Problem 28 - Algebra and Trigonometry

Question

An unknown substance has a hydrogen ion concentration of \(\left[\mathrm{H}^{+}\right]=3.1 \times 10^{-8} \mathrm{M} .\) Find the pH and classify the substance as acidic or basic.

Step-by-Step Solution

Answer

The pH is approximately 7.51; the substance is basic.

Step 1: Understanding pH Formula

The formula to find pH is given by \( \text{pH} = - \log_{10} \left[\text{H}^+\right] \), where \( \left[\text{H}^+\right] \) is the concentration of hydrogen ions.

Step 2: Substituting Hydrogen Ion Concentration

Substitute the value of \( \left[\text{H}^+\right] = 3.1 \times 10^{-8} \) M into the pH formula: \( \text{pH} = - \log_{10}(3.1 \times 10^{-8}) \).

Step 3: Calculating the pH Value

Using a calculator, compute \( - \log_{10}(3.1 \times 10^{-8}) \). This results in \( \text{pH} \approx 7.51 \).

Step 4: Classifying the Substance

A pH value less than 7 indicates an acidic substance, a pH value greater than 7 indicates a basic substance, and a pH of exactly 7 indicates a neutral substance. Since the pH is approximately 7.51, the substance is classified as basic.

Key Concepts

Hydrogen Ion Concentration

Hydrogen ion concentration refers to the number of hydrogen ions present in a solution, commonly represented in moles per liter (M). This concentration is crucial in determining the acidity or basicity of a substance.
The concentration directly influences the pH level, with a higher concentration accounting for a lower pH, indicating a more acidic solution, and a lower concentration leading to a higher pH, indicating a more basic solution.
For instance, if a solution has a hydrogen ion concentration of \(3.1 \times 10^{-8}\ M\), it suggests the presence of very few hydrogen ions. This value will typically yield a pH above 7, pointing to a basic or alkaline nature.
The hydrogen ions (\(\text{H}^+\)) interact with other molecules and influence the chemical properties of the substance, making their concentration key to understanding chemical behaviors.

Logarithmic Function

The logarithmic function is a mathematical tool that simplifies the calculation of pH from hydrogen ion concentration.
In pH calculations, the formula involves a negative logarithm, represented as \( \text{pH} = - \log_{10} \left[\text{H}^+\right] \).
This method transforms multiplicative relationships into additive ones, simplifying the process of handling concentration values, especially those involving scientific notation.
When dealing with concentrations like \(3.1 \times 10^{-8}\ M\), the logarithmic function efficiently computes the pH by converting this small number into a more manageable form.

• The negative sign shows the inverse relationship between hydrogen ion concentration and pH level.
• Logarithmic functions compress wide-ranging numbers, providing a straightforward scale to compare acidity and basicity.

Hence, using the logarithmic approach, the pH reflects the acidity level without overwhelming decimal values.

Acidic and Basic Substances

Acidic and basic substances are classified based on their pH levels, which stem from their hydrogen ion concentration.
Typically, substances with pH levels less than 7 are considered acidic, while those with pH levels greater than 7 are considered basic. A pH of exactly 7 is labeled neutral, as seen with pure water.
Acidic substances, having high hydrogen ion concentrations, may include vinegar or citric acid. These solutions often taste sour and can conduct electricity due to the free ions present.

• Acids have more \(\text{H}^+\) ions, making them reactive and capable of donating protons.
• Basic substances possess fewer \(\text{H}^+\) ions but more \(\text{OH}^-\) ions. Common examples include baking soda and soap, often slipping smoothly between fingers due to saponification.

To classify an unknown substance, like the one with a pH of approximately 7.51, observe the pH value: above 7 indicates it is basic, corroborating the influence of a low hydrogen ion concentration.