Problem 96 - Precalculus

Question

The pH of a solution is a measure of the molar concentration of hydrogen ions, \(H^{+},\) in moles per liter, in the solution, which means that it is a measure of the acidity or basicity of the solution. The letters pH stand for "power of hydrogen," and the numerical value is defined as $$\mathrm{pH}=-\log _{10}\left[H^{+}\right]$$ Very acid corresponds to pH values near \(1,\) neutral corresponds to a pH near 7 (pure water), and very basic corresponds to values near \(14 .\) In the next six exercises you will be asked to calculate the pH value of wine, Pepto- Bismol, normal rainwater, bleach, and fruit. List these six liquids and use your intuition to classify them as neutral, acidic, very acidic, basic, or very basic before you calculate their actual pH values. Bleach has an approximate hydrogen ion concentration of \(5.0 \times 10^{-13} .\) Calculate its pH value.

Step-by-Step Solution

Answer

Bleach has a pH of approximately 12.301, making it very basic.

Step 1: Understand the pH Formula

The pH of a solution is calculated using the formula \(\mathrm{pH} = -\log_{10}[H^+]\), where \([H^+]\) is the hydrogen ion concentration in mol/L. In this problem, the concentration of hydrogen ions for bleach is given as \(5.0 \times 10^{-13}\) mol/L.

Step 2: Apply the pH Formula

Substitute the given hydrogen ion concentration into the pH formula: \[\mathrm{pH} = -\log_{10}(5.0 \times 10^{-13})\].

Step 3: Calculate the pH Value

Use logarithmic properties to evaluate \(-\log_{10}(5.0 \times 10^{-13})\). Using a calculator, compute \(\log_{10}(5.0)\) and add \(13\) (the exponent): \[\mathrm{pH} = -(-12.301) \approx 12.301\].

Step 4: Determine the Classification

With a pH of approximately 12.301, bleach is classified as very basic since typical very basic solutions have pH values closer to 14.

Key Concepts

Hydrogen Ion Concentration

The term "hydrogen ion concentration" refers to the number of hydrogen ions (\( H^+ \) ions) present in a solution. This concentration is usually expressed in moles per liter (mol/L). It is a key factor in determining the acidity or basicity of a solution.
- A high concentration of hydrogen ions typically means that the solution is acidic. - A low concentration suggests that the solution is basic or alkaline.
In the context of pH calculation, knowing the concentration of these ions helps us categorize substances along the acidity-basicity spectrum.

Logarithmic Properties

Understanding logarithmic properties is essential when calculating pH values. The pH scale is logarithmic, meaning that each whole pH value below or above 7 indicates a tenfold change in hydrogen ion concentration. The formula used for calculating pH is: \[ \mathrm{pH} = -\log_{10}[H^+] \] - The negative sign in the formula indicates that as the hydrogen ion concentration increases, the pH value decreases.
- For example, if the hydrogen ion concentration changes from \( [10^{-3}] \) to \( [10^{-4}] \), the pH changes from 3 to 4. Logarithms simplify handling very large or very small numbers, which is why they are used in pH calculations.

Solution Acidity and Basicity

Solution acidity and basicity are critical concepts defined by the pH scale. - Solutions with a pH less than 7 are called acidic. These solutions have higher concentrations of hydrogen ions. - Solutions with a pH greater than 7 are considered basic or alkaline, indicating lower concentrations of hydrogen ions.
The neutral point of 7 on the pH scale is typically represented by pure water.
The pH scale is a measure that helps us understand how acidic or basic a given solution is, considering its hydrogen ion concentration.

Classification of Substances by pH

Substances can be classified based on their pH values into various categories: - **Very Acidic:** Solutions with pH values from 0 to 3, e.g., battery acid or stomach acid.
- **Acidic:** Solutions with pH values from 3 to 6, like wine or vinegar.
- **Neutral:** Solutions with a pH of around 7, such as pure water.
- **Basic:** Solutions with a pH between 8 and 11, like baking soda in water.
- **Very Basic:** Solutions with pH values of 11 to 14, such as bleach.
Each category helps quickly determine a substance's potential effects and suitability for various applications, making comprehension of the pH scale vital for everyday and scientific contexts.