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 \(1.0525 \times 10^{-7} \text{ M}\), which is option (d).

Step 1: Understanding Ionization in Water

Pure water has a self-ionization equilibrium expressed as \([H^+][OH^-] = 10^{-14}\). At 298 K, this means both \([H^+]\) and \([OH^-]\) are \(10^{-7}\) M in pure water.

Step 2: Combining Ion Concentrations

Since \(HCl\) dissociates completely in water, we add the concentration of hydrogen ions from \(HCl\) to that from water. Thus, the total \([H^+]\) is \(10^{-8} + 10^{-7}\) M.

Step 3: Calculating Total Hydrogen Ion Concentration

Compute the total hydrogen ion concentration: \[ [H^+] = 10^{-8} + 10^{-7} = 1 imes 10^{-7} + 1 imes 10^{-8} = 1.0525 imes 10^{-7} \text{ M} \]

Step 4: Conclusion

Comparing the total hydrogen ion concentration to the options given: The correct answer is \(1.0525 \times 10^{-7} \text{ M}\), which matches option (d).

Key Concepts

Self-ionization of water

Water molecules can naturally interact with one another in a fascinating process known as self-ionization. In pure water, two water molecules can react to form a hydronium ion \([H_3O^+]\) and a hydroxide ion \([OH^-]\). This process is essential as it defines the neutral point of the pH scale. The equilibrium constant for this self-ionization process is denoted as \(K_w\) and is given by the product \[ [H^+][OH^-] = 10^{-14} \] at 298 K. This means that in pure water itself, the concentrations of hydrogen ions \[ (H^+) \] and hydroxide ions \[ (OH^-) \] are both \[ 10^{-7} \] M. Understanding this equilibrium helps us appreciate how even in seemingly pure water, there is a delicate balance of ions contributing to its chemical properties.

Ionization equilibrium

Ionization equilibrium refers to the state where the rate of ionizing and recombining in a solution is equal. In aqueous solutions, strong acids, like HCl, dissociate completely in water. This means every molecule of HCl splits into its constituent ions: a hydrogen ion \(H^+\) and a chloride ion \(Cl^-\). However, in the context of very dilute solutions, the additional \[H^+\] from water's self-ionization cannot be ignored. Hence, we consider the original hydrogen ion contribution from the acid, and the self-ionization of water. The combined concentration establishes the ionization equilibrium in the solution, significantly impacting its pH. Recognizing how different sources contribute to the total \[H^+\] concentration is crucial for accurate calculations in chemistry.

Aqueous solutions chemistry

The behavior of chemicals in water, or aqueous solutions, is a cornerstone of chemistry. Water is often considered the universal solvent due to its remarkable ability to dissolve various substances; its polar nature allows it to interact with both negatively and positively charged particles. In the case of \(HCl\), when dissolved in water, the strong acid dissociates entirely, adding to the existing hydrogen ion pools from water's self-ionization. This dual source of ions modifies the overall acidity of the solution, adjusting its pH and, thus, its chemical properties. When calculating concentrations in such solutions, it is crucial to consider all contributing factors, including the solute added and the solvent's inherent reaction properties. Consequently, understanding aqueous solutions requires taking into account both the compound involved and water’s unique characteristics.