Problem 30 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The pH is approximately 7.509, classifying the substance as basic.

Step 1: Understand the Problem

We are given a hydrogen ion concentration \([\mathrm{H}^{+}] = 3.1 \times 10^{-8} \mathrm{M}\) and need to calculate the pH to classify the substance as acidic or basic.

Step 2: Learn the pH Formula

The pH of a solution is calculated using the formula \[ \text{pH} = -\log_{10}[\mathrm{H}^{+}] \] where \([\mathrm{H}^{+}]\) is the hydrogen ion concentration.

Step 3: Calculate the pH

Substitute the given hydrogen ion concentration into the pH formula: \[ \text{pH} = -\log_{10}(3.1 \times 10^{-8}) \] Use a calculator to find the logarithm. Since \(-\log_{10}(3.1) \approx -0.491\), the calculation becomes \(-0.491 + 8 = 7.509\). Thus, the pH is approximately 7.509.

Step 4: Classify the Substance

Compare the calculated pH to the neutral pH of 7. Since 7.509 is greater than 7, the substance is classified as basic.

Key Concepts

pH calculation

The pH of a substance is a measure of its acidity or basicity, expressed on a logarithmic scale. It is calculated using the formula \( \text{pH} = -\log_{10}[\mathrm{H}^{+}] \), where \([\mathrm{H}^{+}]\) represents the concentration of hydrogen ions in moles per liter (M). This equation signifies how pH is inversely related to the hydrogen ion concentration.

In our exercise, we are given a hydrogen ion concentration of \(3.1 \times 10^{-8} \mathrm{M}\). To find the pH, we plug this value into the formula. The logarithmic calculation is done as follows:

• Calculate \(-\log_{10}(3.1)\), which is approximately \(-0.491\).
• Add this value to 8 (since \(10^{-8}\) is involved), resulting in a pH of approximately 7.509.

Logarithms often involve a calculator for accuracy, as estimating can be complex.

In essence, pH calculation using this formula helps us determine the level of acidity or basicity of a solution by converting the concentration of hydrogen ions into a more manageable number on a scale from 0 to 14.

hydrogen ion concentration

Hydrogen ion concentration \([\mathrm{H}^{+}]\) is a crucial measure in determining the acidity or basicity of a substance. It is expressed in moles per liter and signifies the amount of free hydrogen ions present in a solution.

The formula used to calculate pH, \( \text{pH} = -\log_{10}[\mathrm{H}^{+}] \), directly relies on this concentration value. A higher hydrogen ion concentration means an acidic solution, while a lower concentration indicates a basic solution.

In this particular exercise, the given hydrogen ion concentration is \(3.1 \times 10^{-8} \mathrm{M}\). This very small number reflects a low concentration of hydrogen ions, which suggests the solution is likely to be basic due to the sparse presence of these ions.

Understanding hydrogen ion concentration is vital, as it bridges the gap between measurable chemical amounts and the intuitive pH scale that indicates how a solution will behave chemically. Especially in contexts such as chemistry and biochemistry, these calculations form the backbone of many analyses and decisions.

acidic or basic classification

Classifying substances as acidic or basic is a fundamental concept in chemistry. The pH scale, ranging from 0 to 14, provides a straightforward method to make this classification:

• A pH less than 7 indicates an acidic solution.
• A pH of exactly 7 is considered neutral (pure water).
• A pH greater than 7 indicates a basic (alkaline) solution.

In this exercise, after calculating the pH as approximately 7.509, we compare it to these benchmarks. Since 7.509 is greater than 7, the substance is classified as basic, meaning it has more hydroxide ions \((\mathrm{OH}^{-})\) than hydrogen ions \((\mathrm{H}^{+})\).

Understanding this classification is essential since the pH of a solution influences its chemical properties, reactivity, and biological interactions. For instance, everyday substances like lemon juice (acidic) or baking soda (basic) showcase the diversity of pH in everyday materials. This knowledge also guides safe handling practices and predicts possible reactions in various scientific fields.