Problem 95 - 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. Normal rainwater is slightly acidic and has an approximate hydrogen ion concentration of \(10^{-5.6} .\) Calculate its pH value. Acid rain and tomato juice have similar approximate hydrogen ion concentrations of \(10^{-4} .\) Calculate the pH value of acid rain and tomato juice.

Step-by-Step Solution

Answer

Normal rainwater pH: 5.6, Acid rain pH: 4, Tomato juice pH: 4.

Step 1: Understand the pH Formula

The formula for calculating the pH of a solution is given by \( \mathrm{pH} = -\log_{10} \left[H^{+}\right] \). This formula uses the concentration of hydrogen ions \( \left[H^{+}\right] \) to find the pH value.

Step 2: Calculate pH of Normal Rainwater

Normal rainwater has a hydrogen ion concentration of \(10^{-5.6}\). Using the pH formula, we calculate its pH as follows:\[ \mathrm{pH} = -\log_{10}(10^{-5.6}) = 5.6 \] Normal rainwater has a pH of 5.6, indicating it is slightly acidic.

Step 3: Calculate pH of Acid Rain

Acid rain has a hydrogen ion concentration of \(10^{-4}\). Calculate its pH using the formula:\[ \mathrm{pH} = -\log_{10}(10^{-4}) = 4 \] Acid rain has a pH of 4, indicating it is acidic.

Step 4: Calculate pH of Tomato Juice

Tomato juice has the same hydrogen ion concentration as acid rain, \(10^{-4}\). Therefore, its pH is calculated in the same way:\[ \mathrm{pH} = -\log_{10}(10^{-4}) = 4 \] Tomato juice has a pH of 4, indicating it is also acidic.

Key Concepts

Hydrogen Ion Concentration

The concept of hydrogen ion concentration is central to understanding pH.It refers to the amount of hydrogen ions (\( H^+ \)) present in a solution.This concentration is typically expressed in moles per liter.In other words, it indicates how many hydrogen ions are in one liter of the solution.

The concentration of hydrogen ions determines the solution's acidity or basicity.A higher concentration of hydrogen ions signifies a more acidic solution.Conversely, a lower concentration indicates a more basic solution.

When we say a solution has a hydrogen ion concentration of \(10^{-5.6}\)or \(10^{-4}\), we are expressing the concentration in scientific notation.This helps in handling very small numbers more easily.For example:

• Normal rainwater, with a hydrogen ion concentration of \(10^{-5.6}\), indicates it is slightly acidic.
• Acid rain and tomato juice, with a hydrogen ion concentration of \(10^{-4}\), are more acidic.

Recognizing these concentrations is key to understanding the pH scale.

Acidity and Basicity

Acidity and basicity are terms that describe the chemical nature of a solution.They are determined by the concentration of hydrogen ions (\( H^+ \)) present.

A solution's acidity is measured on the pH scale:

• Solutions with a pH less than 7 are considered acidic.
• Solutions with a pH exactly equal to 7 are neutral (like pure water).
• Solutions with a pH greater than 7 are basic (or alkaline).

The pH scale ranges from 0 to 14.

Acidic solutions have a higher concentration of hydrogen ions, while basic solutions have a lower concentration. In our examples:

• Normal rainwater has a pH of 5.6, making it slightly acidic.
• Acid rain and tomato juice have a pH of 4, showing they are more acidic.

Understanding how the pH scale reflects acidity and basicity is crucial for recognizing the behavior of different solutions in chemistry and everyday life.

Logarithms in Chemistry

Logarithms play an essential role in chemistry, especially in calculating pH.The pH of a solution is defined as the negative base-10 logarithm of its hydrogen ion concentration.The formula:\[ \mathrm{pH} = -\log_{10} \left[H^{+}\right] \]might seem complex, but it simplifies the way we express very small concentrations.

Using logarithms, we transform multiplicative relationships into additive ones, making it easier to interpret changes in concentration.For example, a tenfold change in hydrogen ion concentration results in a 1-unit change in pH.

Here's a breakdown:

• If \([H^+] = 10^{-5.6}\), the pH is 5.6, indicating slight acidity.
• If \([H^+] = 10^{-4}\), the pH is 4, indicating a higher acidity.

Logarithms help communicate information about the acidity or basicity of a solution more effectively.Remember, each unit decrease in pH corresponds to a tenfold increase in hydrogen ion concentration in the solution!