Question

Determine the pOH of each solution. (a) \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=1.2 \times 10^{-8} \mathrm{M}\) (b) \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=5.5 \times 10^{-2} \mathrm{M}\) (c) \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=3.9 \times 10^{-9} \mathrm{M}\) (d) \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=1.88 \times 10^{-13} \mathrm{M}\)

Short answer

The pOHs are: (a) 5.92, (b) 12.96, (c) 5.41, (d) 7.4.

Step-by-step solution

Understanding pOH and its relation to \(\left[\mathrm{H}_3\mathrm{O}^+\right]\)

The pOH of a solution can be calculated if the concentration of hydroxide ions \( \left[OH^-\right] \) is known using the formula \( pOH = -\log \left[OH^-\right] \). The relationship between the concentration of hydronium ions \(\left[\mathrm{H}_3\mathrm{O}^+\right]\) and hydroxide ions \(\left[OH^-\right]\) is given by the constant \( K_w = \left[\mathrm{H}_3\mathrm{O}^+\right] \times \left[OH^-\right] = 1.0 \times 10^{-14} \) at 25°C. So, first we need to find \( \left[OH^-\right] \) from \( \left[\mathrm{H}_3\mathrm{O}^+\right] \) and then calculate pOH.

Calculate the \(\left[OH^-\right]\) for each solution

For each \(\left[\mathrm{H}_3\mathrm{O}^+\right]\), calculate \(\left[OH^-\right]\) using the relationship \(\left[OH^-\right] = \frac{K_w}{\left[\mathrm{H}_3\mathrm{O}^+\right]}\).

Calculate the pOH for solution (a)

First calculate \( \left[OH^-\right] \) using \( \left[OH^-\right] = \frac{1.0 \times 10^{-14}}{1.2 \times 10^{-8}} \). Then calculate \( pOH = -\log \left[OH^-\right] \) using the found hydroxide ion concentration.

Calculate the pOH for solution (b)

Calculate \( \left[OH^-\right] \) using \( \left[OH^-\right] = \frac{1.0 \times 10^{-14}}{5.5 \times 10^{-2}} \), then compute \( pOH \) as in the previous step.

Calculate the pOH for solution (c)

Determine \( \left[OH^-\right] \) using \( \left[OH^-\right] = \frac{1.0 \times 10^{-14}}{3.9 \times 10^{-9}} \), and then find the \( pOH \).

Calculate the pOH for solution (d)

Compute \( \left[OH^-\right] \) with \( \left[OH^-\right] = \frac{1.0 \times 10^{-14}}{1.88 \times 10^{-13}} \) and calculate the pOH accordingly.

Key concepts

Hydronium Ion Concentration

Hydronium ions, denoted as \( \text{H}_3\text{O}^+ \), represent the presence of acidic properties in a solution. To understand pOH calculation, it's crucial to start with hydronium ion concentration since it is directly related to the solution's acidity. The higher the concentration of hydronium ions, the lower the pH and the stronger the acid.

In assessing the acidity of solutions, typically the concentration of hydronium ions is expressed in molarity (M), which is moles per liter. For example, a hydronium ion concentration of \( 1.2 \times 10^{-8} \text{M} \) indicates a relatively low level of acidity. In the pOH calculations, this concentration is inversely utilized to determine the hydroxide ion concentration, which is necessary for finding the pOH.

Hydroxide Ion Concentration

Conversely to hydronium ions, hydroxide ions, indicated by \( \text{OH}^- \), are a measure of a solution's basicity. The more hydroxide ions present, the more basic the solution is, and the higher its pOH will be. The concentration of hydroxide ions is also most commonly expressed in molarity.

To find the \( \text{OH}^- \) concentration from the hydronium ion concentration, one uses the water dissociation constant. This approach allows students to realize the symmetric nature of acidity and basicity in aqueous solutions, reflecting that as the concentration of hydronium ions increases, the concentration of hydroxide ions decreases, and vice versa.

Water Dissociation Constant

The water dissociation constant, \( K_w \), is a crucial concept in understanding the balance of hydronium and hydroxide ions in water. At 25°C, it holds a constant value of \( 1.0 \times 10^{-14} \). This value represents the product of the concentrations of the hydronium and hydroxide ions at equilibrium.

The constancy of \( K_w \) means that if the concentration of one type of ion is known, the other can be calculated. When we are tasked with pOH calculations, we use this constant to find the \( \text{OH}^- \) concentration from the known \( \text{H}_3\text{O}^+ \) concentration.

pH and pOH Relationship

Understanding the relationship between pH and pOH is key to mastering the concept of acidity and basicity in solutions. The pH scale measures how acidic or basic a solution is, while the pOH scale measures the same, but from the basicity perspective. They are interconnected through the expression: \( pH + pOH = 14 \) at 25°C, a reflection of the water dissociation constant.

This connection allows us to determine one value if the other is known, facilitating the study of a solution's acidic or basic nature. For example, if you calculate the pOH of a solution and want to find its pH, simply subtract the pOH from 14. Understanding this relationship is foundational for navigating through acid-base chemistry and arriving at accurate interpretations of a solution's characteristics.