Problem 77 - A Complete Resource Book in Chemistry for JEE Main

Question

The hydrogen ion concentration of a \(10^{-8} \mathrm{M} \mathrm{HCl}\) aqueous solution at \(298 \mathrm{~K}\left(\mathrm{Kw}=10^{-14}\right)\) is (a) \(9.525 \times 10^{-8} \mathrm{M}\) (b) \(1.0 \times 10^{-8} \mathrm{M}\) (c) \(1.0 \times 10^{-6} \mathrm{M}\) (d) \(1.0525 \times 10^{-7} \mathrm{M}\)

Step-by-Step Solution

Answer

The hydrogen ion concentration is (d) 1.0525 x 10^{-7} M.

Step 1: Understanding Ion Product of Water

Water itself can dissociate slightly into hydrogen ions \( \text{H}^+ \) and hydroxide ions \( \text{OH}^- \), and the ion product constant of water at 298 K is \( \text{Kw} = [\text{H}^+][\text{OH}^-] = 10^{-14} \). In pure water or neutral solutions, \( [\text{H}^+] = [\text{OH}^-] \), so \( [\text{H}^+] = \sqrt{10^{-14}} = 10^{-7} \). Thus, the concentration of hydrogen ions due to water alone is \( 10^{-7} \) M.

Step 2: Calculate Total Hydrogen Ion Concentration

In a solution of \( 10^{-8} \) M \( \text{HCl} \), which is a strong acid, it dissociates completely, contributing \( 10^{-8} \) M of hydrogen ions. Therefore, the total hydrogen ion concentration \( [\text{H}^+]_{\text{total}} \) is the sum of the hydrogen ions from both the acid and water: \[ [\text{H}^+]_{\text{total}} = [\text{H}^+]_{\text{HCl}} + [\text{H}^+]_{\text{water}} = 10^{-8} \text{ M} + 10^{-7} \text{ M}. \]

Step 3: Simplify the Total Concentration

Now we simplify the expression from Step 2. Since \(10^{-7} \) M is significantly greater than \(10^{-8} \) M, we can approximate the total hydrogen ion concentration as dominated by the contribution from water: \[ [\text{H}^+]_{\text{total}} \approx 10^{-7} + 10^{-8} = 1.0525 \times 10^{-7} \text{ M}. \]

Step 4: Determine the Correct Answer

With the calculated value of \[ [\text{H}^+] \approx 1.0525 \times 10^{-7} \text{ M}, \] the correct option corresponding to this is (d) \(1.0525 \times 10^{-7} \text{ M} \), which incorporates both contributions effectively.

Key Concepts

Ion Product Constant

The ion product constant of water, denoted as \( K_w \), is a fundamental equilibrium constant that reflects how water can slightly dissociate into ions. At 298 K, or standard room temperature, the expression for this dissociation is given by:

• \( K_w = [\text{H}^+][\text{OH}^-] = 10^{-14} \)

In pure water, the concentrations of hydrogen ions \( [\text{H}^+] \) and hydroxide ions \( [\text{OH}^-] \) are equal, leading to:

• \( [\text{H}^+] = [\text{OH}^-] = \sqrt{10^{-14}} = 10^{-7} \text{ M} \)

This relationship means that even without any added acids or bases, water contains a small concentration of these ions. Understanding this balance is crucial for grasping concepts like pH and the neutrality of pure water. It provides the basis for calculating the effects of added substances, such as strong acids.

Dissociation of Water

The dissociation of water is a natural process, though it occurs to a very small extent. Water molecules split into hydrogen ions \( \text{H}^+ \) and hydroxide ions \( \text{OH}^- \). This is why even pure water is not completely devoid of ions. The presence of these ions allows us to calculate how other substances will affect the water's pH.

When an acid or base is added to water, it disrupts the natural balance of \( \text{H}^+ \) and \( \text{OH}^- \) ions. However, the product of their concentrations always equals \( K_w = 10^{-14} \) at room temperature. This property is key for analyzing solutions' acidity or basicity. By knowing one of the ion concentrations, you can always determine the other, maintaining the

• equation \( [\text{H}^+][\text{OH}^-] = 10^{-14} \).

Thus, for weak or strong acids alike, their addition to water affects the ion balance, which we can predict and calculate using \( K_w \).

Strong Acids

Strong acids are distinguished by their ability to completely dissociate into ions in aqueous solutions. In other words, nearly every molecule of a strong acid will release hydrogen ions \( \text{H}^+ \). This complete dissociation leads to a relatively higher concentration of hydrogen ions compared to weak acids.

The example in the exercise features hydrochloric acid \( \text{HCl} \), a prototypical strong acid. When \( 10^{-8} \) M of \( \text{HCl} \) is dissolved in water, it fully dissociates, contributing \( 10^{-8} \) M of \( \text{H}^+ \) ions.

• However, compared to the natural \( 10^{-7} \) M concentration of hydrogen ions from dissociation of water, this contribution appears minimal.
• This illustrates that in such dilute solutions, the water's own dissociation can dominate the overall hydrogen ion concentration.

In summary, understanding the nature of strong acids and their behavior in water empowers us to predict how they alter the solution's characteristics, like its pH.