The half life of radium is 1600 years. The mean life of radium is

Options

A. 800 years
B. 2309 years
C. 3209 years
D. 2903 years

Answer

To find the mean life (also known as the average life) of radium given its half-life, we can use the relationship between half-life (T½) and mean life (τ). The mean life is related to the half-life by the formula:

\[ \tau = \frac{T_{1/2}}{\ln(2)} \]

Where:
- \( \tau \) is the mean life.
- \( T_{1/2} \) is the half-life.
- \( \ln(2) \) is the natural logarithm of 2, approximately equal to 0.693.

### Step-by-Step Solution:

1. **Identify the half-life of radium**:
The half-life \( T_{1/2} \) is given as 1600 years.

2. **Use the formula for mean life**:
Substitute the value of the half-life into the formula:
\[ \tau = \frac{T_{1/2}}{\ln(2)} = \frac{1600 \text{ years}}{0.693} \]

3. **Calculate the mean life**:
Now, perform the division:
\[ \tau \approx \frac{1600}{0.693} \approx 2306.5 \text{ years} \]

4. **Round off the answer**:
The mean life of radium can be rounded to approximately 2307 years.

### Final Answer:
The mean life of radium is approximately 2307 years.

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