Question

Identify the following solutions as either acidic or basic. What is the value of \(\left[\mathrm{OH}^{-}\right]\) in each? (a) Household ammonia, \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=3.1 \times 10^{-12} M\) (b) Vinegar, \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=4.0 \times 10^{-3} \mathrm{M}\)

Short answer

Household ammonia is basic, with \\(\left[\mathrm{OH}^{-}\right] = 3.23 \times 10^{-3} M\\). Vinegar is acidic, with \\(\left[\mathrm{OH}^{-}\right] = 2.5 \times 10^{-12} M\\).

Step-by-step solution

Understand the Relationship

To determine if a solution is acidic or basic, we need to compare the concentrations of \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) and \(\left[\mathrm{OH}^{-}\right]\). A solution is acidic when the concentration of \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) is greater than \(\left[\mathrm{OH}^{-}\right]\), and basic when \(\left[\mathrm{OH}^{-}\right]\) is greater. We will use the ion product of water, \(K_w = 1.0 \times 10^{-14} M^2\), which relates these concentrations: \[\left[\mathrm{H}_{3} \mathrm{O}^{+}\right] \times \left[\mathrm{OH}^{-}\right] = K_w\] to find \(\left[\mathrm{OH}^{-}\right]\).

Calculate \\(\left[\mathrm{OH}^{-}\right]\\) for Household Ammonia

Given \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right] = 3.1 \times 10^{-12} M\) for household ammonia, use the ion product of water to find \(\left[\mathrm{OH}^{-}\right]\): \[\left[\mathrm{OH}^{-}\right] = \frac{K_w}{\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]} = \frac{1.0 \times 10^{-14}}{3.1 \times 10^{-12}}\] Calculating this gives \(\left[\mathrm{OH}^{-}\right] = 3.23 \times 10^{-3} M\). Since \(\left[\mathrm{OH}^{-}\right] > \left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\), the solution is basic.

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

Given \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right] = 4.0 \times 10^{-3} M\) for vinegar, use the ion product of water to find \(\left[\mathrm{OH}^{-}\right]\): \[\left[\mathrm{OH}^{-}\right] = \frac{K_w}{\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]} = \frac{1.0 \times 10^{-14}}{4.0 \times 10^{-3}}\] Calculating this gives \(\left[\mathrm{OH}^{-}\right] = 2.5 \times 10^{-12} M\). Since \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right] > \left[\mathrm{OH}^{-}\right]\), the solution is acidic.

Key concepts

Ion Product of Water

In acid-base chemistry, an essential concept is the ion product of water, denoted as \(K_w\), which is fundamental for understanding the acidity or basicity of a solution. The ion product of water is a constant at room temperature and is expressed as \(K_w = 1.0 \times 10^{-14} M^2\). This constant relates the concentrations of hydrogen ions \([\mathrm{H}_3\mathrm{O}^+]\) and hydroxide ions \([\mathrm{OH}^-]\) in a solution.
Understanding how these concentrations interact is crucial:

• In neutral water at 25°C, both ions are present at concentrations of \(1.0 \times 10^{-7} M\) because \(\left[\mathrm{H}_3\mathrm{O}^+\right] \times \left[\mathrm{OH}^-\right] = K_w\).
• If the concentration of \([\mathrm{H}_3\mathrm{O}^+]\) increases, the concentration of \([\mathrm{OH}^-]\) must decrease to maintain \(K_w\) as a constant, and vice versa.
• This principle allows the identification of a solution as acidic, basic, or neutral based on the concentrations of these ions.

Concentration Calculation

Calculating concentrations of ions is a pivotal step in identifying if a solution is acidic or basic. Given the ion product relationship, if you know the concentration of one ion, the other can be calculated using the equation:\[\left[\mathrm{OH}^-\right] = \frac{K_w}{\left[\mathrm{H}_3\mathrm{O}^+\right]}\]Let's look at two examples:

• For **household ammonia**: - Given \([\mathrm{H}_3\mathrm{O}^+] = 3.1 \times 10^{-12} M\). - Using the formula, \([\mathrm{OH}^-] = \frac{1.0 \times 10^{-14}}{3.1 \times 10^{-12}} = 3.23 \times 10^{-3} M\).
• For **vinegar**: - Given \([\mathrm{H}_3\mathrm{O}^+] = 4.0 \times 10^{-3} M\). - The calculation gives \([\mathrm{OH}^-] = \frac{1.0 \times 10^{-14}}{4.0 \times 10^{-3}} = 2.5 \times 10^{-12} M\).

These calculations illustrate the method to derive \([\mathrm{OH}^-]\) from a known \([\mathrm{H}_3\mathrm{O}^+]\) or vice versa, allowing us to compare concentrations and determine the solution's nature.

pH and pOH

Understanding the concepts of pH and pOH is crucial to comprehend the acidity or basicity of solutions. These are logarithmic scales used to specify the acidity (pH) and basicity (pOH) of a solution.
- **pH** is defined as the negative logarithm of the \([\mathrm{H}_3\mathrm{O}^+]\) concentration: \[ \text{pH} = -\log_{10}\left(\left[\mathrm{H}_3\mathrm{O}^+\right]\right) \]- **pOH** is similarly defined for \([\mathrm{OH}^-]\): \[ \text{pOH} = -\log_{10}\left(\left[\mathrm{OH}^-\right]\right) \]The relationship between pH and pOH is elegantly linked via:

• At 25°C, \(\text{pH} + \text{pOH} = 14\).
• Low pH values indicate acidic solutions, while high pH values indicate basic solutions.
• pOH works similarly, where low values indicate basic solutions, and high values indicate acidity.

By calculating either the pH or pOH, you can determine the acidity or basicity of any solution, which is pivotal for various chemical and biological applications.