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 iron sample? Please show the calculating so I can learn to do it.

k = 0.693/t1/2

Solve for k and substitute into the below.
ln(No/N) = kt
No = 100
N = 25
k from above
Solve for t in days.

To calculate the age of the iron nail, we need to use the concept of half-life. The half-life is the amount of time it takes for half of the radioactive substance to decay.

In this case, the half-life of Iron-59 is given as 45.1 days. Let's break down the problem step by step:

1. Calculate the number of half-lives that have passed:
The decay of Iron-59 by half every 45.1 days means that after 45.1 days, only half of the initial amount will remain. After another 45.1 days, half of that remaining amount will remain, and so on. We can calculate the number of half-lives using the formula:

Number of half-lives = (Total elapsed time) / (Half-life)

In our case, the elapsed time is the age of the iron nail we want to find. Let's denote it as "t". Therefore:

Number of half-lives = t / 45.1

2. Calculate the remaining fraction of Iron-59:
Since the Fe-59 content in the iron nail is given as 25% that of the new iron sample, we can say that the remaining fraction is 0.25. This is equivalent to 25% or 1/4 of the original amount.

Remaining fraction = 1 / 4

3. Set up the equation:
The remaining fraction is related to the number of half-lives using the formula:

Remaining fraction = (1/2)^(Number of half-lives)

Substituting the values we have:

0.25 = (1/2)^(t / 45.1)

4. Solve for t:
To solve for t, we need to isolate it on one side of the equation. We can take the logarithm (base 2) of both sides to simplify the equation:

log2(0.25) = (t / 45.1) * log2(1/2)

Using logarithmic properties, we can simplify further:

log2(0.25) = -2 = (t / 45.1) * (-1)

Finally, solving for t:

t / 45.1 = -2

t = -2 * 45.1

5. Age of the iron nail:
The above calculation gives a negative value for t, indicating that more than double the half-life duration has passed. However, since we cannot have negative time, we can disregard this negative sign and take the absolute value:

t = |-2 * 45.1|

t = 90.2 days

Therefore, the iron nail is approximately 90.2 days old.