Question

Calculate \(\left[\mathrm{OH}^{-}\right]\) in each of the following solutions, and indicate whether the solution is acidic, basic, or neutral. a. \(\left[\mathrm{H}^{+}\right]=4.21 \times 10^{-7} M\) b. \(\left[\mathrm{H}^{+}\right]=0.00035 M\) c. \(\left[\mathrm{H}^{+}\right]=0.00000010 \mathrm{M}\) d. \(\left[\mathrm{H}^{+}\right]=9.9 \times 10^{-6} M\)

Short answer

a. \(\left[\mathrm{OH}^{-}\right] = 2.37 \times 10^{-8}\,M\), acidic b. \(\left[\mathrm{OH}^{-}\right] = 2.86 \times 10^{-11}\,M\), acidic c. \(\left[\mathrm{OH}^{-}\right] = 1.0 \times 10^{-8}\,M\), basic d. \(\left[\mathrm{OH}^{-}\right] = 1.01 \times 10^{-9}\,M\), acidic

Step-by-step solution

Recall the ion product constant for water

The ion product constant for water, \(K_w\), is always a constant value at a given temperature. At \(25\,^\circ\mathrm{C}\), \(K_w\) is equal to \(1.0 \times 10^{-14}\).

Calculate \(\left[\mathrm{OH}^{-}\right]\) for each solution

Given the value of \(K_w\) and the \(\left[\mathrm{H}^{+}\right]\) for each solution, we can calculate \(\left[\mathrm{OH}^{-}\right]\) using the formula \(K_w = \left[\mathrm{H}^{+}\right] \times \left[\mathrm{OH}^{-}\right]\). Solve for \(\left[\mathrm{OH}^{-}\right]\) by dividing \(K_w\) by \(\left[\mathrm{H}^{+}\right]\) for each solution: a. \(\left[\mathrm{OH}^{-}\right] = \frac{K_w}{[\mathrm{H}^{+}]} = \frac{1.0 \times 10^{-14}}{4.21 \times 10^{-7} M} = 2.37 \times 10^{-8}\,M\) b. \(\left[\mathrm{OH}^{-}\right] = \frac{K_w}{[\mathrm{H}^{+}]} = \frac{1.0 \times 10^{-14}}{0.00035\,M} = 2.86 \times 10^{-11}\,M\) c. \(\left[\mathrm{OH}^{-}\right] = \frac{K_w}{[\mathrm{H}^{+}]} = \frac{1.0 \times 10^{-14}}{0.00000010\,M} = 1.0 \times 10^{-8}\,M\) d. \(\left[\mathrm{OH}^{-}\right] = \frac{K_w}{[\mathrm{H}^{+}]} = \frac{1.0 \times 10^{-14}}{9.9 \times 10^{-6} M} = 1.01 \times 10^{-9}\,M\)

Determine whether the solutions are acidic, basic, or neutral

Compare the concentrations of \(\mathrm{H}^{+}\) ions and \(\mathrm{OH}^{-}\) ions for each solution: a. \(\left[\mathrm{H}^{+}\right] = 4.21 \times 10^{-7} M\) and \(\left[\mathrm{OH}^{-}\right] = 2.37 \times 10^{-8} M\). Since \(\left[\mathrm{H}^{+}\right] > \left[\mathrm{OH}^{-}\right]\), the solution is acidic. b. \(\left[\mathrm{H}^{+}\right] = 0.00035 M\) and \(\left[\mathrm{OH}^{-}\right] = 2.86 \times 10^{-11} M\). Since \(\left[\mathrm{H}^{+}\right] > \left[\mathrm{OH}^{-}\right]\), the solution is acidic. c. \(\left[\mathrm{H}^{+}\right] = 0.00000010 M\) and \(\left[\mathrm{OH}^{-}\right] = 1.0 \times 10^{-8} M\). Since \(\left[\mathrm{H}^{+}\right] < \left[\mathrm{OH}^{-}\right]\), the solution is basic. d. \(\left[\mathrm{H}^{+}\right] = 9.9 \times 10^{-6} M\) and \(\left[\mathrm{OH}^{-}\right] = 1.01 \times 10^{-9} M\). Since \(\left[\mathrm{H}^{+}\right] > \left[\mathrm{OH}^{-}\right]\), the solution is acidic. Overall, we have solutions a, b, and d being acidic, while solution c is basic.

Key concepts

Ion Product Constant

The ion product constant for water, often denoted as \( K_w \), is a key concept in understanding acid-base equilibria. Essentially, \( K_w \) expresses the relationship between the concentrations of hydronium ions \( [\text{H}^+] \) and hydroxide ions \( [\text{OH}^-] \) in water at equilibrium. At 25°C, the value of \( K_w \) is quite stable and is equal to \( 1.0 \times 10^{-14} \). This small number reflects the fact that water slightly ionizes into its component ions under normal conditions.

The formula that represents this relationship is:

• \( K_w = [\text{H}^+] \times [\text{OH}^-] \)

This equation shows that the product of the concentrations of hydronium and hydroxide ions in water is always constant at a given temperature. Therefore, if you know the concentration of one type of ion, you can easily calculate the concentration of the other. Ultimately, this equation is fundamental for determining whether a solution is neutral, acidic, or basic.

Hydronium Ion Concentration

Understanding the hydronium ion concentration, \( [\text{H}^+] \), is important for classifying the acidity of a solution. In water and aqueous solutions, hydronium ions form when hydrogen ions (H⁺) combine with water molecules.

The concentration of these ions provides us with a direct indication of how acidic or basic a solution is. More hydronium ions generally mean a more acidic solution. Conversely, fewer hydronium ions and more hydroxide ions will indicate a basic solution. The hydronium ion concentration is typically expressed in molarity (M), which is the number of moles of hydronium ions per liter of solution.

In our exercise, different concentrations of \( [\text{H}^+] \) were provided for various solutions, and these allowed us to subsequently calculate the \( [\text{OH}^-] \) using the ion product constant for water.

pH and pOH

The concepts of pH and pOH offer a convenient way to express the acidity and basicity of a solution. The pH, in particular, is a measure of the hydronium ion concentration and is calculated using the formula:

• \( \text{pH} = -\log_{10} [\text{H}^+] \)

The pH scale ranges from 0 to 14, where a pH of 7 is neutral, below 7 is acidic, and above 7 is basic.

On the other hand, pOH relates to hydroxide ions and is calculated in a similar manner:

• \( \text{pOH} = -\log_{10} [\text{OH}^-] \)

The relationship between pH and pOH is also connected to the ion product constant, which helps us easily convert between the two:

• \( \text{pH} + \text{pOH} = 14 \)

Ultimately, understanding pH and pOH further helps us analyze a solution's characteristics and predict its behavior in various chemical reactions.

Water Ionization

Water ionization is the process by which water molecules dissociate into ions: hydronium \( [\text{H}^+] \) and hydroxide \( [\text{OH}^-] \). This occurs because water is amphoteric, meaning it can act as both an acid and a base.

In pure water, the concentrations of \( [\text{H}^+] \) and \( [\text{OH}^-] \) are equal, each being approximately \( 1.0 \times 10^{-7} \) M at 25°C, resulting in a neutral pH of 7. This balance changes when other acids or bases are added. For example, when an acid is added, the \( [\text{H}^+] \) increases while the \( [\text{OH}^-] \) decreases, making the solution more acidic. The reverse is true for adding a base.

Water's ionization is essential for the self-ionization process that is fundamental to the dynamics of aqueous solutions, highlighting its essential role in maintaining the equilibrium necessary for life and various industrial processes.