Question

The half-life of Carbon-14 is 5, 730 years. you find a sample where 1.5625% of the original Carbon-14 still remains how old is that sample ?

Answer 1

The sample is about 34380 years old.

Step-by-step explanation:

The amount of Carbon-14 mass diminishes exponentially in time, whose model is described below:

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

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

Where:

`m_(o)` - Initial mass, in grams.

`m(t)` - Current mass, in grams.

`t` - Time, in years.

`\tau` - Time constant, in years.

`t_(1/2)` - Half-life, in years.

If we know that
`(m(t))/(m_(o)) = (1.5625)/(100)` and
`t_(1/2) = 5730\,yr`, then the age of the sample is:

`\tau = (5730\,yr)/(\ln 2)`

`\tau = 8266.643\,yr`

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

`t = - (8266.643\,yr)\cdot \ln (1.5625)/(100)`

`t \approx 34380.002\,yr`

The sample is about 34380 years old.