilibrium for the dissociation of the acid \(HQ\) can be described by the expression for the ionization constant \(K_a\):

• \(K_a = \frac{[H^+][Q^-]}{[HQ]}\)

This formula quantifies the strength of the acid and its propensity to donate hydrogen ions. In the exercise, given \([H^+] = 10^{-3}\) and assuming \([Q^-] = [H^+]\), we calculate \(K_a\) considering the initial concentrations. The calculated \(K_a = 10^{-5}\) indicates a weak acid, which does not completely dissociate in water.
Understanding the equilibrium helps predict the behavior of acids and bases in various chemical reactions and is crucial in areas like buffer solution design and titration analysis.