To find the mean life (or average life) of a radioactive substance given its half-life, we use the following relationship between half-life and mean life:
[tex]\[ \text{Mean Life} = \frac{\text{Half-Life}}{\ln 2} \][/tex]
where [tex]\(\ln 2\)[/tex] is the natural logarithm of 2, approximately equal to 0.693.
Given:
[tex]\[ \text{Half-Life} = 1.28 \times 10^9 \, \text{seconds} \][/tex]
Now, substituting the given half-life into the formula, we have:
[tex]\[ \text{Mean Life} = \frac{1.28 \times 10^9}{0.693} \][/tex]
Performing the division:
[tex]\[ \text{Mean Life} \approx 1.847 \times 10^9 \, \text{seconds} \][/tex]
So, the mean life of the radioactive substance is approximately:
[tex]\[ \boxed{1.85 \times 10^9 \, \text{seconds}} \][/tex]
This detailed step-by-step explanation follows from the principles of radioactive decay and the relationship between half-life and mean life.
