- Use the relation \([H^+] = 10^{-pH}\) to determine the concentration of hydrogen ions for each vinegar brand.
- Calculate \([H^+]_1 = 10^{-4.5} \approx 3.162 \times 10^{-5}\) and \([H^+]_2 = 10^{-5.0} = 1.0 \times 10^{-5}\).
- Compute the difference \([H^+]_1 - [H^+]_2 \approx 2.162 \times 10^{-5}\).
- Identify the correct option: \(\boxed{b}\).
Explanation
- Understanding the Problem and Formula
The pH of a substance is defined by the formula \(pH = -\log_{10}[H^+]\), where \([H^+]\) represents the concentration of hydrogen ions in moles per liter. Given two brands of vinegar with \(pH_1 = 4.5\) and \(pH_2 = 5.0\), we need to find the absolute difference in their hydrogen ion concentrations, \(|[H^+]_1 - [H^+]_2|\). To find \([H^+]\), we rearrange the formula to \([H^+] = 10^{-pH}\).
- Calculating Hydrogen Ion Concentrations
Using the formula \([H^+] = 10^{-pH}\), we calculate the concentrations for each brand:
- For the first brand: \([H^+]_1 = 10^{-4.5}\).
- For the second brand: \([H^+]_2 = 10^{-5.0}\).
To evaluate these, note that \(10^{-4.5} = 10^{-5} \times 10^{0.5} = 10^{-5} \times \sqrt{10}\). Since \(\sqrt{10} \approx 3.162\), we have \([H^+]_1 \approx 3.162 \times 10^{-5}\). The concentration for the second brand is \([H^+]_2 = 1.0 \times 10^{-5}\).
- Calculating the Difference
Now, we find the difference:
\[|[H^+]_1 - [H^+]_2| = |3.162 \times 10^{-5} - 1.0 \times 10^{-5}|\]
Factoring out \(10^{-5}\), we get:
\[(3.162 - 1.0) \times 10^{-5} = 2.162 \times 10^{-5}\]
Rounding this to two significant figures gives approximately \(2.2 \times 10^{-5}\), which corresponds to option (b).
Examples
In chemistry and biology, the concentration of hydrogen ions is crucial for maintaining the pH balance of solutions, such as in blood or soil. For instance, farmers might test soil pH to determine if nutrients are available for plant growth. Similarly, doctors might monitor the blood pH of a patient to ensure that their body's metabolic processes are functioning correctly. Understanding how small differences in pH correspond to logarithmic changes in ion concentration helps in accurately measuring and managing these sensitive environments.