For a radioactive substance with half-life period 500 years, the time for complete decay of 100 milligram of it would be

Options

A. 1000 years
B. `100 xx 500` years
C. 500 years
D. Infinite time

Explanation

To solve the question regarding the complete decay of a radioactive substance with a half-life of 500 years, we can follow these steps:

### Step-by-Step Solution:

1. **Understand 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 given as 500 years.

2. **Radioactive Decay Concept**: Radioactive decay is a random process, and theoretically, a radioactive substance never completely decays to zero in a finite amount of time. Instead, it approaches zero asymptotically.

3. **Mathematical Representation**: The amount of substance remaining after a certain time can be represented by the formula:
\[ A_t = A_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \]
where:
- \( A_t \) is the amount remaining after time \( t \),
- \( A_0 \) is the initial amount,
- \( T_{1/2} \) is the half-life.

4. **Complete Decay Definition**: "Complete decay" implies that the amount remaining (\( A_t \)) is 0. However, as per the decay formula, \( A_t \) can never actually reach 0 in a finite time. It can only get infinitely close to 0.

5. **Conclusion**: Since the substance never completely decays to zero in a finite time, we conclude that the time required for complete decay is infinite.

### Final Answer:
The time for complete decay of 100 milligrams of the radioactive substance is **infinite**.

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