a. Without using a calculator, estimate the pH of each of the following solutions:
1. [HO⁻] = 3.2 × 10⁻⁵
2. [H₃O⁺] = 8.3 × 10⁻¹
3. [H₃O⁺] = 1.7 × 10⁻³
b. Determine the exact pH, using a calculator.

Verified step by step guidance

1. Step 1: Recall the formula for calculating pH, which is defined as \( \text{pH} = -\log_{10}[\text{H}_3\text{O}^+] \). This formula applies to all solutions where the concentration of \( \text{H}_3\text{O}^+ \) (or \( \text{HO}^- \) indirectly) is known.
2. Step 2: For the first solution, \( [\text{HO}^-] = 3.2 \times 10^{-5} \), use the relationship between \( [\text{HO}^-] \) and \( [\text{H}_3\text{O}^+] \): \( [\text{H}_3\text{O}^+] \cdot [\text{HO}^-] = 10^{-14} \). Solve for \( [\text{H}_3\text{O}^+] \) by dividing \( 10^{-14} \) by \( 3.2 \times 10^{-5} \). Then, estimate the pH using \( \text{pH} = -\log_{10}[\text{H}_3\text{O}^+] \).
3. Step 3: For the second solution, \( [\text{H}_3\text{O}^+] = 8.3 \times 10^{-1} \), directly apply the formula \( \text{pH} = -\log_{10}[\text{H}_3\text{O}^+] \). Since the concentration is close to 1, the pH will be slightly less than 0. Estimate the value by considering the logarithmic scale.
4. Step 4: For the third solution, \( [\text{H}_3\text{O}^+] = 1.7 \times 10^{-3} \), again use \( \text{pH} = -\log_{10}[\text{H}_3\text{O}^+] \). Recognize that \( \log_{10}(1.7) \) is slightly greater than 0, and \( \log_{10}(10^{-3}) \) is -3. Combine these to estimate the pH as slightly above 3.
5. Step 5: To determine the exact pH values using a calculator, input the concentrations into the formula \( \text{pH} = -\log_{10}[\text{H}_3\text{O}^+] \) for each solution. For the first solution, calculate \( [\text{H}_3\text{O}^+] \) first using \( [\text{H}_3\text{O}^+] = \frac{10^{-14}}{[\text{HO}^-]} \), then find the pH. For the second and third solutions, directly compute the logarithm of the given \( [\text{H}_3\text{O}^+] \) values.

Key Concepts

pH Scale

The pH scale measures the acidity or basicity of a solution, ranging from 0 to 14. A pH less than 7 indicates an acidic solution, while a pH greater than 7 indicates a basic solution. The pH is calculated as the negative logarithm of the hydronium ion concentration: pH = -log[H3O+]. Understanding this scale is crucial for estimating and calculating pH values.

Hydronium Ion Concentration

The concentration of hydronium ions ([H3O+]) in a solution directly influences its pH. A higher concentration of H3O+ corresponds to a lower pH, indicating a more acidic solution. Conversely, a lower concentration results in a higher pH, indicating a more basic solution. This relationship is fundamental for both estimating and calculating pH values in various solutions.

Logarithmic Relationships

The pH scale is logarithmic, meaning each whole number change on the scale represents a tenfold change in H3O+ concentration. For example, a solution with a pH of 3 has ten times more hydronium ions than a solution with a pH of 4. This concept is essential for understanding how small changes in ion concentration can significantly affect pH, which is critical for both estimation and precise calculation.