One brand of vinegar has a pH of 4.5. Another brand has a pH of 5.0. The equation for the pH of a substance is [tex]$pH = - \log [H^+]$[/tex], where [tex]$H^+$[/tex] is the concentration of hydrogen ions. What is the approximate difference in the concentration of hydrogen ions between the two brands of vinegar?
A. [tex]$2.2 \times 10^{-5}$[/tex]
B. [tex]$3.2 \times 10^{-1}$[/tex]
C. [tex]$3.2 \times 10^1$[/tex]
D. [tex]$6.8 \times 10^4$[/tex]
Answer (1)
Define the pH values and the formula relating pH to hydrogen ion concentration.
Calculate the hydrogen ion concentrations for both brands of vinegar using the formula [ H + ] = 1 0 − p H .
Find the absolute difference between the two concentrations: Δ [ H + ] = ∣1 0 − 4.5 − 1 0 − 5.0 ∣ .
Approximate the result to match one of the given options: 2.2 × 1 0 − 5 .
Explanation
Understanding the Problem We are given the pH values of two brands of vinegar and the formula relating pH to the concentration of hydrogen ions, p H = − lo g 10 [ H + ] . We need to find the difference in hydrogen ion concentration between the two brands.
Defining Variables Let p H 1 = 4.5 and p H 2 = 5.0 be the pH values of the two brands. Let [ H + ] 1 and [ H + ] 2 be the corresponding hydrogen ion concentrations.
Calculating Hydrogen Ion Concentrations Using the formula p H = − lo g 10 [ H + ] , we can express the hydrogen ion concentrations as:
[ H + ] 1 = 1 0 − p H 1 = 1 0 − 4.5
[ H + ] 2 = 1 0 − p H 2 = 1 0 − 5.0
Finding the Difference The difference in concentration is:
Δ [ H + ] = ∣ [ H + ] 1 − [ H + ] 2 ∣ = ∣1 0 − 4.5 − 1 0 − 5.0 ∣
Calculating the Numerical Difference Calculating the values:
1 0 − 4.5 ≈ 0.00003162
1 0 − 5.0 = 0.00001
Δ [ H + ] = ∣0.00003162 − 0.00001∣ = 0.00002162
Approximating the Result Expressing the result in scientific notation:
0.00002162 = 2.162 × 1 0 − 5
This is approximately 2.2 × 1 0 − 5 .
Final Answer Therefore, the approximate difference in the concentration of hydrogen ions between the two brands of vinegar is 2.2 × 1 0 − 5 .
Examples
Understanding pH differences is crucial in many real-world applications. For example, in agriculture, knowing the pH of the soil helps farmers choose the right crops and fertilizers. In medicine, maintaining the correct pH balance in the body is essential for various biological processes. In environmental science, monitoring the pH of water sources helps assess pollution levels and protect aquatic life. The concept of pH and its logarithmic scale allows us to quantify acidity and alkalinity, enabling informed decisions in diverse fields.
