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 approximately \(1.0525 \times 10^{-7} M\), which corresponds to option (d).

Step 1: Understand the problem

We have a hydrochloric acid (HCl) solution with a concentration of \(10^{-8} M\). Since HCl is a strong acid, it dissociates completely in water to produce hydronium ions \((H^+)\) and chloride ions \((Cl^-)\). The solution provided is at 298 K, and the ion-product constant of water \((Kw)\) is \(10^{-14}\). Our task is to find the total hydrogen ion concentration in the solution.

Step 2: Consider contributions to hydrogen ion concentration

In pure water, since \(Kw = 10^{-14}\), the concentration of \(H^+\) due to water autodissociation is \([H^+] = [OH^-] = 10^{-7} M\). Therefore, the total \([H^+]\) will be the sum of \([H^+]\) from the water and the \([H^+]\) from the HCl dissociation.

Step 3: Write the expression for total hydrogen ion concentration

The total hydrogen ion concentration \([H^+]_\text{total}\) is the sum of hydrogen ions from HCl and from water. This is: \([H^+]_\text{total} = [H^+]_{HCl} + [H^+]_{water}\).

Step 4: Calculate total hydrogen ion concentration

Substitute the known values: \([H^+]_{HCl} = 10^{-8} M\) and \([H^+]_{water} = 10^{-7} M\). Adding these gives: \([H^+]_\text{total} = 10^{-8} + 10^{-7} = 1.1 \times 10^{-7} M\).

Step 5: Compare with given options

Option (a) is \(9.525 \times 10^{-8} M\), (b) is \(1.0 \times 10^{-8} M\), (c) is \(1.0 \times 10^{-6} M\), and (d) is \(1.0525 \times 10^{-7} M\). Comparing \(1.1 \times 10^{-7} M\) with these options, the closest matching value is option (d), \(1.0525 \times 10^{-7} M\).

Key Concepts

Strong Acid

In chemistry, strong acids are substances that completely dissociate into their ions in water. Hydrochloric acid (HCl) is one such example. When dissolved in water, a strong acid like HCl will break into its component ions: hydrogen ions

• (H⁺)
• chloride ions (Cl^-)

Because strong acids dissociate entirely, they make it straightforward to calculate the concentration of hydrogen ions just from knowing the initial concentration of the acid. When you add HCl with a concentration of \(10^{-8} M\) to water, it fully dissociates, contributing to the hydrogenion concentration. This quality of full ionization is crucial here because it helps us directly equate the initial molarity of the acid to the concentration of hydrogen ions contributed by the acid. Yet, at very low concentrations like \(10^{-8} M\), the influence of water's natural ionization on hydrogen ion concentration becomes significant, as we'll see in the next sections.

Water Autodissociation

Water is unique in its ability to self-ionize, even if it's a very weak process. This process is known as autodissociation, where water molecules

• spontaneously form ions:
  • hydrogen ions (H⁺)
  • hydroxide ions (OH^-)

Autodissociation is important when considering very dilute solutions like \(10^{-8} M\) HCl. Even with no added acid or base, pure water itself has a hydrogen ion concentration of \(10^{-7} M\). So, while the dissociation of strong acids like HCl contributes to hydrogen ion concentration, the hydrogen ions from water itself are already present and cannot be ignored when the acid's concentration is very low. This is because the water's naturally occurring hydrogen ions can overshadow the weakly effective concentration of the acid, thus affecting the final measurable concentration of hydrogen ions in the solution.

Ion-Product Constant of Water

The ion-product constant of water, known as \(K_w\), is a fundamental concept that quantifies the degree of water's autodissociation at a given temperature. At standard temperature and pressure (298 K), \(K_w\) has a value of \(10^{-14}\). This constant expresses the product of the molar concentrations of hydrogen ions (H⁺) and hydroxide ions (OH^-) in water:\[K_w = [H^+][OH^-]\]In pure water at 298 K, both \([H^+]\) and \([OH^-]\) are equal, resulting in \([H^+] = 10^{-7} M\) and \([OH^-] = 10^{-7} M\). The value of \(K_w\) assists in understanding the behavior of acids and bases in aqueous solutions, crucial particularly in explaining why pure water contributes significantly to the hydrogen ion concentration in solutions like the \(10^{-8} M\) HCl. It provides a backdrop to assess how much of the ionic concentration can be attributed to added substances versus innate water dissociation. Therefore, when calculating total hydrogen ion concentration, both the contributions from a strong acid and autodissociation must be summed up, placing the concept of \(K_w\) at the core of these evaluations.