They decay constant of radioactive element is `1.5xx10^(-9)s^(-1)` Its mens-life (in second) will be

Options

A. `1.xx10^(9)`
B. `4.62xx10^(8)`
C. `6.67xx10^(8)`
D. `10.35xx10^(8)`

Answer

To find the mean life of a radioactive element given its decay constant, we can use the formula:

\[ T_{\text{mean}} = \frac{1}{\lambda} \]

where \( T_{\text{mean}} \) is the mean life and \( \lambda \) is the decay constant.

### Step-by-Step Solution:

1. **Identify the decay constant**:
The decay constant \( \lambda \) is given as:
\[ \lambda = 1.5 \times 10^{-9} \, \text{s}^{-1} \]

2. **Substitute the decay constant into the mean life formula**:
Using the formula for mean life:
\[ T_{\text{mean}} = \frac{1}{\lambda} = \frac{1}{1.5 \times 10^{-9}} \]

3. **Calculate the mean life**:
To simplify the calculation, we can rewrite the expression:
\[ T_{\text{mean}} = \frac{1}{1.5} \times 10^{9} \]
Now, calculating \( \frac{1}{1.5} \):
\[ \frac{1}{1.5} = \frac{10}{15} = \frac{2}{3} \]
Therefore:
\[ T_{\text{mean}} = \frac{2}{3} \times 10^{9} \, \text{s} \]

4. **Convert to decimal form**:
To express \( \frac{2}{3} \) in decimal form:
\[ \frac{2}{3} \approx 0.6667 \]
Thus:
\[ T_{\text{mean}} \approx 0.6667 \times 10^{9} \, \text{s} = 6.67 \times 10^{8} \, \text{s} \]

5. **Final result**:
Therefore, the mean life of the radioactive element is:
\[ T_{\text{mean}} \approx 6.67 \times 10^{8} \, \text{s} \]

### Answer:
The mean life of the radioactive element is \( 6.67 \times 10^{8} \, \text{s} \).

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