Problem 100 - Chemistry Matter and Change

Question

What are the concentrations of \(\mathrm{OH}^{-}\) ions in solutions having \(\mathrm{pH}\) values of \(3.00,6.00,9.00,\) and 12.00 at 298 \(\mathrm{K}\) ? What are the pOH values for the solutions?

Step-by-Step Solution

Answer

For pH values of 3.00, 6.00, 9.00, and 12.00, [OH^-] are approximately \(10^{-11}, 10^{-8}, 10^{-5},\) and \(10^{-2} \) M, respectively. Corresponding pOH values are 11.00, 8.00, 5.00, and 2.00.

Step 1: Understanding pH and pOH

The pH of a solution is a measure of the hydrogen ion concentration, defined as \( \text{pH} = -\log[\text{H}^+] \). The pOH is similarly defined for hydroxide ions: \( \text{pOH} = -\log[\text{OH}^-] \). At 298 K, the relationship between pH and pOH is given by \( \text{pH} + \text{pOH} = 14 \).

Step 2: Calculating pOH from pH

Using the formula \( \text{pH} + \text{pOH} = 14 \), calculate the pOH for each given pH value: - For \( \text{pH} = 3.00 \), \( \text{pOH} = 14 - 3.00 = 11.00 \).- For \( \text{pH} = 6.00 \), \( \text{pOH} = 14 - 6.00 = 8.00 \).- For \( \text{pH} = 9.00 \), \( \text{pOH} = 14 - 9.00 = 5.00 \).- For \( \text{pH} = 12.00 \), \( \text{pOH} = 14 - 12.00 = 2.00 \).

Step 3: Calculating Hydroxide Ion Concentration

Use the formula \( \text{pOH} = -\log[\text{OH}^-] \) to find the concentration of \( \text{OH}^- \) ions. Rearrange to determine \( [\text{OH}^-] = 10^{-\text{pOH}} \).- For \( \text{pOH} = 11.00 \), \([\text{OH}^-] = 10^{-11} \approx 1.00 \times 10^{-11} \text{ M} \).- For \( \text{pOH} = 8.00 \), \([\text{OH}^-] = 10^{-8} \approx 1.00 \times 10^{-8} \text{ M} \).- For \( \text{pOH} = 5.00 \), \([\text{OH}^-] = 10^{-5} \approx 1.00 \times 10^{-5} \text{ M} \).- For \( \text{pOH} = 2.00 \), \([\text{OH}^-] = 10^{-2} \approx 1.00 \times 10^{-2} \text{ M} \).

Key Concepts

Hydroxide Ion Concentration

The concentration of hydroxide ions \(([\text{OH}^-])\) in a solution is an important factor in determining the solution's basicity. When calculating hydroxide ion concentration, we frequently use the pOH formula: \[ \text{pOH} = -\log[\text{OH}^-] \] Knowing the pOH, we can rearrange the formula to find \[ [\text{OH}^-] = 10^{-\text{pOH}} \]
This equation tells us the concentration of hydroxide ions. Let's illustrate this with an example for clarity:
- If the pOH is calculated to be 11.00, substituting into the formula yields \[ [\text{OH}^-] = 10^{-11} = 1.00 \times 10^{-11} \] - If the pOH is 2.00, the concentration becomes \[ [\text{OH}^-] = 10^{-2} = 0.01 \]

• Higher concentrations of \([\text{OH}^-]\) correspond to a more basic solution.
• As the pOH increases, indicating a lower hydroxide concentration, the solution becomes less basic.

Understanding hydroxide ion concentration helps to unravel other concepts associated with basic and acidic solutions.

Relationship Between pH and pOH

pH and pOH are interconnected properties that describe the acidity and basicity of a solution. Their relationship is governed by the expression:
\[ \text{pH} + \text{pOH} = 14 \]
This formula is valid at 25°C (or 298 K) and highlights how, as one value increases, the other decreases. For a neutral solution at this temperature, both the pH and pOH are exactly 7.
Let's explore this relationship further:

• For a solution with a pH of 3.00, the pOH is 11.00, hinting at its acidic nature with many hydrogen ions present.
• Conversely, a solution with a pH of 12.00 will have a pOH of 2.00, indicating high basicity due to a larger \([\text{OH}^-]\) presence.

Recognizing this relationship aids in seamlessly switching between analyzing a solution’s acidity or basicity. This understanding makes it easy to calculate one value when the other is known, thus offering insights into the nature of the solution.

pH Values

pH values signify how acidic or basic a solution is. Calculating pH involves the concentration of hydrogen ions in the solution using the formula
\[ \text{pH} = -\log[\text{H}^+] \]
A pH less than 7 signifies an acidic solution, while a pH greater than 7 signifies a basic one. A pH equal to 7 indicates a neutral solution, such as pure water at 298 K.
Here's a simple breakdown:

• A pH of 3.00 suggests a very acidic environment with high \([\text{H}^+]\) concentration, typical for substances like vinegar.
• Meanwhile, a pH of 12.00 indicates a strong base with a very low hydrogen ion presence, common in substances like bleach.

Understanding pH values gives us a quick glance into how a solution can behave chemically. Moreover, it's instrumental in determining \([\text{OH}^-]\) when combined with the understanding of pOH calculations, completing the acid-base equation puzzle.