Final answer:
To determine how much Sodium-24 will remain after 3.7 days, we calculate the number of half-lives that occur in that period and then continuously halve the original amount for each half-life. After 3.7 days (6 half-lives), about 12.27 mg of the original 785 mg sample of Sodium-24 would remain.
Step-by-step explanation:
The student is asking about the calculation of the remaining amount of a radioactive isotope, Sodium-24, after a given time interval, considering its half-life. To solve this, we will use the half-life formula to determine how much substance remains after a certain number of half-lives have passed.
Sodium-24 has a half-life of 14.8 hours. The question is asking how much of a 785.0 mg sample would remain after 3.7 days. To begin, we must convert 3.7 days into hours, which is 3.7 days times 24 hours/day, equaling 88.8 hours. Now, we divide 88.8 hours by the half-life of Sodium-24, which is 14.8 hours, giving us 6 half-lives (88.8 hours ÷ 14.8 hours/half-life = 6 half-lives).
To find the final amount remaining, we then take the original amount and halve it for each half-life that has passed:
- After 1 half-life: 785 mg / 2 = 392.5 mg
- After 2 half-lives: 392.5 mg / 2 = 196.25 mg
- After 3 half-lives: 196.25 mg / 2 = 98.125 mg
- After 4 half-lives: 98.125 mg / 2 = 49.0625 mg
- After 5 half-lives: 49.0625 mg / 2 = 24.53125 mg
- After 6 half-lives: 24.53125 mg / 2 = 12.265625 mg
Therefore, after 3.7 days, approximately 12.27 mg of Sodium-24 would remain from the original 785 mg sample.