Problem 38 - Chemistry: The Central Science in SI Units, Global Edition

Question

Carbon dioxide in the atmosphere dissolves in raindrops to produce carbonic acid \(\left(\mathrm{H}_{2} \mathrm{CO}_{3}\right)\), causing the pH of clean, unpolluted rain to range from about 5.2 to 5.6 . What are the ranges of \(\left[\mathrm{H}^{+}\right]\) and \(\left[\mathrm{OH}^{-}\right]\) in the raindrops?

Step-by-Step Solution

Answer

The ranges of \(\left[\mathrm{H}^{+}\right]\) and \(\left[\mathrm{OH}^{-}\right]\) in the raindrops are approximately \(2.51 \times 10^{-6} \ \mathrm{M}\) to \(6.31 \times 10^{-6} \ \mathrm{M}\) and \(1.59 \times 10^{-9} \ \mathrm{M}\) to \(3.98 \times 10^{-9} \ \mathrm{M}\), respectively.

Step 1: Recall the definition of pH and relation to concentration of H+ ions

pH is the negative logarithm of the hydrogen ion concentration, i.e., \(\mathrm{pH} = -\log_{10}([\mathrm{H}^+])\). We can rewrite this equation to solve for the concentration of \(\mathrm{H}^{+}\) ions: \([\mathrm{H}^{+}] = 10^{-\mathrm{pH}}\).

Step 2: Calculate the range of H+ ion concentration

Given the pH range of clean, unpolluted raindrops is between 5.2 and 5.6, we can use the equation from step 1 to find the range of \(\mathrm{H}^{+}\) ion concentration: Lower bound: \([\mathrm{H}^{+}]_{lower} = 10^{-5.6} \approx 2.51 \times 10^{-6} \ \mathrm{M}\) Upper bound: \([\mathrm{H}^{+}]_{upper} = 10^{-5.2} \approx 6.31 \times 10^{-6} \ \mathrm{M}\) So, the range of \(\mathrm{H}^{+}\) ion concentration in the raindrops is approximately \(2.51 \times 10^{-6} \ \mathrm{M}\) to \(6.31 \times 10^{-6} \ \mathrm{M}\).

Step 3: Use the ion product constant of water to find the OH- ion concentration

The ion product constant of water (\(K_w\)) is given by the equation: \(K_w = [\mathrm{H}^+][\mathrm{OH}^{-}]\), where \(K_w = 1.0 \times 10^{-14}\) at 25°C. We can use this equation to find the range of \(\mathrm{OH}^{-}\) ion concentration: Lower bound: \([\mathrm{OH}^{-}]_{lower} = \frac{K_w}{[\mathrm{H}^{+}]_{upper}} = \frac{1.0 \times 10^{-14}}{6.31 \times 10^{-6} \ \mathrm{M}} \approx 1.59 \times 10^{-9} \ \mathrm{M}\) Upper bound: \([\mathrm{OH}^{-}]_{upper} = \frac{K_w}{[\mathrm{H}^{+}]_{lower}} = \frac{1.0 \times 10^{-14}}{2.51 \times 10^{-6} \ \mathrm{M}} \approx 3.98 \times 10^{-9} \ \mathrm{M}\) So, the range of \(\mathrm{OH}^{-}\) ion concentration in the raindrops is approximately \(1.59 \times 10^{-9} \ \mathrm{M}\) to \(3.98 \times 10^{-9} \ \mathrm{M}\). In conclusion, the ranges of \(\left[\mathrm{H}^{+}\right]\) and \(\left[\mathrm{OH}^{-}\right]\) in the raindrops are approximately \(2.51 \times 10^{-6} \ \mathrm{M}\) to \(6.31 \times 10^{-6} \ \mathrm{M}\) and \(1.59 \times 10^{-9} \ \mathrm{M}\) to \(3.98 \times 10^{-9} \ \mathrm{M}\), respectively.

Key Concepts

pH Calculation

In the study of acid-base chemistry, understanding and calculating pH is essential. The pH scale measures how acidic or basic a solution is. It ranges from 0, which is very acidic, to 14, which is very basic. A neutral substance, like pure water, has a pH of 7.
To calculate pH, use the formula:

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

This formula indicates that pH is the negative logarithm of the hydrogen ion concentration \([\text{H}^+]\). If you know the pH of a solution, you can determine the concentration of hydrogen ions by rearranging the formula to:

• \[ [\text{H}^+] = 10^{-\text{pH}} \]

Consider rainwater, which under unpolluted conditions, has a pH range of 5.2 to 5.6. When you apply the formula, calculate the hydrogen ion concentration easily:

• For pH 5.6, \([\text{H}^+] = 10^{-5.6} \approx 2.51 \times 10^{-6} \ M\)
• For pH 5.2, \([\text{H}^+] = 10^{-5.2} \approx 6.31 \times 10^{-6} \ M\)

This gives you the range of hydrogen ion concentration in rainwater, reflecting varying acidity levels.

Ion Concentration

In aqueous solutions, ion concentrations are significant to identify solution properties. Specifically, in our atmosphere-related exercise, we're interested in both \([\text{H}^+]\) and \([\text{OH}^-]\) concentrations, which determine whether a solution is acidic or basic.
When you know the hydrogen ion concentration, you can find the hydroxide ion concentration using the ion product constant of water \(K_w\). At 25°C, \(K_w\) is a constant value:

• \(K_w = [\text{H}^+][\text{OH}^-] = 1.0 \times 10^{-14}\)

From this relationship, calculate \([\text{OH}^-]\) by rearranging:

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

For rainwater with a pH range of 5.2 to 5.6, use the calculated \([\text{H}^+]\) values:

• For \([\text{H}^+] = 6.31 \times 10^{-6}\ M\), \([\text{OH}^-] = \frac{1.0 \times 10^{-14}}{6.31 \times 10^{-6}} \approx 1.59 \times 10^{-9} \ M\)
• For \([\text{H}^+] = 2.51 \times 10^{-6}\ M\), \([\text{OH}^-] = \frac{1.0 \times 10^{-14}}{2.51 \times 10^{-6}} \approx 3.98 \times 10^{-9} \ M\)

These calculations highlight how ion concentration interacts with pH, emphasizing the balance between hydrogen and hydroxide ions.

Chemical Equilibrium

Chemical equilibrium involves a state where the concentrations of reactants and products remain constant over time in a closed system. This concept is crucial to the study of acid-base chemistry, especially when dealing with dissociation processes and reactions in solutions.
In the equilibrium involving water, the self-ionization of water reflects the balance:

• \(2\ H_2O \leftrightarrow H_3O^+ + OH^-\)

Here, the equilibrium constant \(K_w\) represents the product of the concentrations of hydronium \([H_3O^+]\) and hydroxide ions \([OH^-]\), which is:

• \(K_w = [H_3O^+][OH^-]\) and is always \(1.0 \times 10^{-14}\) at 25°C

By understanding chemical equilibrium, you can predict how changes in one component of a reaction influence the overall system. Applying this to rain's pH, remember that the presence of carbonic acid \((H_2CO_3)\) in raindrops leads to equilibrium between hydrogen ions and the acid. This concept helps explain different \(pH\) values in natural rain under normal conditions. Keep in mind:

• A shift in equilibrium, such as increased \([CO_2]\), can lower the pH, making rain more acidic.
• Maintaining equilibrium ensures that the expressions \([H_3O^+][OH^-] = K_w\) remain constant unless temperature or conditions change.

Chemical equilibrium is a foundational principle, critical for exploring how substances react and interact in solution.