Problem 94 - 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. Pepto-Bismol has a hydrogen ion concentration of about \(5.01 \times 10^{-11} .\) Calculate its pH value.

Step-by-Step Solution

Answer

Pepto-Bismol has a pH of 10.3010 and is basic.

Step 1: Understanding the pH formula

The formula to calculate the pH of a solution is \( \mathrm{pH} = -\log_{10}\left[H^{+}\right] \). Our task is to apply this to the given hydrogen ion concentration for Pepto-Bismol, which is \(5.01 \times 10^{-11}\).

Step 2: Substitute the hydrogen ion concentration

We substitute the hydrogen ion concentration into the pH formula: \( \mathrm{pH} = -\log_{10}\left(5.01 \times 10^{-11}\right) \).

Step 3: Calculate the logarithm

Use a calculator to compute the logarithm: \( \log_{10}(5.01 \times 10^{-11}) = \log_{10}(5.01) + \log_{10}(10^{-11}) = 0.6990 - 11 = -10.3010 \).

Step 4: Calculate the pH value

Calculate \( \mathrm{pH} = -(-10.3010) = 10.3010 \).

Step 5: Classify Pepto-Bismol based on pH

Since a pH of 10.3010 is greater than 7, Pepto-Bismol is classified as basic.

Key Concepts

Hydrogen Ion Concentration

The hydrogen ion concentration, often expressed as \([H^{+}]\), is a crucial component in determining the pH level of a solution. It denotes the amount of hydrogen ions present in a solution, which directly influences its acidity or basicity.

• A high concentration of hydrogen ions means the solution is more acidic.
• A low concentration of hydrogen ions indicates the solution is more basic.

For example, in the case of Pepto-Bismol with a hydrogen ion concentration of \(5.01 \times 10^{-11}\), this very low concentration suggests that Pepto-Bismol is basic. In acidic solutions, the presence of more free hydrogen ions (\(H^{+}\)) typically leads to increased acidity, which is exhibited by a lower pH value.

Logarithms

Logarithms are a mathematical concept that helps in calculating values that span several orders of magnitude, such as the pH of solutions. When determining the pH, the logarithmic scale is particularly useful because it allows us to represent very large or very small numbers with ease.

• The pH formula \(\mathrm{pH} = -\log_{10}[H^{+}]\) utilizes the base-10 logarithm.
• The negative sign indicates an inverse relationship between pH and \([H^{+}]\); as the hydrogen ion concentration increases, the pH decreases.

Taking the logarithm of the hydrogen ion concentration simplifies calculations, letting us express the wide range of solute concentrations in a more manageable form. When you compute \(\log_{10}(5.01 \times 10^{-11})\), as seen in the exercise for Pepto-Bismol, you break down the expression into manageable parts: \([\log_{10}(5.01) + \log_{10}(10^{-11})\). This method helps obtain the pH efficiently.

Acidic and Basic Solutions

Solutions are classified as acidic, neutral, or basic based on their pH level:

• Acidic solutions have a pH less than 7.
• Neutral solutions have a pH of approximately 7.
• Basic, or alkaline, solutions have a pH greater than 7.

Pepto-Bismol, with a pH calculated to be 10.3010, is thus considered basic because its pH exceeds 7. The scale is logarithmic; therefore, each whole pH value below 7 is ten times more acidic than the next higher value. This helps explain why substances such as lemon juice, with a pH around 2-3, are very acidic compared to Pepto-Bismol.
Understanding the pH scale and the role of hydrogen ion concentration is essential in classifying solutions in our daily lives, from the drinks we consume to the household cleaning products we use.