Question

Given the molar concentration of hydroxide ion, calculate the concentration of hydrogen ion: (a) \(\left[\mathrm{OH}^{-}\right]=8.8 \times 10^{-8}\) (b) \(\left[\mathrm{OH}^{-}\right]=4.6 \times 10^{-13}\)

Short answer

(a) \( [H^+] = 1.14 \times 10^{-7} \text{ M} \), (b) \( [H^+] = 2.17 \times 10^{-2} \text{ M} \).

Step-by-step solution

Understand the Concept

To find the concentration of hydrogen ions \([H^+]\), we use the relationship between hydroxide ions \([OH^-]\) and hydrogen ions in water. This relationship is given by the water dissociation constant \(K_w\), which is \(1.0 \, \times \, 10^{-14}\) at 25°C. We know \(K_w = [H^+][OH^-]\).

Solve for [H⁺] in Part (a)

Rearrange the formula \(K_w = [H^+][OH^-]\) to solve for \([H^+]\) in part (a):\[[H^+] = \frac{K_w}{[OH^-]}\]\[ = \frac{1.0 \times 10^{-14}}{8.8 \times 10^{-8}}\]\[ = 1.14 \times 10^{-7} \text{ M}.\]

Solve for [H⁺] in Part (b)

Similarly, for part (b), use the formula: \( [H^+] = \frac{K_w}{[OH^-]} \) :\[[H^+] = \frac{1.0 \times 10^{-14}}{4.6 \times 10^{-13}}\]\[ = 2.17 \times 10^{-2} \text{ M}.\]

Conclusion

For both concentrations of hydroxide ions provided, the concentrations of hydrogen ions can be calculated using the water dissociation constant. The results indicate how acidic or basic the solutions are.

Key concepts

Hydrogen Ion Concentration

In acid-base chemistry, hydrogen ion concentration \([H^+]\) is a crucial measure. It determines the acidity of a solution. The concentration of these ions directly affects the pH level, which indicates how acidic or basic a solution is. A higher concentration of hydrogen ions means the solution is more acidic, while a lower concentration means it is more basic.
The relationship between hydrogen ions and pH is given by the formula: \[ pH = -\log_{10}[H^+] \]

Understanding how to find the concentration of hydrogen ions is essential. This is especially important when given the concentration of hydroxide ions, as it allows us to better understand the solution's nature. In pure water, hydrogen ions and hydroxide ions exist in a balance. When one increases, the other decreases to maintain this equilibrium.

Water Dissociation Constant

The water dissociation constant, denoted as \({K_w}\), is a fundamental concept in understanding the behavior of water as a solvent. It represents the product of the concentrations of hydrogen ions \([H^+]\) and hydroxide ions \([OH^-]\) in water. At 25°C, \({K_w}\) is equal to \(1.0 \times 10^{-14}\). This constant is crucial because it forms the basis for calculating ion concentrations in water solutions.

\[{K_w} = [H^+][OH^-]\]

Since the value of \({K_w}\) is constant at a given temperature, if we know one ion concentration, we can always find the other. This is possible by rearranging the formula to find \([H^+]\) if \([OH^-]\) is known, or vice versa. Understanding how these concentrations relate helps us predict how substances will behave when dissolved in water.

Hydroxide Ion Concentration

The concentration of hydroxide ions \([OH^-]\) is a key aspect of determining the basicity of a solution. Hydroxide ions are associated with basic, or alkaline, solutions. A higher concentration of \([OH^-]\) leads to a more basic environment, whereas a lower concentration indicates the solution is less basic or more acidic.

It is important to measure \([OH^-]\) concentration accurately because it directly impacts the calculation of \([H^+]\) via the water dissociation constant \({K_w}\).
Using the formula:

• \([H^+] = \frac{K_w}{[OH^-]} \)

This allows us to determine the hydrogen ion concentration and, in essence, the pH of the solution. Calculating these values helps us understand the chemical nature of the solution, whether you're dealing with household items, in a laboratory, or studying natural bodies of water.