`N` atoms of a radioactive element emit `n` alpha particles per second. The half-life of tge element is.

Options

A. `n/N`
B. `N/n`
C. `0.693N/n`
D. `0.693 n/N`

Answer

To find the half-life of a radioactive element given that `N` atoms emit `n` alpha particles per second, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Activity**:
The activity \( A \) of a radioactive substance is defined as the number of decay events (or emissions) per unit time. In this case, we are given that the activity is equal to \( n \) alpha particles per second. Therefore, we can write:
\[ A = n \]

2. **Relating Activity to Decay Constant**:
The activity \( A \) is also related to the decay constant \( \lambda \) and the number of radioactive atoms \( N \) present:
\[ A = \lambda \cdot N \]
From the above two equations, we can equate them:
\[ n = \lambda \cdot N \]

3. **Finding the Decay Constant**:
Rearranging the equation gives us the decay constant \( \lambda \):
\[ \lambda = \frac{n}{N} \]

4. **Using the Relationship Between Decay Constant and Half-Life**:
The decay constant \( \lambda \) is also related to the half-life \( t_{1/2} \) of the radioactive substance by the formula:
\[ \lambda = \frac{\ln 2}{t_{1/2}} \]
We can substitute this expression for \( \lambda \) into our earlier equation:
\[ \frac{n}{N} = \frac{\ln 2}{t_{1/2}} \]

5. **Solving for Half-Life**:
Rearranging this equation to solve for the half-life \( t_{1/2} \) gives:
\[ t_{1/2} = \frac{N \cdot \ln 2}{n} \]

6. **Substituting the Value of \( \ln 2 \)**:
The natural logarithm of 2 is approximately \( 0.693 \). Therefore, we can write:
\[ t_{1/2} = \frac{N \cdot 0.693}{n} \]

### Final Answer:
Thus, the half-life of the radioactive element is given by:
\[ t_{1/2} = \frac{0.693 \cdot N}{n} \]