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

Step-by-Step Solution

Answer

The pH of the sodium hydroxide solution is 13.5.

Step 1: Understanding the Formula for pH

The pH of a solution is calculated using the formula: \[ \text{pH} = -\log_{10} \left( [\mathrm{H}_{3} \mathrm{O}^{+}] \right) \]Here, \([\mathrm{H}_{3} \mathrm{O}^{+}]\) represents the concentration of hydronium ions in the solution.

Step 2: Plugging Values into the Formula

Insert the given hydronium ion concentration, \(3.2 \times 10^{-14}\), into the pH formula:\[ \text{pH} = -\log_{10} \left( 3.2 \times 10^{-14} \right) \]

Step 3: Calculate the Logarithm

Use a calculator to find the logarithm: \[ \log_{10} (3.2 \times 10^{-14}) \approx -13.5 \]This means that: \[ \text{pH} = -(-13.5) = 13.5 \]

Step 4: Confirm the pH Classification

Since a pH greater than 7 indicates a basic (or alkaline) solution, and sodium hydroxide is a strong base, the calculated pH of 13.5 is consistent with expectations for a basic substance.

Key Concepts

Hydronium Ion Concentration

Hydronium ion concentration, indicated as \([\mathrm{H}_{3} \mathrm{O}^{+}]\), plays a crucial role in determining the acidity or basicity of a solution. In water, hydrogen ions (\mathrm{H}^{+}) are often represented as hydronium ions for accuracy. These ions form when water molecules (\mathrm{H_2 O}) gain an extra hydrogen ion, thus becoming \mathrm{H_3 O^+}. The concentration of hydronium ions in a solution is a direct measure of its acidic property. More hydronium ions mean a stronger acidic solution.To illustrate, consider the concentration of hydronium ions in various substances:

• A high concentration, like in lemon juice, signifies strong acidity.
• A low concentration, such as in soap, indicates basicity.

Knowing the hydronium ion concentration allows us to determine the pH value easily using logarithms. This is vital for classifying substances as acidic, neutral, or basic in nature.

Basic Solution

A basic solution, also known as an alkaline solution, is one where the concentration of hydroxide ions is greater than that of hydronium ions. This results in a pH value greater than 7. Basic solutions are characterized by their ability to neutralize acids and their slippery feel. Common examples include baking soda and household ammonia. For instance, in sodium hydroxide (lye), the concentration of hydronium ions is very low. This is why it exhibits a pH of 13.5, according to the given exercise, and affirms its position as a strong base. A basic solution typically has properties like:

• They turn red litmus paper blue.
• They taste bitter when appropriate to test (e.g., when testing with substances like baking soda).
• They conduct electricity when dissolved in water.

Understanding what constitutes a basic solution helps predict reactions in chemistry, such as neutralization reactions where acids and bases combine to form water and salt.

pH Formula

The pH formula is a mathematical tool used to quantify the acidity or basicity of an aqueous solution. This is given by the formula:\[\text{pH} = -\log_{10} \left( [\mathrm{H}_{3} \mathrm{O}^{+}] \right)\]This formula is pivotal because it translates the hydronium ion concentration into a simple and comparable scale, ranging from 0 to 14.Here's how it works:

• A pH value of less than 7 indicates an acidic solution.
• A pH value of exactly 7 suggests a neutral solution (like pure water).
• A pH value greater than 7 signifies a basic solution.

For the exercise with sodium hydroxide, the low hydronium ion concentration made the pH extremely high at 13.5, verifying it as a basic substance. Calculating pH using the formula involves taking the negative logarithm of the hydronium ion concentration. This step-by-step computational process lets us classify substances quickly. Hence, the pH formula is an essential aspect of understanding chemical properties.