Question

Calculate the \(\left[\mathrm{H}^{+}\right]\) in each of the following solutions, and indicate whether the solution is acidic or basic. a. \(\left[\mathrm{OH}^{-}\right]=2.32 \times 10^{-4} M\) b. \(\left[\mathrm{OH}^{-}\right]=8.99 \times 10^{-10} M\) c. \(\left[\mathrm{OH}^{-}\right]=4.34 \times 10^{-6} M\) d. \(\left[\mathrm{OH}^{-}\right]=6.22 \times 10^{-12} M\)

Short answer

a. \([\mathrm{H}^{+}] = 4.31 \times 10^{-11} M\), basic. b. \([\mathrm{H}^{+}] = 1.11 \times 10^{-6} M\), acidic. c. \([\mathrm{H}^{+}] = 2.30 \times 10^{-9} M\), basic. d. \([\mathrm{H}^{+}] = 1.61 \times 10^{-3} M\), acidic.

Step-by-step solution

Calculate the concentration of \(\mathrm{H}^{+}\)

Using the ion product constant for water (\(K_w = 1.0 \times 10^{-14} \, \text{M}^2\)), we can calculate the concentration of \(\mathrm{H}^{+}\) with the following equation: \( [\mathrm{H}^{+}] = \frac{K_w}{[\mathrm{OH}^{-}]} \) For a. we have \([\mathrm{OH}^{-}] = 2.32 \times 10^{-4} M\), so: \( [\mathrm{H}^{+}] = \frac{1.0 \times 10^{-14}}{2.32 \times 10^{-4}} \) \( [\mathrm{H}^{+}] = 4.3103448 \times 10^{-11} M \)

Determine if the solution is acidic or basic

Compare the concentrations of \(\mathrm{H}^{+}\) and \(\mathrm{OH}^{-}\): \( [\mathrm{H}^{+}] < [\mathrm{OH}^{-}] \) Thus, the solution is basic. b.

Calculate the concentration of \(\mathrm{H}^{+}\)

Using the same equation, we have: \( [\mathrm{H}^{+}] = \frac{1.0 \times 10^{-14}}{8.99 \times 10^{-10}} \) \( [\mathrm{H}^{+}] = 1.112346 \times 10^{-6} M \)

Determine if the solution is acidic or basic

Comparing the concentrations of \(\mathrm{H}^{+}\) and \(\mathrm{OH}^{-}\): \( [\mathrm{H}^{+}] > [\mathrm{OH}^{-}] \) Thus, the solution is acidic. c.

Calculate the concentration of \(\mathrm{H}^{+}\)

Using the equation: \( [\mathrm{H}^{+}] = \frac{1.0 \times 10^{-14}}{4.34 \times 10^{-6}} \) \( [\mathrm{H}^{+}] = 2.304147\times 10^{-9} M \)

Determine if the solution is acidic or basic

Comparing the concentrations of \(\mathrm{H}^{+}\) and \(\mathrm{OH}^{-}\): \( [\mathrm{H}^{+}] < [\mathrm{OH}^{-}] \) Thus, the solution is basic. d.

Calculate the concentration of \(\mathrm{H}^{+}\)

Using the equation: \( [\mathrm{H}^{+}] = \frac{1.0 \times 10^{-14}}{6.22 \times 10^{-12}} \) \( [\mathrm{H}^{+}] = 1.608038\times10^{-3} M \)

Determine if the solution is acidic or basic

Comparing the concentrations of \(\mathrm{H}^{+}\) and \(\mathrm{OH}^{-}\): \( [\mathrm{H}^{+}] > [\mathrm{OH}^{-}] \) Thus, the solution is acidic.

Key concepts

pH Calculations

Understanding pH calculations is essential for determining how acidic or basic a solution is. The pH scale ranges from 0 to 14. A pH below 7 indicates acidity, while a pH above 7 indicates a basic solution. Neutral solutions have a pH of exactly 7.

The formula for calculating pH is:

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

When you know the concentration of hydrogen ions \([\mathrm{H}^{+}]\), you can easily find the pH by substituting this value into the formula. The negative logarithm lets you express the concentration in a manageable, smaller number.

In our calculations, we derived \([\mathrm{H}^{+}]\) from given \([\mathrm{OH}^{-}]\) using the ion product of water. This helps us determine if the solution is more on the acidic or basic side. Remember that lower \([\mathrm{H}^{+}]\) concentration means a higher pH and a more basic solution.

Being comfortable with logarithms and understanding the pH scale will make these calculations much easier.

Ion Product Constant of Water

The ion product constant of water, symbolized as \(K_w\), is fundamental in acid-base chemistry. It represents the relationship between hydrogen ions (\(\mathrm{H}^{+}\)) and hydroxide ions (\(\mathrm{OH}^{-}\)) in water. At 25°C, \(K_w\) is given as \(1.0 \times 10^{-14} \, \text{M}^2\). This value serves as the foundation for understanding the neutral point of water and how it reacts in acid-base solutions.

In pure water, \([\mathrm{H}^{+}]\) is equal to \([\mathrm{OH}^{-}]\), so both are equal to \(1.0 \times 10^{-7} \, \text{M}\), which results in a neutral pH of 7.

Using \(K_w\) in Calculations

By knowing \(K_w\), you can find either \([\mathrm{H}^{+}]\) or \([\mathrm{OH}^{-}]\) if the other is known:

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

This formula is crucial for finding the missing ion concentration and helps determine the acidity or basicity of a solution. Understanding \(K_w\) makes working with these equations straightforward and manageable.

Acidic and Basic Solutions

Understanding what makes a solution acidic or basic is critical in chemistry. Solutions are classified based on the balance of hydrogen ions \([\mathrm{H}^{+}]\) and hydroxide ions \([\mathrm{OH}^{-}]\).

Identifying Acidity or Basicity

- **Acidic Solutions**: Have more \([\mathrm{H}^{+}]\) than \([\mathrm{OH}^{-}]\). These solutions have a pH less than 7.- **Basic Solutions**: Have more \([\mathrm{OH}^{-}]\) than \([\mathrm{H}^{+}]\). These solutions have a pH greater than 7.- **Neutral Solutions**: Equal concentrations of \([\mathrm{H}^{+}]\) and \([\mathrm{OH}^{-}]\), typically with a pH of 7.

In the exercise, converting \([\mathrm{OH}^{-}]\) to \([\mathrm{H}^{+}]\) with \(K_w\) guides us in identifying how the solution behaves. Comparing these ion concentrations leads us to categorize each solution correctly. Whether you determine a solution to be acidic or basic depends solely on which ion, \(\mathrm{H}^{+}\) or \(\mathrm{OH}^{-}\), is more prevalent.

This understanding forms the backbone of acid-base chemistry and extends to real-world applications like chemical manufacturing, agriculture, and environmental science.