If you accidentally spill phosphorus-32 onto your shoe, how long would it take before 99.9 of the radioactive material has decayed so that you can safely wear the shoes again?
The half-life of phosphorus-32 is approximately 14.28 days. This means that every 14.28 days, half of the radioactive material will decay. To determine how long it takes for 99.9% of the material to decay, we can use the concept of half-life.
1 half-life: 50% remains
2 half-lives: 25% remains (50% of 50%)
3 half-lives: 12.5% remains (50% of 25%)
4 half-lives: 6.25% remains (50% of 12.5%)
5 half-lives: 3.125% remains (50% of 6.25%)
After 5 half-lives, approximately 3.125% of the radioactive material remains. To ensure that only 0.1% (or 0.001) remains, we need to continue until 99.9% has decayed:
6 half-lives: 1.5625% remains (50% of 3.125%)
7 half-lives: 0.78125% remains (50% of 1.5625%)
8 half-lives: 0.390625% remains (50% of 0.78125%)
After 8 half-lives, approximately 0.390625% of the radioactive material remains.
To calculate the total time, we multiply the half-life time (14.28 days) by the number of half-lives:
Time = 14.28 days * 8 half-lives
= 114.24 days
Therefore, it would take approximately 114.24 days or around 3.8 months for 99.9% of the radioactive material (phosphorus-32) to decay, allowing you to safely wear the shoes again. However, it is important to consult professionals or authorities specialized in handling radioactive material for proper guidance and disposal methods.
To determine how long it would take for 99.9% of phosphorus-32 to decay, we need to know the half-life of this radioactive material. The half-life is the time it takes for half of the radioactive material to decay.
Phosphorus-32 has a half-life of approximately 14.3 days. This means that every 14.3 days, the amount of radioactive material is halved.
To calculate how long it would take for 99.9% of the material to decay, we can use the equation:
N = N0 * (1/2)^(t/h),
where N is the final amount of radioactive material (0.1 * N0, where N0 is the initial amount), t is the time in days, and h is the half-life of the material.
Substituting the values into the equation:
0.1 * N0 = N0 * (1/2)^(t/14.3).
By canceling out N0 on both sides, we get:
0.1 = (1/2)^(t/14.3).
To solve for t, we can take the logarithm of both sides:
log(0.1) = (t/14.3) * log(1/2).
Rearranging the equation and solving for t:
t = (log(0.1) / log(1/2)) * 14.3.
Using a calculator, we can find:
t ≈ 30.1 days.
Therefore, it would take approximately 30.1 days for 99.9% of the phosphorus-32 to decay, allowing you to safely wear the shoes again.
Is that 99.9% or some other value? You have no units.
k = 0.693/t1/2
Substitute and solve for k, then substitute k into the equation below.
ln(No/N) = kt
No = some convenient number like 100.
N will be 0.1 if 99.9% of the atoms have decayed.
k from above.
Solve for t (in the same units as the half life is given).