Problem 49 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 47-50: Chemists use a number denoted by \(\mathrm{pH}\) to describe quantitatively the acidity or basicity of solutions. By definition, \(\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right]\), where \(\left[\mathrm{H}^{+}\right]\)is the hydrogen ion concentration in moles per liter. A solution is considered basic if \(\left[\mathrm{H}^{+}\right]<10^{-7}\) or acidic if \(\left[\mathrm{H}^{+}\right]>10^{-7}\). Find the corresponding inequalities involving pH.

Step-by-Step Solution

Answer

A solution is acidic if pH < 7 and basic if pH > 7.

Step 1: Understand the definition of pH

According to the definition, \( \text{pH} = -\log [\text{H}^+] \). This equation relates the pH value to the concentration of hydrogen ions in the solution.

Step 2: Rewrite the inequality for acidity and basicity

We are given two conditions: a solution is acidic if \([\text{H}^+] > 10^{-7}\) and basic if \([\text{H}^+] < 10^{-7}\). We need to express these conditions in terms of pH.

Step 3: Convert hydrogen ion concentration to pH

Use the definition of pH to express \([\text{H}^+] > 10^{-7}\) as \( \text{pH} < -\log(10^{-7}) \) and \([\text{H}^+] < 10^{-7}\) as \( \text{pH} > -\log(10^{-7}) \).

Step 4: Simplify the inequalities

Since \(-\log(10^{-7}) = 7\), the inequality for an acidic solution becomes \( \text{pH} < 7 \) and for a basic solution becomes \( \text{pH} > 7 \).

Key Concepts

Acidity and Basicity

In chemistry, understanding whether a solution is acidic or basic is crucial for various applications. Acidity and basicity are determined by the concentration of hydrogen ions, denoted as \([ ext{H}^+]\), present in a solution. An acidic solution has a higher concentration of hydrogen ions, which leads to a lower pH value. In contrast, a basic solution has a lower concentration of hydrogen ions, resulting in a higher pH value.

Here are some key points to remember:

• Acidic solutions have pH values less than 7.
• Basic or alkaline solutions have pH values greater than 7.
• A neutral solution, such as pure water, has a pH value of approximately 7.

Knowing these distinctions assists in predicting how substances will react and behave in different environments.

Hydrogen Ion Concentration

The concentration of hydrogen ions in a solution is a fundamental concept in chemistry. It is represented as \([ ext{H}^+]\) and is a key indicator of the solution's acidity or basicity.

• The hydrogen ion concentration is measured in moles per liter.
• It provides insight into the acidic or basic nature of a solution.
• High \([ ext{H}^+]\) indicates an acidic solution (pH < 7).
• Low \([ ext{H}^+]\) indicates a basic solution (pH > 7).

Understanding \([ ext{H}^+]\) is essential for applications such as balancing chemical equations, controlling reaction conditions, and environmental analysis.

Logarithmic Functions

Logarithmic functions are vital in translating hydrogen ion concentrations into pH values. The relationship is mathematically expressed as \( ext{pH} = - ext{log} [ ext{H}^+] \). This means the pH scale is a logarithmic scale, not a linear one.

• Each unit change in pH reflects a tenfold change in \([ ext{H}^+]\).
• For example, moving from pH 3 to 4 indicates the hydrogen ion concentration has decreased ten times.
• This scale helps to compactly represent the wide range of hydrogen ion concentrations found in various solutions.

Logarithms simplify the otherwise vast range of concentrations, making it easier to compare acidity and basicity across different situations.

Inequalities in Chemistry

In the context of pH, inequalities help in establishing the condition under which a solution is considered acidic or basic. The definition \( [ ext{H}^+] < 10^{-7} \) for basic and \( [ ext{H}^+] > 10^{-7} \) for acidic is translated to inequalities involving pH as follows:

• If \( [ ext{H}^+] > 10^{-7} \), then \( ext{pH} < 7 \).
• If \( [ ext{H}^+] < 10^{-7} \), then \( ext{pH} > 7 \).
• This creates two distinct ranges on the pH scale, with 7 being neutral.

Inequalities simplify the categorization of solutions based on their hydrogen ion concentrations, therefore aiding in various analyses and practical applications in chemistry.