Calculate the \(\mathrm{pH}\) of (a) orange juice, \(3.7 \times 10^{-4} \mathrm{M} \mathrm{H}^{+}\) (b) vinegar, \(2.8 \times 10^{-3} \mathrm{M} \mathrm{H}^{+}\) (c) shampoo, \(2.4 \times 10^{-6} \mathrm{M} \mathrm{H}^{+}\) (d) dishwashing detergent, \(3.6 \times 10^{-8} \mathrm{M} \mathrm{H}^{+}\)

When working with logarithms in chemistry, especially for pH calculations, it's critical to understand how they function. A logarithm answers the question: 'To what exponent must we raise a specific base to get a given number?' For pH, we use base 10 logarithms.
The formula \( \text{pH} = - \text{log}[\text{H}^+] \) is fundamental. It means you are finding the exponent that 10 must be raised to, multiplied by -1, to match the hydrogen ion concentration (\text{H}^+). As a result, any number in scientific notation, like \(3.7 \times 10^{-4}\), can be simplified using logarithmic properties.
Logarithms help handle very small and very large numbers, which are common with ion concentrations in chemistry. By converting them into simpler values to work with, they allow us to better understand and communicate chemical properties like acidity.

The concentration of hydrogen ions (\text{H}\text{+}) in a solution directly influences its pH value. The concentration is often given in moles per liter (M). For example, in the provided exercise:

• Orange juice: \(3.7 \times 10^{-4} \text{ M} \text{ H}^+ \)
• Vinegar: \(2.8 \times 10^{-3} \text{ M} \text{ H}^+ \)
• Shampoo: \(2.4 \times 10^{-6} \text{ M} \text{ H}^+ \)
• Dishwashing detergent: \(3.6 \times 10^{-8} \text{ M} \text{ H}^+ \)

When calculating pH, remember that higher \text{H}\text{+} concentrations mean more acidic solutions (lower pH). Conversely, lower \text{H}\text{+} concentrations lead to less acidic or more basic solutions (higher pH). This understanding is essential when interpreting lab results or solving textbook exercises.

Acidity and alkalinity are measured using the pH scale, which ranges from 0 to 14. pH values:

• Below 7 indicate acidic solutions
• Above 7 indicate basic (alkaline) solutions
• 7 is neutral, like pure water.

To calculate pH, you use the formula \( \text{pH} = -\text{log}[\text{H}^+]\). Here's how it applies:
For a given \text{H}\text{+} concentration, such as \(3.7 \times 10^{-4} \text{ M} \) for orange juice, you find the log value and then take the negative. \( -\text{log}(3.7 \times 10^{-4}) = 3.43 \). This value tells you that orange juice is acidic. Similarly:
Vinegar with \(2.8 \times 10^{-3} \text{ M} \text{ H}^+ \) gives pH = 2.55, also acidic. Shampoo with \(2.4 \times 10^{-6} \text{ M} \text{ H}^+ \) results in pH = 5.62, mildly acidic. Dishwashing detergent with \(3.6 \times 10^{-8} \text{ M} \text{ H}^+ \) has pH = 7.44, slightly basic. Understanding the pH scale helps you know how acidic or basic a solution is, which is crucial in both academic and real-world chemistry.