Problem 60 - 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. Sodium hydroxide, \(3.2 \times 10^{-14}\)

Step-by-Step Solution

Answer

The pH is 13.49.

Step 1: Understanding the pH Formula

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

Step 2: Substitute Given Values

We have been given the hydronium ion concentration as \(3.2 \times 10^{-14}\). Substitute this value into the pH formula: \( pH = -\log_{10}(3.2 \times 10^{-14}) \).

Step 3: Calculate the Logarithm

Use a calculator to compute \(-\log_{10}(3.2 \times 10^{-14})\). This requires calculating the logarithm of \(3.2\) and accounting for the exponent \(-14\).

Step 4: Simplify the Calculation

The pH calculation can be broken down as follows: \( pH \approx -(\log_{10}(3.2) + \log_{10}(10^{-14})) = -\log_{10}(3.2) + 14 \).

Step 5: Final Calculation

Calculate \(\log_{10}(3.2)\) which is approximately \(0.5051\). The pH is then \(-0.5051 + 14\).

Step 6: Result

By subtracting, the pH is \(13.4949\), rounded to two decimal places as \(13.49\).

Key Concepts

Hydronium Ion Concentration

In chemistry, the concentration of hydronium ions \([H_3O^+]\) in a solution is of great importance, as it is directly related to the acidity of the solution. High hydronium ion concentration means the solution is acidic, while a low concentration indicates basicity. Hydronium ions are formed when an acid dissolves in water, donating a proton (H⁺) to a water molecule, thus creating \([H_3O^+]\).

The concentration of hydronium ions is usually given in moles per liter (M), commonly denoted using scientific notation due to their often very small values. In the given exercise, the hydronium ion concentration for sodium hydroxide is \(3.2 \times 10^{-14}\), which suggests a very basic solution.

• More hydronium ions = more acidic solution.
• Fewer hydronium ions = more basic solution.
• The pH scale, ranging from 0 to 14, provides a simple way to express solution acidity based on hydronium ion concentration.

Logarithm

Logarithms are mathematical functions used to simplify the process of working with large or small numbers, such as those encountered in hydronium ion concentrations. For pH calculations, the logarithm base 10, written as \(\log_{10}\), is employed. The formula for calculating pH is given by:\[ pH = -\log_{10}[H_3O^+] \]

To compute the pH from the hydronium ion concentration, one takes the negative logarithm of that concentration. For the exercise on sodium hydroxide, the concentration is \(3.2 \times 10^{-14}\), and you calculate:\[ pH = - (\log_{10}(3.2) + \log_{10}(10^{-14})) \]

• Logarithms transform multiplicative relationships into additive ones, making complex calculations more manageable.
• Using a calculator for logarithmic calculations ensures accuracy in finding pH values.
• The learning process can be complicated, but understanding how logarithms relate to pH is crucial for chemistry studies.

Sodium Hydroxide

Sodium hydroxide, often known as lye or caustic soda, is a strong base commonly used in chemistry and industry. When dissolved in water, it dissociates to form sodium ions (Na⁺) and hydroxide ions (OH⁻). These OH⁻ ions react to reduce hydronium ion concentration in the solution, resulting in a higher pH value, indicating basicity.

In the exercise, sodium hydroxide's impact is shown by its very low hydronium ion concentration \(3.2 \times 10^{-14}\), producing a high pH value of approximately 13.49. This reflects sodium hydroxide's strength as a base.

• Sodium hydroxide is commonly used in household cleaning products due to its ability to break down organic materials.
• Its high pH level implies strong basic properties, which can neutralize acids effectively.
• Handling sodium hydroxide requires caution as it can cause chemical burns and reactions with water and other substances.

Understanding the basics of sodium hydroxide helps grasp broader chemical concepts, including the balance of pH in solutions.