Question

14. A sample that originally contained 2.5 g of rubidium-87 now contains 1.25 g. The half-life of rubidium-87 is 6 x 10^10 years. How old is the sample? Is this possible? Why or why not?​

Answer 1

The age of the sample is
`t \approx 6* 10^(10)\,yr`. It is possible, since the definition of half-life is the time taken by the isotope to halve its mass.

Step-by-step explanation:

All radioactive isotopes decays exponentially, the decay is represented by the following formula:

`m(t) = m_(o)\cdot e^{-(t)/(\tau) }` (1)

Where:

`m_(o)` - Initial mass of the isotope, measured in grams.

`m(t)` - Current mass of the isotope, measured in grams.

`t` - Time, measured in years.

`\tau` - Time constant, measured in years.

Now we clear the time of the isotope within the formula:

`t = -\tau\cdot \ln (m(t))/(m_(o))`

In addtion, the time constant can be calculated in terms of the half-life (
`t_(1/2)`), measured in years:

`\tau = (t_(1/2))/(\ln 2)` (2)

If we know that
`m_(o) = 2.5\,g`,
`m(t) = 1.25\,g` and
`t_(1/2) = 6* 10^(10)\,yr`, then the age of the isotope is:

`\tau = (6* 10^(10)\,yr)/(\ln 2)`

`\tau \approx 8.656* 10^(10)\,yr`

`t = -(8.656* 10^(10)\,yr)\cdot \ln (1.25\,g)/(2.5\,g)`

`t \approx 6* 10^(10)\,yr`

The age of the sample is
`t \approx 6* 10^(10)\,yr`. It is possible, since the definition of half-life is the time taken by the isotope to halve its mass.