Problem 3 - Campbell Biology

Question

Measurements show that the \(\mathrm{pH}\) of a particular lake is 4.0 . What is the hydrogen ion concentration of the lake? (A) \(4.0 M\) (C) \(10^{-4} M\) (B) \(10^{-10} M\) (D) \(10^{4} M\)

Step-by-Step Solution

Answer

(C) \(10^{-4} \text{ M} \)

Step 1: Understand the relationship between \( \text{pH} \) and hydrogen ion concentration

The \( \text{pH} \) of a solution is defined as \( \text{pH} = -\text{log}[H^+] \), where \( [H^+] \) denotes the hydrogen ion concentration in moles per liter.

Step 2: Apply the pH formula

Given \( \text{pH} = 4.0 \), substitute this value into the pH formula: \( 4.0 = -\text{log}[H^+] \).

Step 3: Solve for hydrogen ion concentration \( [H^+] \)

To find \( [H^+] \), rewrite the equation: \( -\text{log}[H^+] = 4.0 \). This can be rearranged to \( \text{log}[H^+] = -4.0 \). To solve for \( [H^+] \), take the antilog (base 10) of both sides: \( [H^+] = 10^{-4} \text{ M} \).

Step 4: Select the correct answer

Compare the calculated hydrogen ion concentration with the given choices: (A) \(4.0 \text{ M} \) (B) \(10^{-10} \text{ M} \) (C) \(\boldsymbol{10^{-4} \text{ M}} \) (D) \(10^{4} \text{ M} \). The correct answer is (C) \(10^{-4} \text{ M} \).

Key Concepts

acid-base chemistry

Acid-base chemistry is an essential area of chemistry that helps explain the behavior of acids and bases. Acids are substances that increase the hydrogen ion concentration \(\text{[H}^+]\) of a solution. Bases, in contrast, reduce \(\text{[H}^+]\) by increasing the concentration of hydroxide ions \(\text{[OH}^-]\).
The pH scale is used to measure the acidity or basicity of a solution. The scale ranges from 0 to 14, where values less than 7 represent acidic solutions, and values greater than 7 represent basic or alkaline solutions. A pH of 7 is neutral, like pure water.
Understanding the properties of acids and bases helps us in many applications, from industrial processes to environmental science.

pH calculation

Calculating the pH of a solution is a fundamental skill in chemistry. The formula used is:
\[ \text{pH} = -\text{log}[\text{H}^+] \]
This means that the pH is the negative logarithm (base 10) of the hydrogen ion concentration. If you know the \(\text{[H}^+]\), you can use this formula to find the pH. For example, if the \(\text{[H}^+]\) of a solution is \(1 \times 10^{-4} \text{ M}\), its pH would be calculated as follows:
\[ \text{pH} = -\text{log}(1 \times 10^{-4}) = 4 \]
This tells us that the solution is acidic because the pH is less than 7. Similarly, you can reverse the calculation to find \(\text{[H}^+]\) from the pH.

solution concentration

The concentration of a solution refers to the amount of solute that is dissolved in a solvent. In acid-base chemistry, it's crucial to know the concentration of hydrogen ions \(\text{[H}^+]\) to understand the solution's properties.
Concentration is usually expressed in moles per liter (Molarity, M). For the given problem, if the pH is known, the hydrogen ion concentration can be determined.
For example, if a lake has a pH of 4.0, the hydrogen ion concentration is calculated as:
Given \(\text{pH} = 4.0\)
\[ 4.0 = -\text{log}[\text{H}^+] \]
Rearranging the equation to solve for \(\text{[H}^+]\):
\[ \text{log}[\text{H}^+] = -4.0 \]
Taking the antilog of both sides:
\[ [\text{H}^+] = 10^{-4} \text{ M} \]
This concentration tells us that for every liter of the lake water, there are \(10^{-4}\) moles of hydrogen ions present, confirming its acidic nature.