To determine how long the waste material containing the radioactive isotope P32 must be stored before it is safe to dispose of, we can use the concept of radioactive decay. The half-life of P32 is 14.3 days, which means that every 14.3 days, the activity of the isotope will reduce to half of its previous value. We start with an initial activity of 1 milli curie (mCi) and want to find out how long it will take for the activity to decrease to 0.01 micro curie (µCi).
Understanding Radioactive Decay
Radioactive decay follows an exponential decay model, which can be expressed mathematically. The formula to calculate the remaining activity after a certain time is:
A(t) = A0 * (1/2)^(t/T)
Where:
- A(t) = remaining activity after time t
- A0 = initial activity
- T = half-life of the isotope
- t = time elapsed
Setting Up the Problem
In this case:
- A0 = 1 mCi = 1000 µCi
- A(t) = 0.01 µCi
- T = 14.3 days
Calculating the Time Required
We need to find the time t when the activity drops from 1000 µCi to 0.01 µCi. Plugging the values into the decay formula:
0.01 = 1000 * (1/2)^(t/14.3)
To isolate the exponential term, divide both sides by 1000:
(1/2)^(t/14.3) = 0.01 / 1000 = 0.00001
Next, we can take the logarithm of both sides to solve for t:
log((1/2)^(t/14.3)) = log(0.00001)
Using the property of logarithms, we can bring down the exponent:
(t/14.3) * log(1/2) = log(0.00001)
Now, we can solve for t:
t = 14.3 * (log(0.00001) / log(1/2))
Calculating the Logarithms
Using a calculator, we find:
- log(0.00001) ≈ -5
- log(1/2) ≈ -0.301
Now substituting these values back into the equation:
t ≈ 14.3 * (-5 / -0.301) ≈ 14.3 * 16.61 ≈ 237.5 days
Final Thoughts
Therefore, the waste material must be stored in a lead container for approximately 238 days before it is safe to dispose of. This calculation highlights the importance of understanding radioactive decay and the safety measures necessary when handling radioactive materials.
