Problem 48 - 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 hydrogen ion concentration \(\left[\mathrm{H}^{+}\right]\)of each substance. (a) apples: \(\mathrm{pH} \approx 3.0\) (b) beer: \(\mathrm{pH} \approx 4.2\) (c) milk: \(\mathrm{pH} \approx 6.6\)

Step-by-Step Solution

Answer

Apples: \(1 \times 10^{-3}\), Beer: \(6.3 \times 10^{-5}\), Milk: \(2.5 \times 10^{-7}\) moles per liter.

Step 1: Understand the pH formula

The pH of a solution is defined as \( \mathrm{pH} = -\log [\mathrm{H}^+] \). The task is to find the hydrogen ion concentration \( [\mathrm{H}^+] \) given the pH.

Step 2: Rearrange the formula to solve for hydrogen ion concentration

To find \( [\mathrm{H}^+] \), rearrange the formula: \( [\mathrm{H}^+] = 10^{-\mathrm{pH}} \). This equation allows us to calculate the concentration directly from the pH value.

Step 3: Calculate \( [\mathrm{H}^+] \) for apples

For apples, with \( \mathrm{pH} \approx 3.0 \), use the equation: \( [\mathrm{H}^+] = 10^{-3.0} = 1 \times 10^{-3} \) moles per liter.

Step 4: Calculate \( [\mathrm{H}^+] \) for beer

For beer, with \( \mathrm{pH} \approx 4.2 \), use the equation: \( [\mathrm{H}^+] = 10^{-4.2} \approx 6.3 \times 10^{-5} \) moles per liter.

Step 5: Calculate \( [\mathrm{H}^+] \) for milk

For milk, with \( \mathrm{pH} \approx 6.6 \), use the equation: \( [\mathrm{H}^+] = 10^{-6.6} \approx 2.5 \times 10^{-7} \) moles per liter.

Key Concepts

Hydrogen Ion Concentration

In chemistry, hydrogen ion concentration \([\mathrm{H}^+]\) provides a way to quantify how acidic or basic a solution is. It is expressed in moles per liter, which indicates the number of hydrogen ions present in a given volume of solution.
The hydrogen ion concentration is crucial because it directly influences a solution's pH. The pH scale is a measure used to specify the acidity or basicity of a solution, where each whole number change represents a tenfold change in hydrogen ion concentration.
For instance, a solution with a hydrogen ion concentration of \(1 \, \times \, 10^{-3}\) moles per liter has a higher acidity than one with a concentration of \(1 \, \times \, 10^{-6}\). This makes it important for determining the characteristics of various substances, such as food items or beverages.

Logarithms in Chemistry

Logarithms, particularly the base-10 logarithm or common log, play a vital role in chemistry. When determining pH, we use the formula: \(\mathrm{pH} = -\log [\mathrm{H}^+] \). This formula uses logarithms to convert hydrogen ion concentration into a more manageable scale.
When we take the log of a number, we determine the power to which the base (in this case, 10) must be raised to produce that number. Hence, the negative logarithm of the hydrogen ion concentration directly gives us the pH value. This log transformation turns incredibly small numbers, like \(1 \, \times \, 10^{-3}\), into more intuitive numbers, aiding our understanding of a substance's acidity or basicity.
In practice, if apples have a pH of approximately 3.0, their hydrogen ion concentration is about \(10^{-3.0} = 1 \, \times \, 10^{-3}\) moles per liter. This illustrates how logarithms can scale down intricate calculations to manageable figures.

Acidic and Basic Solutions

Acidic and basic solutions are fundamental concepts in chemistry, indicating a substance’s position on the pH scale. Solutions are categorized based on their pH, which ranges from 0 to 14.

• A pH less than 7 indicates an acidic solution.
• A pH equal to 7 is considered neutral, like pure water.
• A pH greater than 7 signifies a basic or alkaline solution.

The pH value of a solution not only describes its acidity or basicity but also affects its chemical reactivity, taste, solubility, and many other properties.
For example, common substances include apples with a pH of around 3.0, indicating high acidity. In contrast, milk with a pH of about 6.6 is slightly acidic but closer to neutral, which is why it tastes more neutral compared to apples. By understanding the pH scale, we can better predict how substances will behave in different chemical contexts.