Problem 97 - 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. An apple has an approximate hydrogen ion concentration of \(10^{-3.6} .\) Calculate its pH value.

Step-by-Step Solution

Answer

The pH value of the apple is 3.6, indicating it is acidic.

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 molar concentration of hydrogen ions in the solution.

Step 2: Identify the hydrogen ion concentration

We are given that an apple has an approximate hydrogen ion concentration of \( 10^{-3.6} \).

Step 3: Apply the pH formula

Substitute the given hydrogen ion concentration into the pH formula: \( \mathrm{pH} = -\log_{10}(10^{-3.6}) \).

Step 4: Simplify using logarithm properties

Using the property of logarithms that states \( -\log_{10}(10^{x}) = x \), we calculate \( \mathrm{pH} = 3.6 \).

Key Concepts

Understanding Logarithms

Logarithms play a vital role in calculating the pH of a solution. They help us translate large and small numbers into a more manageable scale. A logarithm in simple terms represents the power to which a number must be raised to get another number. Specifically, the base-10 logarithm (commonly used in pH calculations) answers the question: "10 raised to which power equals this number?"
For instance, the logarithm of 1000 is 3 because 10 raised to the power of 3 gives 1000: \( \log_{10}(1000) = 3 \). When calculating pH, we often deal with negative exponents related to the concentration of hydrogen ions, which can be tiny decimal numbers or even fractions, like the example of an apple's hydrogen ion concentration \( 10^{-3.6} \).
Through logarithms, these concentrations are transformed into simple pH values, making it more intuitive to understand the chemical behavior of the solutions. Properties of logarithms, such as \( \log_{10}(10^{x}) = x \), simplify calculations significantly by allowing us to directly pull out the exponent as the pH value.

Hydrogen Ion Concentration

Hydrogen ion concentration indicates the quantity of \( H^{+} \) ions present in a solution, measured in moles per liter. The concentration of these ions is crucial because they determine the solution's acidity or basicity. In more acidic solutions, there is a higher concentration of \( H^{+} \) ions. Conversely, basic solutions have a lower concentration of these ions.
Measurement of \( H^{+} \) is often done using scientific notation, which allows for representation of very small numbers efficiently. For example, a hydrogen ion concentration of \( 10^{-3.6} \) as found in an apple means we have \( 0.0002511 \) moles of \( H^{+} \) ions per liter, denoting more acidity.

• The more negative the exponent in the hydrogen ion concentration, the more acidic the solution.
• The fewer \( H^{+} \) ions, the more basic (or alkaline) the solution becomes.
• Neutral solutions, like pure water, have a balanced \( H^{+} \) concentration expressed as \( 10^{-7} \).

Acidity and Basicity

Acidity and basicity of a solution are determined by the pH scale, which ranges from 0 to 14. This scale is a practical way to express the concentration of hydrogen ions in a given solution.
- Solutions with a pH less than 7 are considered acidic. They have a higher \( H^{+} \) concentration.
- A neutral solution has a pH of 7, meaning it has equal amounts of hydrogen and hydroxide ions, like pure water.
- Solutions with a pH greater than 7 are classified as basic (or alkaline), meaning they have a lower \( H^{+} \) concentration.

Let's break it down further:

• "Very acidic" solutions can have pH values approaching 0 or 1, signaling very high acidity with high \( H^{+} \) ion concentration. Examples would be battery acid or lemon juice.
• "Basic" or "alkaline" solutions often have pH values that are between 7 and 14. Well-known examples include household bleach and soap.
• On this scale, slight changes in pH can indicate significant changes in \( H^{+} \) ion concentration, reflecting different chemical environments.

Understanding these distinctions helps us predict and explain chemical reactions and their impacts in various environments, be it in nature or our bodies.