Certainly! Let's break down the problem step-by-step:
### Step 1: Understand the Concept of Half-Life
The half-life of a substance is the time it takes for half of the sample to decay. For Uranium-232, the half-life is 68.8 years.
### Step 2: Determine the Time Elapsed
In this problem, we are given that 344.0 years have elapsed.
### Step 3: Calculate the Number of Half-Lives
To find out how many half-lives have passed in 344.0 years, we divide the total time elapsed by the half-life of Uranium-232.
[tex]\[ \text{Number of half-lives} = \frac{\text{Time elapsed}}{\text{Half-life}} \][/tex]
[tex]\[ \text{Number of half-lives} = \frac{344.0 \text{ years}}{68.8 \text{ years}} = 5.0 \][/tex]
### Step 4: Calculate the Remaining Mass
We start with an initial mass of 100.0 grams. After each half-life, the remaining mass is reduced to half of its previous amount. Since 5 half-lives have passed, we can calculate the remaining mass using the relationship:
[tex]\[ \text{Remaining mass} = \text{Initial mass} \times \left( \frac{1}{2} \right)^{\text{Number of half-lives}} \][/tex]
Substitute the values we have:
[tex]\[ \text{Remaining mass} = 100.0 \text{ grams} \times \left( \frac{1}{2} \right)^5 \][/tex]
[tex]\[ \text{Remaining mass} = 100.0 \times \frac{1}{32} \][/tex]
[tex]\[ \text{Remaining mass} = 3.125 \text{ grams} \][/tex]
### Step 5: Compare with Given Options
From the calculation, the remaining mass of Uranium-232 after 344.0 years from a 100.0-gram sample is 3.125 grams. This rounds to 3.13 grams, matching the first option provided in the question:
[tex]\[ \boxed{3.13 \text{ grams}} \][/tex]
So, the correct answer is 3.13 grams.
