Problem 29 - Chemistry: The Central Science

Question

Calculate \(\left[\mathrm{H}^{+}\right]\)for each of the following solutions, and indicate whether the solution is acidic, basic, or neutral: (a) \([\mathrm{OH}]=0.00045 \mathrm{M} ;\) (b) \([\mathrm{OH}]=8.8 \times 10^{-9} \mathrm{M}\); (c) a solution in which \(\left[\mathrm{OH}^{-}\right]\)is 100 times greater than \(\left[\mathrm{H}^{+}\right]\).

Step-by-Step Solution

Answer

For the given solutions: (a) \([\mathrm{H}^{+}] = 2.22 \times 10^{-11} \mathrm{M}\) and the solution is basic, (b) \([\mathrm{H}^{+}] = 1.14 \times 10^{-6} \mathrm{M}\) and the solution is acidic, (c) \([\mathrm{H}^{+}] = 1 \times 10^{-8} \mathrm{M}\) and the solution is basic.

Step 1: Calculate \([\mathrm{H}^{+}]\) for solution (a) using the ion product constant of water#

To find the concentration of hydrogen ions in solution (a), we can use the ion product constant for water, which states that: \(K_w = [\mathrm{H}^{+}] [\mathrm{OH}^{-}]\). We know that \(K_w = 1 \times 10^{-14}\) at 25°C and that \([\mathrm{OH}^{-}] = 0.00045 \mathrm{M}\). We can rearrange the equation to solve for \([\mathrm{H}^{+}]\): \[[\mathrm{H}^{+}] = \frac{K_w}{[\mathrm{OH}^{-}]} = \frac{1 \times 10^{-14}}{0.00045}\]

Step 2: Evaluate \([\mathrm{H}^{+}]\) for solution (a) and determine its acidity#

Using a calculator, we find that: \[[\mathrm{H}^{+}] = \frac{1 \times 10^{-14}}{0.00045} = 2.22 \times 10^{-11} \mathrm{M}\] Since \([\mathrm{H}^{+}] < [\mathrm{OH}^{-}]\), solution (a) is basic.

Step 3: Calculate \([\mathrm{H}^{+}]\) for solution (b) using the ion product constant of water#

Similar to solution (a), we can use the ion product constant for water to find the concentration of hydrogen ions in solution (b). We know \([\mathrm{OH}^{-}] = 8.8 \times 10^{-9}\mathrm{M}\), so: \[[\mathrm{H}^{+}] = \frac{K_w}{[\mathrm{OH}^{-}]} = \frac{1 \times 10^{-14}}{8.8 \times 10^{-9}}\]

Step 4: Evaluate \([\mathrm{H}^{+}]\) for solution (b) and determine its acidity#

Using a calculator, we find that: \[[\mathrm{H}^{+}] = \frac{1 \times 10^{-14}}{8.8 \times 10^{-9}} = 1.14 \times 10^{-6} \mathrm{M}\] Since \([\mathrm{H}^{+}] > [\mathrm{OH}^{-}]\), solution (b) is acidic.

Step 5: Calculate \([\mathrm{H}^{+}]\) for solution (c) using the given relationship between \([\mathrm{H}^{+}]\) and \([\mathrm{OH}^{-}]\)#

We're given that \(\left[\mathrm{OH}^{-}\right]\) in solution (c) is 100 times greater than \(\left[\mathrm{H}^{+}\right]\), so we can write: \[[\mathrm{OH}^{-}] = 100 [\mathrm{H}^{+}]\] Now, we can use the ion product constant for water to find the concentration of hydrogen ions: \[[\mathrm{H}^{+}] [\mathrm{OH}^{-}] = K_w\] Substitute the relationship between the concentrations: \[[\mathrm{H}^{+}] (100 [\mathrm{H}^{+}]) = 1 \times 10^{-14}\]

Step 6: Solve for \([\mathrm{H}^{+}]\) in solution (c) and determine its acidity#

Solve the equation for \([\mathrm{H}^{+}]\): \[[\mathrm{H}^{+}] = \sqrt{\frac{1 \times 10^{-14}}{100}} = \sqrt{1 \times 10^{-16}} = 1 \times 10^{-8} \mathrm{M}\] Since \(\left[\mathrm{OH}^{-}\right] = 100 [\mathrm{H}^{+}] = 1 \times 10^{-6} \mathrm{M}\) and \(\left[\mathrm{H}^{+}\right] = 1 \times 10^{-8} \mathrm{M}\), we can conclude that \([\mathrm{H}^{+}] < [\mathrm{OH}^{-}]\), making solution (c) basic.

Key Concepts

pH Calculations

Understanding pH calculations is crucial in acid-base chemistry. pH is a measure of how acidic or basic a solution is. The formula for calculating pH is:
\[\text{pH} = -\log_{10} [\mathrm{H}^+]\]
This formula tells us the pH based on the concentration of hydrogen ions \(\mathrm{H}^+\) in a solution. A lower pH value (below 7) indicates acidity, while a higher pH value (above 7) indicates basicity.
For neutral water at 25°C, the pH is 7. By computing the pH from the \(\mathrm{H}^+\) concentration, we can determine the nature of the solution:

• pH < 7: Acidic
• pH = 7: Neutral
• pH > 7: Basic

Ion Product Constant of Water

The ion product constant of water, denoted as \(K_w\), is a fundamental concept in acid-base chemistry.
At 25°C, \(K_w\) is valued at \(1 \times 10^{-14}\), signifying the product of the concentrations of hydrogen ions \(\mathrm{H}^+\) and hydroxide ions \(\mathrm{OH}^-\) in pure water:
\[K_w = [\mathrm{H}^+][\mathrm{OH}^-] = 1 \times 10^{-14}\]
This constant helps in identifying the concentration of either ion if the other is known. For instance, if \(\mathrm{OH}^-\) is known, you can find \(\mathrm{H}^+\) by rearranging the equation:
\[[\mathrm{H}^+] = \frac{K_w}{[\mathrm{OH}^-]}\]
This relationship is essential for calculations related to solution acidity and basicity.

Solution Acidity Determination

Determining solution acidity involves analyzing \(\mathrm{H}^+\) and \(\mathrm{OH}^-\) concentrations. By comparing these concentrations, we can classify a solution as acidic, basic, or neutral.
Here's how you can determine acidity based on concentrations:

• If \(\mathrm{H}^+ > \mathrm{OH}^-\): The solution is acidic.
• If \(\mathrm{H}^+ < \mathrm{OH}^-\): The solution is basic.
• If \(\mathrm{H}^+ = \mathrm{OH}^-\): The solution is neutral.

This analysis provides insight into the relative strength of acids and bases. It underscores the central role of hydrogen and hydroxide ions in determining a solution's character. Understanding and applying these principles allow you to predict how a solution will behave chemically.