Problem 59 - A Graphical Approach to Precalculus with Limits

Question

Find the \(p H\) 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 of crackers is approximately 8.41.

Step 1: Understanding pH Calculation Formula

The formula to calculate the pH from the hydronium ion concentration is \( pH = -\log_{10}[H_3O^+] \). Here, \([H_3O^+]\) is the concentration of hydronium ions in moles per liter.

Step 2: Substitute the Given Concentration

Given \([H_3O^+] = 3.9 \times 10^{-9}\), we substitute this value into the formula: \( pH = -\log_{10}(3.9 \times 10^{-9}) \).

Step 3: Calculate the Negative Logarithm

Use a calculator to compute the negative logarithm: \( -\log_{10}(3.9 \times 10^{-9}) \approx 8.41 \). This is the pH of the substance.

Key Concepts

Understanding Hydronium Ion Concentration

Hydronium ion concentration is a fundamental concept in acid-base chemistry. It refers to the number of hydronium ions \([H_3O^+]\) present in a solution. When an acid dissolves in water, it donates hydrogen ions \((H^+)\), which combine with water molecules to form hydronium ions. This concentration is typically measured in moles per liter (M). Understanding this concentration helps us determine the acidity of a solution. For example, a higher concentration of hydronium ions indicates a more acidic solution, while a lower concentration suggests a more basic or neutral solution.

• A lower hydronium ion concentration means higher pH (less acidic)
• A higher hydronium ion concentration means lower pH (more acidic)

Remember: the hydronium ion concentration is inversely related to pH, which is why it is an essential aspect of pH calculation.

The Role of the Logarithmic Function in pH Calculation

The logarithmic function is used to convert hydronium ion concentration into a more manageable number, known as pH. The formula \( pH = -\log_{10}[H_3O^+] \) allows us to easily express the acidity on a scale that is simple to understand. This scale usually ranges from 0 to 14, where lower values indicate greater acidity and higher values indicate basicity.Logarithms help scale down the wide range of hydronium ion concentrations in a precise manner:

• Using a base-10 logarithm simplifies calculations, especially when dealing with small numbers like \(3.9 \times 10^{-9}\).
• The negative sign in front of the logarithm reorients the scale: more acidity means a lower pH number.

Think of it as a tool to convert the multiplicative scale of ion concentration into a more intuitive additive scale. It's important in scientific calculations to efficiently express these variations in acidity.

Basics of Acid-Base Chemistry

Acid-base chemistry is a key area in general chemistry that revolves around reactions involving acids and bases. An acid is typically characterized by its ability to donate protons (or hydrogen ions \(H^+\)). When these protons are released into water, hydronium ions \((H_3O^+)\) form, increasing the acidity of the solution. On the other hand, a base accepts protons and reduces the concentration of hydronium ions.Understanding the behavior of acids and bases helps us predict their impact on their environment:

• Acids tend to taste sour and can corrode metals.
• Bases tend to taste bitter and feel slippery.

Understanding the pH scale, derived from hydronium ion concentration and logarithmic calculations, assists in identifying the strength of an acid or base. The scale embodies a spectrum where 7 is neutral, less than 7 denotes acidic, and more than 7 indicates basic (alkaline). Well-grounded knowledge in acid-base chemistry allows learners to grasp broader chemical processes and industrial applications that rely on controlling acidity or basicity in substances.