A certain nuclide has a half-life period of 30 minutes. If a sample containing 600 atoms is allowed to decay for 90 minutes, how many atoms will remain.

Options

A. 200 atoms
B. 450 atoms
C. 75 atoms
D. 500 atoms

Answer

To solve the problem of how many atoms remain after a certain time period given the half-life of a nuclide, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Half-Life Concept**:
The half-life of a nuclide is the time taken for half of the radioactive atoms in a sample to decay. In this case, the half-life is given as 30 minutes.

2. **Identify the Initial Conditions**:
We start with an initial sample containing 600 atoms of the nuclide.

3. **Determine the Total Decay Time**:
The sample is allowed to decay for 90 minutes.

4. **Calculate the Number of Half-Lives**:
To find out how many half-lives fit into the total decay time, we divide the total time by the half-life:
\[ \text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} = \frac{90 \text{ minutes}}{30 \text{ minutes}} = 3 \]

5. **Use the Half-Life Formula**:
The formula to calculate the remaining amount of a radioactive substance after a certain number of half-lives is:
\[ N_t = N_0 \left(\frac{1}{2}\right)^n \]
where:
- \( N_t \) = number of atoms remaining
- \( N_0 \) = initial number of atoms (600 atoms)
- \( n \) = number of half-lives (3 in this case)

6. **Substitute the Values into the Formula**:
\[ N_t = 600 \left(\frac{1}{2}\right)^3 \]

7. **Calculate the Remaining Atoms**:
\[ N_t = 600 \times \frac{1}{8} = 600 \div 8 = 75 \]

8. **Conclusion**:
After 90 minutes, the number of atoms remaining is 75.

### Final Answer:
The number of atoms that will remain after 90 minutes is **75 atoms**.

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