Iron -59 has a half life of 45.1 days. How old is an iron nail if the Fe-59 content is 25% that a new sample of iron? Show all calculations leading to a solution. Can you work the problem? I don't know how to solve for t!!! Please.
9 years ago
9 years ago
It's half life - try looking in your book.
11 months ago
Sure! I'd be happy to help you with the calculations.
To find the age of the iron nail, we can use the concept of half-life and the given information.
The half-life of Iron-59 (Fe-59) is 45.1 days. This means that after every 45.1 days, half of the Fe-59 atoms decay or transform into another element. In this case, we are given that the Fe-59 content is 25% of that in a new sample of iron.
To solve for the age of the iron nail (t), we can use the following formula:
(Fe-59 Final Amount) = (Fe-59 Initial Amount) * (0.5)^(t / half-life)
In this formula:
- (Fe-59 Final Amount) represents the current amount of Fe-59 in the iron nail.
- (Fe-59 Initial Amount) represents the initial amount of Fe-59 in the iron nail (which was 100% in the new sample).
- t represents the age of the iron nail in days.
- half-life represents the half-life of Fe-59, which is 45.1 days.
Since we know that the Fe-59 content is currently 25% of the initial amount, we can substitute these values into the formula:
0.25 = 1 * (0.5)^(t / 45.1)
To solve for t, we need to isolate the variable t. We can start by taking the natural logarithm (ln) of both sides of the equation to eliminate the exponential:
ln(0.25) = ln((0.5)^(t / 45.1))
Using logarithm properties, we can simplify the equation further:
ln(0.25) = (t / 45.1) * ln(0.5)
Now, we can solve for t by multiplying both sides of the equation by (45.1 / ln(0.5)):
t = (45.1 / ln(0.5)) * ln(0.25)
Using a calculator, we can find ln(0.25) and ln(0.5) to substitute in the formula:
t ≈ (45.1 / -0.693) * -1.386
Simplifying further:
t ≈ 31.8 days
Therefore, the age of the iron nail is approximately 31.8 days.