The half-life period `t_(1//2)` of a radioactive element is N years. The period of its complete decays is

Options

A. `N^(2)` years
B. 2N years
C. `(1)/(2)N^(2)` years
D. Infinity

Explanation

To solve the problem regarding the complete decay of a radioactive element with a half-life period of \( t_{1/2} = N \) years, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Half-Life**: The half-life of a radioactive element is the time required for half of the radioactive atoms in a sample to decay. For our element, this is given as \( N \) years.

2. **Decay Constant Calculation**: The decay constant \( \lambda \) can be calculated using the formula:
\[ \lambda = \frac{0.693}{t_{1/2}} = \frac{0.693}{N} \]

3. **Complete Decay Concept**: Complete decay implies that all radioactive atoms have decayed. However, in radioactive decay, it is important to note that the process theoretically never reaches complete decay in a finite amount of time.

4. **Mathematical Representation**: The amount of substance remaining after time \( t \) can be expressed as:
\[ A_t = A_0 e^{-\lambda t} \]
where \( A_0 \) is the initial amount and \( A_t \) is the amount remaining after time \( t \).

5. **Setting Up for Complete Decay**: For complete decay, we would want \( A_t \) to approach 0. However, as \( t \) approaches infinity, \( A_t \) approaches 0 but never actually reaches it in finite time.

6. **Conclusion on Time for Complete Decay**: Thus, the time required for complete decay is theoretically infinite. Therefore, the period of complete decay of the radioactive element is:
\[ \text{Complete decay time} = \infty \]

### Final Answer:
The period of its complete decays is infinite.

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