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]$pH =$ $-\log \left[ H ^{+}\right]$[/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?
[tex]$2.2 \times 10^{-5}$[/tex]
[tex]$3.2 \times 10^{-1}$[/tex]
[tex]$3.2 \times 10^1$[/tex]
[tex]$6.8 \times 10^4$[/tex]
Answer (2)
Determine the hydrogen ion concentrations H 1 and H 2 using the pH formula: H 1 = 1 0 − 4.5 and H 2 = 1 0 − 5.0 .
Calculate the difference in hydrogen ion concentrations: D i ff ere n ce = ∣ H 1 − H 2 ∣ = ∣1 0 − 4.5 − 1 0 − 5.0 ∣ .
Approximate the difference: D i ff ere n ce ≈ 2.16227766 × 1 0 − 5 .
Choose the closest answer from 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. We need to find the difference in the concentration of hydrogen ions between the two brands.
Setting up the Equations Let H 1 be the concentration of hydrogen ions in the first brand of vinegar (pH 4.5), and H 2 be the concentration of hydrogen ions in the second brand of vinegar (pH 5.0). We have the following equations:
4.5 = − lo g 10 ( H 1 ) 5.0 = − lo g 10 ( H 2 )
Solving for Hydrogen Ion Concentrations To find the hydrogen ion concentrations, we can rewrite the equations as:
H 1 = 1 0 − 4.5 H 2 = 1 0 − 5.0
Calculating the Values Now, we calculate the values of H 1 and H 2 :
H 1 = 1 0 − 4.5 ≈ 3.16227766 × 1 0 − 5 H 2 = 1 0 − 5.0 = 1.0 × 1 0 − 5
Finding the Difference The difference in the concentration of hydrogen ions is:
D i ff ere n ce = ∣ H 1 − H 2 ∣ = ∣1 0 − 4.5 − 1 0 − 5.0 ∣ D i ff ere n ce = ∣3.16227766 × 1 0 − 5 − 1.0 × 1 0 − 5 ∣ D i ff ere n ce = 2.16227766 × 1 0 − 5
Final Answer Comparing this result to the given options, the closest value 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 select the right crops and manage soil acidity. In environmental science, monitoring the pH of water bodies is essential for assessing water quality and the health of aquatic ecosystems. In the food industry, controlling pH levels is important for food preservation and ensuring food safety. The relationship between pH and hydrogen ion concentration allows us to quantify these differences and make informed decisions.
The difference in the concentration of hydrogen ions between the two brands of vinegar is approximately 2.1623 × 1 0 − 5 . The closest answer choice from the given options is 2.2 × 1 0 − 5 . Therefore, the answer is 2.2 × 1 0 − 5 .
;
