Problem 246 - A Complete Resource Book in Chemistry for JEE Main

Question

A monoprotic acid in a \(0.1 \mathrm{M}\) solution ionizes to \(0.001 \%\). If its ionization constant is \(\frac{10^{-\mathrm{x}}}{100}\), the value of \(\mathrm{x}\) is

Step-by-Step Solution

Answer

The value of \(x\) is 9.

Step 1: Understand the Ionization

The problem states that a monoprotic acid ionizes in a solution. A monoprotic acid ionizes according to the equation \( HA \leftrightarrow H^+ + A^- \). Since the acid ionizes \(0.001\%\), this means that \(0.001\%\) of the acid's concentration forms \(H^+\) and \(A^-\) ions.

Step 2: Calculate the Concentration of Ionized Acid

Given the initial concentration of the acid is \(0.1\, \mathrm{M}\), calculate the ionized concentration: \(\text{Ionized } = 0.1\, \mathrm{M} \times \frac{0.001}{100} = 0.000001\, \mathrm{M} \).

Step 3: Write the Expression for the Ionization Constant \(K_a\)

The ionization constant formula for a weak acid is \( K_a = \frac{[H^+][A^-]}{[HA]} \). Here, \([H^+] = [A^-] = 0.000001\, \mathrm{M} \) and \([HA] = 0.1 - 0.000001\, \mathrm{M} \approx 0.1\, \mathrm{M}\) for small ionization percentage.

Step 4: Substitute Values to Find \(K_a\)

Substituting the values into the formula, we get: \[ K_a = \frac{(0.000001)(0.000001)}{0.1} = \frac{1 \times 10^{-12}}{0.1} = 1 \times 10^{-11}. \]

Step 5: Relate \(K_a\) to the Given Ionization Constant Expression

We know the expression of the ionization constant is \(\frac{10^{-x}}{100}\). From our calculation, \(K_a = 1 \times 10^{-11}\) can be written as \(\frac{10^{-11}}{1}\).

Step 6: Equate and Solve for \(x\)

Equating the expressions, \(\frac{10^{-x}}{100} = \frac{10^{-11}}{1} \). Solving for \(x\), \(10^{-x} = 100 \times 10^{-11} = 10^{-9} \). Thus, \(x = 9\).

Key Concepts

Monoprotic Acid

A monoprotic acid is a type of acid that can donate only one proton (or hydrogen ion, \(H^+\)) per molecule during the process of ionization. This distinguishes it from polyprotic acids, which can release more than one \(H^+\). The general ionization reaction for a monoprotic acid \(HA\) can be represented by:

• \( HA \leftrightarrow H^+ + A^- \)

During this process, the acid dissociates into hydrogen ions and its conjugate base \(A^-\). This concept is crucial because it determines how the equilibrium of the reaction is approached and calculated for balancing chemical equations.

Weak Acid

Weak acids are acids that do not completely dissociate in solution. Unlike strong acids, which ionize almost completely, weak acids partially ionize when dissolved in water. This means that at any given time, there will be a mix of ionized and non-ionized acid in the solution. This is why calculating the ionization constant \(K_a\) is important for understanding weak acids; it indicates the strength of the acid's ionization.

• The smaller the \(K_a\), the weaker the acid, indicating limited proton donation.
• For the specific exercise, the ionization constant was derived as \(1 \times 10^{-11}\), suggesting a very weak acid.

Weak acids are common in everyday life, and include acetic acid (found in vinegar) and citric acid (found in citrus fruits). Understanding their properties can help in various chemical applications.

Ionization Percentage

Ionization percentage is a measure of the extent to which an acid ionizes in a solution. It reflects the strength of the acid in the solution. To calculate it, you compare the concentration of ionized acid to its initial concentration:

• \(\text{Ionization percentage} = \left(\frac{\text{Ionized Acid}}{\text{Initial Acid Concentration}}\right) \times 100\%\)

For example, in the given problem, the monoprotic acid ionizes at \(0.001\%\). This means only this small fraction of the acid forms \(H^+\) and \(A^-\) ions, which is typical for weak acids. Often, weak acids have a low ionization percentage, reflecting limited dissociation.

Equilibrium Concentration

The concept of equilibrium concentration arises when a reaction has reached a state where the concentrations of reactants and products remain constant over time. In the context of weak acids, this means calculating the concentrations of \([H^+]\), \([A^-]\), and \([HA]\) when the reaction has stabilized. It is essential for predicting how the acid behaves in the solution and for calculating the ionization constant \(K_a\).

• The equilibrium concentration of \([H^+]\) and \([A^-]\) is the concentration of ionized acid. For this case, it was found to be \(0.000001 \, \mathrm{M}\).
• The equilibrium concentration of \([HA]\) is the initial concentration minus the ionized part, hence approximately \(0.1 \, \mathrm{M}\) for a very small ionization percentage.

These concentrations are crucial in understanding the point at which the forward and reverse reactions balance each other in a chemical solution.