Problem 34 - Precalculus Mathematics for Calculus

Question

These exercises deal with logarithmic scales. The pH readings for wines vary from 2.8 to 3.8. Find the corresponding range of hydrogen ion concentrations.

Step-by-Step Solution

Answer

The range of hydrogen ion concentrations is from \(1.58 \times 10^{-3}\, \text{mol/L}\) to \(1.58 \times 10^{-4}\, \text{mol/L}\).

Step 1: Understanding pH and Hydrogen Ion Concentration

The pH scale is logarithmic and measures the concentration of hydrogen ions (H+) in a solution. The formula relating pH and hydrogen ion concentration is given by: \[ \text{pH} = -\log[H^+] \]. To find the hydrogen ion concentration from pH, we rearrange the formula to \[ [H^+] = 10^{-\text{pH}} \].

Step 2: Calculate Hydrogen Ion Concentration for pH 2.8

Using the formula \([H^+] = 10^{-\text{pH}}\), substitute pH = 2.8 into the equation: \([H^+] = 10^{-2.8}\). Calculate this value: \([H^+] \approx 1.58 \times 10^{-3}\, \text{mol/L}\).

Step 3: Calculate Hydrogen Ion Concentration for pH 3.8

Similarly, use the formula \([H^+] = 10^{-\text{pH}}\) and substitute pH = 3.8: \([H^+] = 10^{-3.8}\). Calculate this concentration: \([H^+] \approx 1.58 \times 10^{-4}\, \text{mol/L}\).

Step 4: Determine Range of Hydrogen Ion Concentrations

Now that we have calculated hydrogen ion concentrations for pH 2.8 and pH 3.8, the range of hydrogen ion concentrations is from approximately \(1.58 \times 10^{-3}\, \text{mol/L}\) to \(1.58 \times 10^{-4}\, \text{mol/L}\).

Key Concepts

Understanding the pH Scale

The pH scale is a vital part of understanding acidity and basicity in solutions, particularly in chemistry and environmental sciences. It is a logarithmic scale used to specify the acidity or basicity of an aqueous solution. The calculation is based on the concentration of hydrogen ions (H+) in the solution. A lower pH value signifies a higher concentration of hydrogen ions, which correlates to greater acidity, while a higher pH value indicates lower acidity (more basic).
For context, the pH scale usually ranges from 0 to 14, where 7 is neutral; values below 7 denote acidic solutions, and values above 7 signify basic solutions. In practical terms, each pH unit change represents a tenfold change in hydrogen ion concentration.
For example, a wine with a pH of 2.8 is ten times more acidic than a solution with a pH of 3.8, which explains why the calculation of hydrogen ion concentration varies exponentially across seemingly small shifts in pH values.

Hydrogen Ion Concentration and its Calculations

Understanding hydrogen ion concentration is crucial for assessing the strength of acids and bases. The concentration of hydrogen ions is represented in moles per liter (mol/L), and it determines how acidic or basic a solution is.
By using the relationship between pH and hydrogen ion concentration, \[ [H^+] = 10^{-\text{pH}} \], we can convert pH values into concrete measures of hydrogen ion concentration. This is essential, especially in industries like winemaking, where precision in acidity directly influences product quality.
For example, finding hydrogen ion concentrations for pH 2.8 involves computing: \[ [H^+] = 10^{-2.8} \] which equals approximately \(1.58 \times 10^{-3}\, \text{mol/L}\). Similarly, for pH 3.8, \[ [H^+] = 10^{-3.8} \] provides a concentration of about \(1.58 \times 10^{-4}\, \text{mol/L}\).
Therefore, precise calculations of hydrogen ion concentrations can help create products with desired taste profiles, safety standards, and other qualitative features.

The Role of Logarithms in Scientific Measurements

Logarithms play a crucial role in scientific calculations, making it easier to handle extremely large or small numbers. They are particularly useful in chemistry, where the effects of concentration changes can be massive, even with small numeric shifts.
The relationship between pH and hydrogen ion concentration is a prime example of logarithmic application. The formula, \[ \text{pH} = -\log[H^+] \], directly uses logarithms to translate hydrogen ion concentrations into more manageable pH values. Using logarithms simplifies complex multiplicative relations into additive ones, making them easier to understand and compute.
This transformation is powerful because it allows us to intuitively perceive the acidity or basicity of a solution through pH values, which would be incredibly cumbersome if expressed directly as hydrogen ion concentrations. Thus, logarithms are indispensable tools in the scientific community for scaling data accurately and efficiently.