Problem 47 - 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. Approximate the \(\mathrm{pH}\) of each substance. (a) vinegar: \(\left[\mathrm{H}^{+}\right] \approx 6.3 \times 10^{-3}\) (b) carrots: \(\left[\mathrm{H}^{+}\right] \approx 1.0 \times 10^{-5}\) (c) sea water: \(\left[\mathrm{H}^{+}\right] \approx 5.0 \times 10^{-9}\)

Step-by-Step Solution

Answer

Vinegar: pH ≈ 2.2, Carrots: pH = 5.0, Sea Water: pH ≈ 8.3

Step 1: Understanding the Problem

The problem requires us to find the pH of different substances, given their hydrogen ion concentrations. The formula to calculate pH is \( \mathrm{pH} = -\log \left[ \mathrm{H}^{+} \right] \).

Step 2: Calculating pH for Vinegar

For vinegar, the hydrogen ion concentration is \( \left[ \mathrm{H}^{+} \right] = 6.3 \times 10^{-3} \). Using the pH formula, we calculate: \[ \mathrm{pH} = -\log (6.3 \times 10^{-3}) \approx 2.2 \]

Step 3: Calculating pH for Carrots

For carrots, the hydrogen ion concentration is \( \left[ \mathrm{H}^{+} \right] = 1.0 \times 10^{-5} \). Using the pH formula, we calculate: \[ \mathrm{pH} = -\log (1.0 \times 10^{-5}) = 5.0 \]

Step 4: Calculating pH for Sea Water

For sea water, the hydrogen ion concentration is \( \left[ \mathrm{H}^{+} \right] = 5.0 \times 10^{-9} \). Using the pH formula, we calculate: \[ \mathrm{pH} = -\log (5.0 \times 10^{-9}) \approx 8.3 \]

Key Concepts

Hydrogen Ion Concentration

Hydrogen ion concentration is a crucial element in the world of chemistry, specifically when discussing the acidity or basicity of solutions. It is represented by \( \left[ \mathrm{H}^{+} \right] \), which indicates the number of hydrogen ions in a solution, measured in moles per liter.
Understanding this concept is essential because the concentration of hydrogen ions determines a solution's pH level.
To visualize this:

• High \( \left[ \mathrm{H}^{+} \right] \) means the solution is more acidic.
• Low \( \left[ \mathrm{H}^{+} \right] \) means the solution is more basic, or alkaline.

Hydrogen ions come from acids when they are dissolved in water, which enhances the ion concentration and impacts the solution's pH.

Logarithms in Chemistry

Logarithms are a handy mathematical tool used extensively in chemistry to simplify calculations involving large ranges of quantities, like hydrogen ion concentration. In pH calculations, the logarithm is crucial because it helps transform exponential growth or concentration levels into manageable numbers.
The formula for pH, \( \mathrm{pH} = -\log \left[ \mathrm{H}^{+} \right] \), illustrates how logarithms are used to calculate a measure of acidity.
Applying logarithms allows chemists to:

• Convert tiny concentrations (like those of hydrogen ions in dilute solutions) into numbers close to our everyday experience.
• Understand relationships where each unit change in pH reflects a tenfold change in \( \left[ \mathrm{H}^{+} \right] \).

This makes the concept of pH practical and integral in chemistry.

Acidity and Basicity

Acidity and basicity are two fundamental concepts that describe how substances behave in solutions. They are crucial for a variety of applications, from biological systems to industrial processes.
Here’s a simple way to understand them:

• An acidic solution has a high concentration of hydrogen ions, resulting in a low pH (less than 7).
• A basic or alkaline solution has a low concentration of hydrogen ions, resulting in a high pH (greater than 7).
• A neutral solution, like pure water, has a pH of 7.

By using the pH scale, chemists can easily communicate how acidic or basic a solution is. This information is key to predicting reactions and understanding chemical interactions.

Chemistry Education

Chemistry education provides students with a foundational framework for understanding the natural world. Concepts such as pH, hydrogen ion concentration, and the role of logarithms lay the groundwork for more advanced studies.
Here’s why chemistry education is important:

• It enhances problem-solving skills through practical labs and theoretical exercises.
• Promotes critical thinking and analytical skills.
• Helps students make informed decisions about everyday chemical interactions they encounter.

Effective chemistry education not only informs students about scientific principles but also encourages exploration and curiosity about the diverse field of chemistry, laying the groundwork for future scientific endeavors.