Recent nuclear reactor accidents have contaminated many soils around the world. One of the isotopes deposited in soils is cesium-137. If the activity must fall by a factor of for contamination to disappear, how long will it take before uncontaminated crops may be grown in these soils? The half-life of cesium-137 is 30.2 years.
A) 301 years
B) 0.23 year
C) 33.1 years
D) 0.03 year
The correct answer and explanation is :
To solve this problem, we need to understand the concept of radioactive decay and how the activity (or amount) of a radioactive isotope like cesium-137 decreases over time based on its half-life.
Key Concepts:
Half-life (t½):
The time it takes for half of a radioactive substance to decay. For cesium-137, this is 30.2 years.
Decay Formula:
The activity (or amount) remaining at time t can be calculated using the formula:
$$
A = A_0 \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}
$$
Where:
- $A$ = remaining activity
- $A_0$ = initial activity
- $t$ = time elapsed
- $t_{1/2}$ = half-life of the substance
We are told the activity must fall by a factor of 1000 (this is implied though missing in your phrasing, but it’s a common value in environmental decontamination problems), so:
$$
\frac{A}{A_0} = \frac{1}{1000}
$$
Solving the Problem:
We use the decay formula:
$$
\frac{1}{1000} = \left(\frac{1}{2}\right)^{\frac{t}{30.2}}
$$
Take the logarithm of both sides:
$$
\log\left(\frac{1}{1000}\right) = \frac{t}{30.2} \cdot \log\left(\frac{1}{2}\right)
$$
$$
-3 = \frac{t}{30.2} \cdot (-0.3010)
$$
$$
\frac{t}{30.2} = \frac{3}{0.3010} \approx 9.97
$$
$$
t = 30.2 \times 9.97 \approx \boxed{301 \text{ years}}
$$
Correct Answer:
A) 301 years
Explanation:
Cesium-137 is a dangerous radioactive isotope that decays slowly. Its half-life of 30.2 years means that after each 30.2-year period, only half of the radioactive material remains. To reduce its activity by a factor of 1000, nearly 10 half-lives must pass. This is because:
- After 1 half-life (30.2 years), 50% remains.
- After 2 half-lives (60.4 years), 25% remains.
- After 10 half-lives (10 × 30.2 = 302 years), only 0.1% remains — a reduction by a factor of 1000.
Thus, it takes about 301 years before the soil is safe enough to grow uncontaminated crops. This emphasizes the long-term environmental consequences of nuclear contamination.