Iron-59 has a half-life of 45.1 days. How old is an iron nail if the Fe-59 content is 25% that of a new sample of iron? Show all calculations leading to a solution.
Is this how you would do this?
(100/25) = 0.693 times (45.1)
4= 31.2543
final answer is 7.8136
25=100 e^(-.693 t/45.1)
take the ln of each side...
Ln(25)=ln(100) -.693 t/45.1
ln(25)-ln(100)=-.693 t/45.1
ln(25/100)=-.693t/45.1
t= - 45.1/.693 ( ln .25)
putting that into the goggle calculator..
t=90.2 days.
Now the easy way. to reduct to 1/4, it takes two half lives, or you do it.
To solve this problem, we need to use the concept of radioactive decay and the formula for calculating the age of a sample given its half-life.
The half-life of Iron-59 is given as 45.1 days. This means that after every 45.1 days, the amount of Iron-59 will decrease by half.
Let's assume the initial amount of Iron-59 in the new sample is "100 units".
Now, if the Fe-59 content is 25% of the initial sample, that means only 25 units of Iron-59 remain in the nail.
To calculate the age of the iron nail, we need to find out how many half-lives have passed. We can do this by dividing the remaining amount of Iron-59 (25 units) by the initial amount (100 units). This gives us:
25 units / 100 units = 0.25
This means that we have reached 0.25 (which is equivalent to 25%) of the original sample. Now, we need to find the number of half-lives that correspond to this fraction.
To calculate the number of half-lives, we can use the formula:
number of half-lives = (ln(remaining fraction)) / ln(0.5)
In this case, the remaining fraction is 0.25, and ln represents the natural logarithm.
number of half-lives = ln(0.25) / ln(0.5)
Using a scientific calculator, we can calculate the natural logarithm of 0.25 as -1.3863 and the natural logarithm of 0.5 as -0.6931.
number of half-lives = -1.3863 / -0.6931 ≈ 2
Therefore, approximately 2 half-lives have passed.
To find the age of the iron nail, we can multiply the number of half-lives by the half-life of Iron-59:
age = number of half-lives x half-life
age = 2 x 45.1 days
age ≈ 90.2 days
So, the iron nail is approximately 90.2 days old based on the given information.