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. A basic solution has a pH of 11.2. An acidic solution has a pH of 2.4. What is the approximate difference in the concentration of hydrogen ions between the two solutions?
A. [tex]$1.6 \times 10^{-9}$[/tex]
B. [tex]$4.0 \times 10^{-3}$[/tex]
C. [tex]$6.7 \times 10^{-1}$[/tex]
D. [tex]$1.6 \times 10^{11}$[/tex]
Answer (2)
Calculate the hydrogen ion concentration for the basic solution: H b + = 1 0 − 11.2 .
Calculate the hydrogen ion concentration for the acidic solution: H a + = 1 0 − 2.4 .
Approximate the difference in concentrations: Δ H + ≈ 1 0 − 2.4 .
Calculate the approximate difference: Δ H + ≈ 4.0 × 1 0 − 3 .
Explanation
Understanding the Problem We are given the formula for pH: p H = − lo g [ H + ] , where H + is the concentration of hydrogen ions. We have a basic solution with p H = 11.2 and an acidic solution with p H = 2.4 . Our goal is to find the approximate difference in the concentration of hydrogen ions between the two solutions.
Calculating Basic Solution Concentration First, we need to find the hydrogen ion concentration for each solution. For the basic solution, we have 11.2 = − lo g [ H b + ] . Solving for H b + , we get H b + = 1 0 − 11.2 .
Calculating Acidic Solution Concentration Similarly, for the acidic solution, we have 2.4 = − lo g [ H a + ] . Solving for H a + , we get H a + = 1 0 − 2.4 .
Finding the Difference Now we find the difference in the hydrogen ion concentrations: Δ H + = ∣ H a + − H b + ∣ = ∣1 0 − 2.4 − 1 0 − 11.2 ∣ . Since 1 0 − 11.2 is much smaller than 1 0 − 2.4 , we can approximate the difference as Δ H + ≈ 1 0 − 2.4 .
Calculating the Approximate Difference We calculate 1 0 − 2.4 to find the approximate difference. 1 0 − 2.4 ≈ 0.003981 . We can express this in scientific notation as 3.981 × 1 0 − 3 . Looking at the answer choices, the closest value is 4.0 × 1 0 − 3 .
Final Answer Therefore, the approximate difference in the concentration of hydrogen ions between the two solutions is 4.0 × 1 0 − 3 .
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. If the soil is too acidic, they can add lime to raise the pH. Similarly, in water treatment, maintaining the correct pH is essential for effective disinfection and preventing corrosion of pipes. By understanding the relationship between pH and hydrogen ion concentration, we can better manage these processes and ensure the health of our environment.
The approximate difference in the concentration of hydrogen ions between a basic solution with a pH of 11.2 and an acidic solution with a pH of 2.4 is about 4.0 × 1 0 − 3 M . Thus, the correct answer is B. This calculation shows how much more concentrated hydrogen ions are in the acidic solution compared to the basic one.
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