Problem 31 - Algebra and Trigonometry

Question

The hydrogen ion concentrations in cheeses range from \(4.0 \times 10^{-7} \mathrm{M}\) to \(1.6 \times 10^{-5} \mathrm{M} .\) Find the corresponding range of \(\mathrm{pH}\) readings.

Step-by-Step Solution

Answer

The pH range is from 4.80 to 6.40.

Step 1: Understand the concept of pH

pH is a measure of the acidity or basicity of a solution. It is calculated using the formula \( ext{pH} = - ext{log}_{10}[ ext{H}^+] \), where \([\text{H}^+]\) is the hydrogen ion concentration.

Step 2: Determine the pH for the lowest concentration

Calculate the pH for the lowest hydrogen ion concentration, \(4.0 \times 10^{-7} \ \mathrm{M}\), using the formula: \( \text{pH} = -\log_{10}(4.0 \times 10^{-7}) \). This simplifies to: \( \text{pH} \approx 6.40 \).

Step 3: Determine the pH for the highest concentration

Calculate the pH for the highest hydrogen ion concentration, \(1.6 \times 10^{-5} \ \mathrm{M}\), using the formula: \( \text{pH} = -\log_{10}(1.6 \times 10^{-5}) \). This simplifies to: \( \text{pH} \approx 4.80 \).

Step 4: Define the range of pH

The pH range spans from the pH of the lowest hydrogen ion concentration to the pH of the highest hydrogen ion concentration. Therefore, the range of pH readings is from 4.80 to 6.40.

Key Concepts

Hydrogen Ion Concentration

Hydrogen ion concentration is a fundamental concept in chemistry that helps us understand the acidity of a solution. It is often denoted by \([\text{H}^+]\), indicating the amount of hydrogen ions present in a given volume of a solution. \([\text{H}^+]\) is measured in moles per liter (M).
Understanding hydrogen ion concentration is crucial because it directly affects the pH of a solution. High concentrations of hydrogen ions correspond to low pH values, making the solution more acidic. Conversely, low concentrations result in higher pH values, indicating a more basic or alkaline solution.
To calculate pH, we utilize the logarithmic formula given by \(\text{pH} = -\log_{10}[\text{H}^+]\). This formula helps translate hydrogen ion concentration into a more understandable scale for evaluating acidity and basicity.

Acidity and Basicity

Acidity and basicity are chemical properties that describe a solution's potential to donate or accept hydrogen ions. When a solution has high hydrogen ion concentration, it is considered acidic, whereas a solution with low hydrogen ion concentration is considered basic.
The acidity or basicity of a solution affects many biological and chemical processes:

• An acidic solution often has a sour taste and can react vigorously with metals.
• A basic solution typically feels slippery and can interact with oils and fats.

These properties are important in various fields, from food science to medicine. For example, the pH of a substance can affect enzyme activity, the shelf life of food products, and even the effectiveness of pharmaceuticals. Understanding the balance of acidity and basicity is key to navigating these applications effectively.

pH Scale

The pH scale is a numerical scale ranging from 0 to 14 used to quantify the acidity or basicity of a solution. It provides a simpler way to express hydrogen ion concentration:

• A pH less than 7 indicates an acidic solution.
• A pH of exactly 7 is considered neutral, like pure water.
• A pH greater than 7 denotes a basic solution.

This scale is logarithmic, meaning each whole number change represents a tenfold change in hydrogen ion concentration. For instance, if a solution has a pH of 3, it's ten times more acidic than a solution with a pH of 4.
The pH scale is essential in many areas of study and practice. It helps researchers and practitioners assess and compare the chemical properties of different solutions easily. In our exercise, determining the pH range of cheese helps producers monitor and control the product's quality, ensuring it remains safe and palatable.