Final answer:
Using the concept of radioactive half-life, we calculate that it will take approximately 22.86 days for a 26 ounce sample of Element X to decay to 3 ounces. This calculation is based on the half-life of the element, which is 7 days.
Step-by-step explanation:
The student's question pertains to the concept of radioactive half-life, which is a key topic in Chemistry. To find how long it will take for a 26 ounce sample of Element X with a half-life of 7 days to decay to 3 ounces, we use the half-life decay formula. After each half-life period, the amount of the substance halves.
First, we determine the number of half-lives required to go from 26 ounces to 3 ounces:
- After 1 half-life (7 days): 13 ounces remain.
- After 2 half-lives (14 days): 6.5 ounces remain.
- After 3 half-lives (21 days): 3.25 ounces remain.
- After slightly more than 3 half-lives: approximately 3 ounces remain.
So, the total time is a bit more than 21 days. To get a more precise answer, we can use the formula:
N = N0 (1/2)^(t/T)
Where:
- N is the final amount of the substance.
- N0 is the initial amount of the substance.
- t is the time elapsed.
- T is the half-life of the substance.
Plugging the values in:
3 = 26 (1/2)^(t/7)
To solve for t, we take the logarithm of both sides:
t/7 = log2(26/3)
t = 7 * log2(26/3)
Calculating this, we get:
t ≈ 22.86 days
Therefore, after about 22.86 days, only 3 ounces will remain.