Problem 23 - Chemistry

Question

Calculate the concentration of \(\mathrm{OH}^{-}\) ions in a \(1.4 \times 10^{-3} \mathrm{M}\) \(\mathrm{HCl}\) solution.

Step-by-Step Solution

Answer

The concentration of \(\mathrm{OH}^{-}\) ions is approximately \(7.1 \times 10^{-12} \mathrm{M}\).

Step 1: Understand the Problem

We are given a hydrochloric acid (HCl) solution with a concentration of \(1.4 \times 10^{-3} \mathrm{M}\). \(\mathrm{HCl}\) is a strong acid, which means it completely dissociates in water to produce \(\mathrm{H}^{+}\) ions. We need to find the concentration of hydroxide ions \(\mathrm{OH}^{-}\) in this solution.

Step 2: Calculate \(\mathrm{H}^{+}\) Concentration

Since \(\mathrm{HCl}\) is a strong acid, the concentration of \(\mathrm{H}^{+}\) ions is equal to the concentration of \(\mathrm{HCl}\). Therefore, \([\mathrm{H}^{+}] = 1.4 \times 10^{-3} \mathrm{M}\).

Step 3: Use Ion Product of Water

Water autoionizes into \(\mathrm{H}^{+}\) and \(\mathrm{OH}^{-}\) ions and has an ion product \(K_w = 1.0 \times 10^{-14}\) at 25°C. The relation is \([\mathrm{H}^{+}][\mathrm{OH}^{-}] = K_w\). We can use this to calculate \([\mathrm{OH}^{-}]\).

Step 4: Calculate \(\mathrm{OH}^{-}\) Concentration

Substitute \([\mathrm{H}^{+}] = 1.4 \times 10^{-3} \mathrm{M}\) into the equation \([\mathrm{H}^{+}][\mathrm{OH}^{-}] = 1.0 \times 10^{-14}\) to find \([\mathrm{OH}^{-}]\): \([\mathrm{OH}^{-}] = \frac{1.0 \times 10^{-14}}{1.4 \times 10^{-3}}\).

Step 5: Perform the Division

Calculate \([\mathrm{OH}^{-}] = \frac{1.0 \times 10^{-14}}{1.4 \times 10^{-3}}\): \([\mathrm{OH}^{-}] \approx 7.1 \times 10^{-12} \mathrm{M}\).

Key Concepts

Strong Acid

Hydrochloric acid (HCl) is known as a strong acid. This means it completely dissociates into its ions when dissolved in water.
For HCl:

• The ionization produces hydrogen ions ( H^+ ) and chloride ions ( Cl^- ).
• This complete dissociation implies that the concentration of H^+ ions equals the initial concentration of the HCl solution.

In the context of our problem, a 1.4 x 10^{-3} M HCl solution will also have a 1.4 x 10^{-3} M concentration of H^+ ions. This characteristic of strong acids is crucial as it simplifies the calculation of other ion concentrations in the solution.

Hydroxide Ion Concentration

Hydroxide ions ( OH^- ) play a key role in determining the basic nature of a solution. But how do we find their concentration in an acidic solution like HCl?
We use the relationship between H^+ and OH^- ions.

• In water, the product of the concentrations of H^+ and OH^- ions is constant at a given temperature.
• This constant is known as the ion product of water (K_w temporarily not mentioned).

From the problem, the concentration of H^+ ions is 1.4 x 10^{-3} M. Using the ion product of water, we can calculate the hydroxide ion concentration.

Ion Product of Water

The ion product of water (K_w) is a fundamental concept in acid-base chemistry. At 25°C, K_w equals 1.0 x 10^{-14}. This comes from the natural autoionization of water:

• Water dissociates minimally into H^+ and OH^- ions.
• The relation is represented by [H^+][OH^-] = K_w.

In our scenario, knowing [H^+] from the given HCl concentration, we can calculate [OH^-] using:\[[OH^-] = \frac{K_w}{[H^+]}\]Substituting the values, we find the concentration of hydroxide ions in the solution, ensuring our understanding of equilibrium remains clear.