Suppose we consider a large number of continers each containing initially `10000` atoms of a radioactive material with a half life of `1` year. After `1` year.

Options

A. all the containers will have 5000 atoms of the material
B. all the containers will contain the same number of atoms of the material but that number will only be approximately 5000.
C. the containers will in general have different numbers of the atoms of the material but their average will be close to 5000
D. none of the containers can have more than 5000 atoms

Explanation

To solve the problem, we need to determine how many atoms of a radioactive material will remain after one year, given that the initial number of atoms is 10,000 and the half-life is 1 year.

### Step-by-Step Solution:

1. **Understanding Half-Life**: The half-life of a radioactive substance is the time required for half of the radioactive atoms in a sample to decay. In this case, the half-life is 1 year.

2. **Initial Number of Atoms**: We start with an initial number of atoms, \( N_0 = 10,000 \).

3. **Applying the Half-Life Formula**: The formula to calculate the remaining number of atoms after a certain time is:
\[ N = N_0 \left( \frac{1}{2} \right)^{t/T_{1/2}} \]
where:
- \( N \) is the number of atoms remaining,
- \( N_0 \) is the initial number of atoms,
- \( t \) is the time elapsed,
- \( T_{1/2} \) is the half-life.

4. **Substituting Values**: In this case, since the half-life \( T_{1/2} = 1 \) year and \( t = 1 \) year, we can substitute these values into the formula:
\[ N = 10,000 \left( \frac{1}{2} \right)^{1/1} = 10,000 \left( \frac{1}{2} \right) = 10,000 \times 0.5 = 5,000 \]

5. **Conclusion**: After one year, approximately 5,000 atoms will remain in each container. However, since we have a large number of containers with different radioactive materials, the exact number of remaining atoms in each container may vary. But on average, it will be around 5,000 atoms.

### Final Answer:
The containers will, in general, have different numbers of atoms of the material, but their average will be close to 5,000.

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