Problem 151 - A Complete Resource Book in Chemistry for JEE Main

Question

At what concentration of \(\mathrm{CH}_{3} \mathrm{COOH}\) will the \(\left[\mathrm{H}^{+}\right]\) obtained will be same as that obtained from \(10^{-2} \mathrm{M}\) \(\mathrm{HCOOH},\left(\mathrm{Ka}\left(\mathrm{CH}_{3} \mathrm{COOH}\right)=10^{-5}, \mathrm{Ka}(\mathrm{HCOOH})=10^{-4}\right)\) (a) \(10 \mathrm{M}\) (b) \(5 \mathrm{M}\) (c) \(10^{-1} \mathrm{M}\) (d) \(6 \mathrm{M}\)

Step-by-Step Solution

Answer

The concentration of \( \mathrm{CH}_{3} \mathrm{COOH} \) is \( 10^{-1} \mathrm{M} \) (option c).

Step 1: Understand the Problem

We need to find the concentration of \( \mathrm{CH}_3\mathrm{COOH} \) that results in the same \( [\mathrm{H}^+] \) as that produced by \( 10^{-2} \mathrm{M} \) \( \mathrm{HCOOH} \). We will use the ionization constant \( K_a \) for each acid.

Step 2: Calculate [H⁺] for HCOOH

For \( \mathrm{HCOOH} \), \( K_a = 10^{-4} \) and initial concentration \([HCOOH]_i = 10^{-2} \mathrm{M} \). We assume that \( x \) is the concentration of \( \mathrm{H}^+ \) dissociated. Thus,\[ K_a = \frac{x^2}{[HCOOH]_i - x} \approx \frac{x^2}{[HCOOH]_i} \] Because \( x \) is small compared to \( [HCOOH]_i \), \( 10^{-4} \approx \frac{x^2}{10^{-2}} \). Solving this, we find\[ x = \sqrt{10^{-4} \times 10^{-2}} = 10^{-3} \mathrm{M} \] So, \( [\mathrm{H}^+] = 10^{-3} \mathrm{M} \).

Step 3: Set Up the Equation for CH₃COOH

Now we need \([\mathrm{H}^+] = 10^{-3} \) M for \( \mathrm{CH}_3\mathrm{COOH} \) as well. Given \( K_a = 10^{-5} \), if \( c \) is the initial concentration of \( \mathrm{CH}_3\mathrm{COOH} \), we use:\[ K_a = 10^{-5} = \frac{(10^{-3})^2}{c - 10^{-3}} \approx \frac{10^{-6}}{c} \]

Step 4: Solve for Concentration c

Rearranging the formula, we get\[ 10^{-5}c = 10^{-6} \] \[ c = \frac{10^{-6}}{10^{-5}} = 10^{-1} \mathrm{M} \] Thus, the required concentration of \( \mathrm{CH}_3\mathrm{COOH} \) is \( 10^{-1} \mathrm{M} \).

Key Concepts

Acid Dissociation Constant (Ka)

The acid dissociation constant, commonly represented as \( K_a \), is a crucial parameter illustrating the strength of an acid in aqueous solution. It quantifies the degree of ionization or dissociation of an acid into its hydrogen ion (\( \mathrm{H}^+ \)) and its conjugate base. A higher \( K_a \) value indicates a stronger acid, which dissociates more completely in solution.

For example:

• Formic acid (HCOOH) has a \( K_a \) of \( 10^{-4} \), indicating a relatively higher degree of ionization compared to acetic acid.
• Acetic acid (CH₃COOH) with a \( K_a \) of \( 10^{-5} \), is weaker since it dissociates less.

The understanding of \( K_a \) helps to predict and compare the behavior of different acids in solutions, especially for determining pH and related calculations.

Ionization of Weak Acids

Weak acids do not completely ionize in aqueous solutions. Instead, they reach an equilibrium where only a fraction of the acid dissociates into \( \mathrm{H}^+ \) ions and its conjugate base.

The ionization can be expressed as:

\[\text{HA} \rightleftharpoons \mathrm{H}^+ + \mathrm{A}^-\]

For weak acids like acetic and formic acid:

• The equilibrium position lies towards the left, meaning most acid molecules remain undissociated.
• Calculation of \( [\mathrm{H}^+] \) requires using the \( K_a \) expression: \( K_a = \frac{[\mathrm{H}^+][\mathrm{A}^-]}{[\text{HA}]} \).

The ionization constant, \( K_a \), is vital to understand how a weak acid reacts in solution and is essential for determining the acidic strength and calculating the concentration of ions.

Concentration of Hydrogen Ions

The concentration of hydrogen ions, \( [\mathrm{H}^+] \), in solution is a direct measure of the acidity. It is commonly calculated using the formula derived from the ionization constant for weak acids.

Considerations include:

• For \( 10^{-2} \) M formic acid, we compute \( [\mathrm{H}^+] \) as \( 10^{-3} \) M, using its specific \( K_a \) value.
• To find the corresponding concentration for acetic acid yielding the same \( [\mathrm{H}^+] \), we equate this with its \( K_a \) and solve for its concentration.

Understanding \( [\mathrm{H}^+] \) helps predict the resulting pH of the solution, impacting how the solution behaves in chemical reactions and biological systems.

Formic Acid Dissociation

Formic acid (HCOOH) dissociates in water to produce\( \mathrm{H}^+ \) ions and its conjugate base, formate ions (\( \mathrm{HCOO}^- \)).
This process is governed by its acid dissociation constant \( (K_a = 10^{-4}) \), reflecting its relative strength as a weak acid.

During dissociation:

• The balance of formic acid molecules and ionized forms is maintained dynamically, influencing \( [\mathrm{H}^+] \).
• When fully dissociated, the \( [\mathrm{H}^+] \) can be calculated using the expression \( K_a = \frac{[\mathrm{H}^+][\mathrm{HCOO}^-]}{[ ext{HCOOH}]} \).

This computation provides insights into the behavior of formic acid in different concentration situations, essential for applications ranging from industrial processes to biological systems.