To determine the activity of a mixed sample of radioisotopes after a certain period, we need to consider the half-lives of each isotope, their initial activity, and how they decay over time. In this case, we have two isotopes, P-32, with half-lives of 14 days and 25 days, mixed in a ratio of 4:1. Let's break this down step by step.
Understanding Initial Conditions
The initial activity of the mixed sample is given as 3.0 mCi. Since the isotopes are mixed in a ratio of 4:1, we can calculate the initial activities of each isotope.
- Let the activity of the first isotope (with a half-life of 14 days) be 4x.
- Let the activity of the second isotope (with a half-life of 25 days) be x.
From the ratio, we can express the total initial activity as:
4x + x = 3.0 mCi
5x = 3.0 mCi
Thus, x = 0.6 mCi.
Now we can find the initial activities:
- Activity of the first isotope = 4x = 4 * 0.6 mCi = 2.4 mCi
- Activity of the second isotope = x = 0.6 mCi
Calculating Activity After 60 Years
Next, we need to calculate the activity of each isotope after 60 years. First, we convert 60 years into days:
60 years = 60 * 365 = 21,900 days.
Decay Formula
The activity of a radioactive substance after a certain time can be calculated using the formula:
A = A₀ * (1/2)^(t/T₁/2)
Where:
- A = remaining activity after time t
- A₀ = initial activity
- t = total time elapsed
- T₁/2 = half-life of the isotope
Activity of the First Isotope (14 days half-life)
For the first isotope:
- A₀ = 2.4 mCi
- T₁/2 = 14 days
- t = 21,900 days
Now, we calculate the number of half-lives that have passed:
Number of half-lives = t / T₁/2 = 21,900 days / 14 days ≈ 1564.29
Now we can find the remaining activity:
A₁ = 2.4 mCi * (1/2)^(1564.29) ≈ 0 mCi (essentially zero due to extensive decay)
Activity of the Second Isotope (25 days half-life)
For the second isotope:
- A₀ = 0.6 mCi
- T₁/2 = 25 days
Calculating the number of half-lives for the second isotope:
Number of half-lives = 21,900 days / 25 days ≈ 876
Now we find the remaining activity:
A₂ = 0.6 mCi * (1/2)^(876) ≈ 0 mCi (again, essentially zero)
Final Activity of the Mixed Sample
After 60 years, both isotopes have decayed to a point where their activities are negligible. Therefore, the total activity of the mixed sample after 60 years is:
A_total = A₁ + A₂ ≈ 0 mCi + 0 mCi = 0 mCi.
In summary, after 60 years, the activity of the mixed sample of P-32 isotopes will be virtually zero due to the extensive decay of both isotopes. This illustrates the significant impact of half-lives on the stability and activity of radioactive materials over long periods.
