The mean of life of a radioactive sample is 100 years. Then after 100 years, about-

Options

A. 0% of the sample remains active
B. 37% of the simple remains active
C. 63% of the sample remain active
D. 50 % of the sample remains active

Explanation

To solve the problem, we need to determine how much of a radioactive sample remains after a certain period, given its mean life. Here’s a step-by-step solution:

### Step 1: Understand the Mean Life
The mean life (T_mean) of a radioactive sample is the average time it takes for half of the radioactive atoms to decay. In this case, T_mean is given as 100 years.

### Step 2: Calculate the Decay Constant (λ)
The decay constant (λ) is related to the mean life by the formula:
\[ \lambda = \frac{1}{T_{mean}} \]
Substituting the value of T_mean:
\[ \lambda = \frac{1}{100 \text{ years}} = 0.01 \text{ years}^{-1} \]

### Step 3: Use the Exponential Decay Formula
The number of radioactive particles remaining after time t is given by the formula:
\[ N = N_0 e^{-\lambda t} \]
where:
- \(N_0\) is the initial amount of the substance,
- \(N\) is the amount remaining after time t,
- \(t\) is the time elapsed.

### Step 4: Substitute the Values
In this problem, we want to find out how much remains after 100 years (t = 100 years). Substituting the values into the equation:
\[ N = N_0 e^{-\lambda t} = N_0 e^{-0.01 \times 100} \]
This simplifies to:
\[ N = N_0 e^{-1} \]

### Step 5: Calculate \(N/N_0\)
Now, we need to find the ratio \(N/N_0\):
\[ \frac{N}{N_0} = e^{-1} \approx 0.3679 \]
This means that approximately 36.79% of the original sample remains after 100 years.

### Step 6: Convert to Percentage
To express this as a percentage:
\[ \frac{N}{N_0} \times 100 \approx 0.3679 \times 100 \approx 36.79\% \]
Rounding this gives us approximately 37%.

### Conclusion
After 100 years, about **37%** of the radioactive sample remains active.

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