A flask contains an acidic solution with a concentration of $7.1 \times 10^{15}$ hydrogen ions per milliliter. If $4.8 \times 10^{23}$ hydrogen ions have a total mass of 0.8 grams, which of the following is closest to the concentration, in grams per liter, of the acidic solution?
(A) $1.2 \times 10^{-5}$
(B) $1.2 \times 10^{-8}$
(C) $1.5 \times 10^{-5}$
(D) $1.5 \times 10^{-8}$
Answer (2)
Calculate the mass of a single hydrogen ion: 4.8 × 1 0 23 0.8 .
Calculate the mass of hydrogen ions per milliliter: ( 7.1 × 1 0 15 ) × 4.8 × 1 0 23 0.8 .
Convert the concentration to grams per liter by multiplying by 1000: ( ( 7.1 × 1 0 15 ) × 4.8 × 1 0 23 0.8 ) × 1000 .
The concentration in grams per liter is approximately 1.2 × 1 0 − 5 . 1.2 × 1 0 − 5
Explanation
Problem Analysis We are given the concentration of hydrogen ions in an acidic solution as 7.1 × 1 0 15 ions per milliliter. We also know that 4.8 × 1 0 23 hydrogen ions have a total mass of 0.8 grams. Our goal is to find the concentration of the solution in grams per liter.
Calculating Mass per Ion First, we need to find the mass of a single hydrogen ion. We can calculate this by dividing the total mass of the hydrogen ions by the number of ions: Mass per ion = Number of ions Total mass = 4.8 × 1 0 23 ions 0.8 grams
Calculating Mass per Milliliter Next, we calculate the mass of hydrogen ions per milliliter by multiplying the number of ions per milliliter by the mass of a single ion: Mass per ml = ( 7.1 × 1 0 15 ions/ml ) × ( 4.8 × 1 0 23 ions 0.8 grams )
Converting to Grams per Liter Now, we convert the concentration from grams per milliliter to grams per liter. Since there are 1000 milliliters in a liter, we multiply the concentration in grams per milliliter by 1000: Concentration in g/L = Mass per ml × 1000 = ( ( 7.1 × 1 0 15 ) × 4.8 × 1 0 23 0.8 ) × 1000
Calculating Final Concentration Let's calculate the final concentration: Concentration in g/L = 4.8 × 1 0 23 7.1 × 1 0 15 × 0.8 × 1000 = 4.8 × 1 0 23 7.1 × 0.8 × 1 0 18 = 4.8 7.1 × 0.8 × 1 0 − 5 ≈ 1.183 × 1 0 − 5
Final Answer The concentration of the acidic solution is approximately 1.183 × 1 0 − 5 grams per liter. Looking at the answer choices, the closest value is 1.2 × 1 0 − 5 grams per liter.
Examples
Understanding concentrations is crucial in many real-world applications. For example, in environmental science, it's used to measure pollutants in water or air. In medicine, drug dosages are determined based on concentration to ensure patient safety and effectiveness. Similarly, in cooking, the concentration of salt or sugar affects the taste of the dish. Knowing how to calculate and interpret concentrations helps us make informed decisions and maintain safety in various aspects of life.
The concentration of the acidic solution is approximately 1.183 × 1 0 − 5 grams per liter. Therefore, the closest answer from the options provided is (A) 1.2 × 1 0 − 5 .
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