Problem 98 - 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 orange has an approximate hydrogen ion concentration of \(10^{-4.2} .\) Calculate its pH value.

Step-by-Step Solution

Answer

The pH of the orange is 4.2, indicating it is acidic.

Step 1: Understanding the Formula

The pH formula is given by \[ \text{pH} = -\log_{10}[H^+] \] where \([H^+]\) is the hydrogen ion concentration. We need to find pH using this formula.

Step 2: Substituting the Given Concentration

We know the hydrogen ion concentration for the orange is \[ [H^+] = 10^{-4.2} \] Substituting this concentration into the pH formula yields:\[ \text{pH} = -\log_{10}(10^{-4.2}) \]

Step 3: Applying the Logarithm Rules

Using the property of logarithms,\[ -\log_{10}(10^{x}) = x \]we can simplify \(-\log_{10}(10^{-4.2})\) to get\[ \text{pH} = 4.2 \]

Step 4: Classification of the pH Value

A pH value of 4.2 indicates that an orange is acidic because it falls between 0 and 7. Thus, oranges are neither neutral nor basic.

Key Concepts

Hydrogen Ion Concentration

The hydrogen ion concentration is a vital part of understanding the pH of a solution. It is represented as \([H^+]\), indicating the amount of hydrogen ions present in a given volume, typically measured in moles per liter. This concentration gives us insight into how acidic or basic a solution is.

When a solution has a high concentration of hydrogen ions, it is considered acidic. Conversely, a low concentration implies a basic nature. For instance, the hydrogen ion concentration in an orange is approximately \(10^{-4.2}\). This value helps us determine that the solution leans towards the acidic side. The concentration itself is pivotal because each increment or decrement affects the pH calculation significantly. Hence, understanding \([H^+]\) is crucial for any pH-related calculations.

Logarithmic Scale

The pH scale operates on a logarithmic scale, a concept essential for quantifying acidity and basicity. A logarithmic scale means each unit on the scale is a power of ten. This characteristic is evident in the pH formula: \[ \text{pH} = -\log_{10}[H^+] \]

This formula demonstrates that even small changes in hydrogen ion concentration can lead to significant changes in pH. For example, a concentration of \(10^{-4.2}\) results in a pH of 4.2—indicating acidity. The logarithmic nature makes the pH scale deeply effective for comparing different concentrations:

• A concentration of \(10^{-3}\) produces a pH of 3.
• A concentration of \(10^{-4}\) results in a pH of 4.

These differences illustrate why small changes on a logarithmic scale correlate to more noticeable shifts in perception of acidity or basicity.

Acidity and Basicity

Acidity and basicity describe where a substance falls on the pH scale between acidic, neutral, and basic. Substances are assigned a pH value based on their hydrogen ion concentration, and consequently, classified as either acidic, basic, or neutral.

• Acidic solutions have a pH less than 7.
• Basic solutions, or alkaline, have a pH greater than 7.
• Neutral solutions have a pH of exactly 7, like pure water.

The pH value of 4.2 for an orange clearly falls in the acidic range. Thus, the orange's acidic nature aligns with its taste. Understanding acidity and basicity is central to predicting the behavior of solutions in various chemical and biological contexts. It's also essential in everyday scenarios, from agriculture to food sciences, where knowing the pH can affect processes and outcomes.