To determine the half-life of Fe-61, we need to use the information provided:
After 10 minutes, a 100g sample of Fe-61 decays down to 31.5g.
The half-life of a radioactive substance is the time it takes for half of the original sample to decay. In this case, the initial sample was 100g, and after 10 minutes it has decreased to 31.5g.
To find the half-life, we can use the formula:
N = N₀ * (1/2)^(t / T)
Where:
- N is the final amount of the substance (31.5g in this case)
- N₀ is the initial amount of the substance (100g)
- t is the time passed (10 minutes)
- T is the half-life we want to find.
Plugging in the values:
31.5g = 100g * (1/2)^(10 minutes / T)
To solve for T, we need to isolate it:
(1/2)^(10 minutes / T) = 31.5g / 100g
Taking the logarithm of both sides:
log((1/2)^(10 minutes / T)) = log(31.5g / 100g)
Using the logarithmic property log(a^b) = b * log(a):
(10 minutes / T) * log(1/2) = log(31.5g / 100g)
Now, rearranging the equation to solve for T:
T = (10 minutes) / (log(1/2) / log(31.5g / 100g))
Using a calculator, we find:
T ≈ 24.92 minutes
Therefore, the half-life of Fe-61 is approximately 24.92 minutes.