Problem 59 - A Graphical Approach to College Algebra

Question

Find the pH for each substance with the given hydronium ion \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) concentration. Crackers, \(3.9 \times 10^{-9}\)

Step-by-Step Solution

Answer

The pH is approximately 8.41.

Step 1: Understanding the pH formula

The pH of a solution is calculated using the formula: \[pH = -\log_{10}[\mathrm{H}_3\mathrm{O}^+]\]where \([\mathrm{H}_3\mathrm{O}^+]\) is the concentration of hydronium ions.

Step 2: Inserting the given values

Insert the given hydronium ion concentration for the crackers into the formula. The concentration is \(3.9 \times 10^{-9} \text{ M}\).

Step 3: Calculating the pH

Calculate the pH by taking the negative logarithm (base 10) of the hydronium ion concentration:\[pH = -\log_{10}(3.9 \times 10^{-9})\]Use a calculator to perform this calculation.

Step 4: Evaluating the logarithm

Using a calculator, compute the logarithm:\[\log_{10}(3.9 \times 10^{-9}) \approx -8.409\]Thus,\[pH = -(-8.409) = 8.409\]

Step 5: Conclusion

The pH of the solution, which represents the hydronium ion concentration of the crackers, is approximately 8.41.

Key Concepts

Hydronium Ion Concentration

Hydronium ion concentration is pivotal in understanding the acidity or basicity of a solution. Hydronium ions \(\text{H}_3\text{O}^+\) are formed when water molecules combine with hydrogen ions. This process is crucial to determining how acidic or basic a solution is.

• The concentration is typically expressed in moles per liter (M), which shows how many hydronium ions are present per liter of solution.
• The concentration directly influences the pH, as seen in the formula \(pH = -\log_{10}[\mathrm{H}_3\mathrm{O}^+]\). A higher concentration means more acidity.

In the example of crackers, the hydronium ion concentration is given as \(3.9 \times 10^{-9} \text{ M}\). This relatively low concentration indicates a weakly acidic or neutral solution.
Understanding this concept lays the groundwork for appreciating how substances interact within solutions.

Logarithmic Functions

Logarithms are mathematical functions that simplify the process of working with very large or very small numbers. When calculating pH, the logarithmic function \(\log_{10}\) is used. This is the logarithm to the base 10, which aligns perfectly with the pH scale.

• A logarithm is the exponent that indicates the power to which a base number must be raised to yield a specific result.
• In pH calculation, it helps in determining how concentrated hydronium ions are by converting multiplication into addition.

For instance, with the crackers, you solve \(\log_{10}(3.9 \times 10^{-9}) \) to find the power needed to express that concentration in a more manageable form.
This results in a friendly number like \(-8.409\), making it easier to assess the acidity level.

pH Scale

The pH scale is a numeric representation of solution acidity or basicity, ranging from 0 to 14. It is a logarithmic scale, meaning each unit change represents a tenfold change in acidity or basicity.

• Solutions with a pH less than 7 are acidic, while those with a pH greater than 7 are basic. A pH of exactly 7 indicates a neutral solution.
• The scale is based on the concentration of hydronium ions \([\text{H}_3\text{O}^+]\), where a lower concentration reflects a higher pH.

For our example with crackers, the calculated pH is 8.41. This means that the solution is slightly basic, suggesting that it's on the higher end of the scale.
Understanding the pH scale is essential in fields such as chemistry, biology, and environmental science, due to its impact on chemical reactivity and organism survival.