Question

The concentration of \(\mathrm{H}_{3} \mathrm{O}^{+}\) ions in the runoff from a coal mine is \(1.4 \times 10^{-4} \mathrm{M}\). Calculate the concentration of \(\mathrm{OH}^{-}\) ions, and classify the solution as acidic, neutral, or basic.

Short answer

The concentration of OH- is \(7.14 \times 10^{-11}\, \text{M}\), and the solution is acidic.

Step-by-step solution

Understanding the relationship between H3O+ and OH- concentrations

The relationship between the concentrations of \([H_3O^+]\) and \([OH^-]\) in a solution at 25°C is given by the water dissociation constant: \[K_w = [H_3O^+][OH^-] = 1.0 \times 10^{-14}\] This equation shows how the concentration of hydronium ions is related to the concentration of hydroxide ions.

Substitute the known concentration of H3O+

We are given that \([H_3O^+] = 1.4 \times 10^{-4}\ \text{M}\). Substitute this value into the water dissociation equation: \[1.0 \times 10^{-14} = (1.4 \times 10^{-4})[OH^-]\] Now, we will solve for \([OH^-]\).

Solve for the concentration of OH- ions

To find \([OH^-]\), rearrange the equation: \[ [OH^-] = \frac{1.0 \times 10^{-14}}{1.4 \times 10^{-4}} \] Now perform the division: \[ [OH^-] = 7.14 \times 10^{-11} \text{M} \] This gives us the concentration of hydroxide ions.

Classify the solution as acidic, neutral, or basic

Compare the concentrations of \([H_3O^+]\) and \([OH^-]\). Since \[1.4 \times 10^{-4} \text{ M } > 7.14 \times 10^{-11} \text{ M } \] the concentration of \([H_3O^+]\) is greater than \([OH^-]\). This means the solution is acidic.

Key concepts

Water Dissociation Constant

In the world of acid-base chemistry, the water dissociation constant, represented as \(K_w\), is a key concept for understanding how water ionizes into hydrogen and hydroxide ions. At standard temperature, which is 25°C, the value of \(K_w\) is given as:\[K_w = [H_3O^+][OH^-] = 1.0 \times 10^{-14}\]This equilibrium constant is a product of the molar concentrations of hydronium ions \([H_3O^+]\) and hydroxide ions \([OH^-]\).

• This means that in pure water at 25°C, both \([H_3O^+]\) and \([OH^-]\) are equal and approximately \(1.0 \times 10^{-7}\; \text{M}\).
• The constant applies to any aqueous solution, not just pure water.

Partial ionization of water can impact the pH level of the solution, affecting whether it is acidic, neutral, or basic. This principle is fundamental for calculating ion concentrations in different solutions.

Hydronium Ion Concentration

Hydronium ion concentration, \([H_3O^+]\), is vital in determining the acidity of a solution. It reflects the amount of hydrogen ions in water, which influences the solution's pH. A high \([H_3O^+]\) implies a low pH and, therefore, an acidic solution.

• For instance, if the concentration of \([H_3O^+]\) is significantly higher than \(1.0 \times 10^{-7}\; \text{M}\), the solution is considered acidic.
• In the exercise, a given \([H_3O^+] = 1.4 \times 10^{-4}\; \text{M}\) clearly suggests that the solution is acidic since this value is much larger than the neutral concentration.

Understanding \([H_3O^+]\) helps predict and compare how acidic different solutions are, an essential tool in chemistry, biology, and environmental science.

Hydroxide Ion Concentration

Hydroxide ion concentration, \([OH^-]\), complements our understanding of a solution's basicity. This concentration tells us how many hydroxide ions are present, with a high \([OH^-]\) signifying a basic or alkaline solution.Using the water dissociation constant, one can calculate \([OH^-]\) when \([H_3O^+]\) is known, and vice versa, as shown in the exercise:\[[OH^-] = \frac{1.0 \times 10^{-14}}{[H_3O^+]}\]

• In the given problem, \([OH^-] = 7.14 \times 10^{-11}\; \text{M}\), derived from \([H_3O^+] = 1.4 \times 10^{-4}\; \text{M}\).
• Since \([OH^-]\) is much less than \([H_3O^+]\), this supports the conclusion that the solution is acidic.

Analyzing \([OH^-]\) gives insight into the solution's properties, an important factor when dealing with chemical reactions or environmental assessments.