The half-life of a radioactive decay is x times its mean life. The value of zx is

Options

A. `0.3010`
B. `0.6930`
C. `0.6020`
D. `(1)/(0.6930)`

Explanation

To solve the problem, we need to establish the relationship between the half-life (T_half) and the mean life (τ) of a radioactive substance. The half-life is defined as the time required for half of the radioactive nuclei in a sample to decay, while the mean life is the average time a nucleus exists before decaying.

### Step-by-Step Solution:

1. **Understanding the Definitions**:
- The half-life (T_half) is given by the formula:
\[ T_{half} = \frac{\ln 2}{\lambda} \]
- The mean life (τ) is given by:
\[ \tau = \frac{1}{\lambda} \]

2. **Setting Up the Relationship**:
- According to the problem, the half-life is x times the mean life:
\[ T_{half} = x \cdot \tau \]

3. **Substituting the Formulas**:
- Substitute the expressions for T_half and τ into the equation:
\[ \frac{\ln 2}{\lambda} = x \cdot \frac{1}{\lambda} \]

4. **Simplifying the Equation**:
- Since λ is in the denominator on both sides, we can cancel it out (assuming λ ≠ 0):
\[ \ln 2 = x \]

5. **Calculating the Value of x**:
- The value of ln 2 is approximately:
\[ \ln 2 \approx 0.693 \]
- Therefore, we conclude that:
\[ x \approx 0.693 \]

6. **Final Answer**:
- The value of x is approximately 0.693, which corresponds to option 2 in the given choices.