Question

A radioactive isotope has a half-life of 10,000 years. How old is the rock if the ratio of radioactive parent isotope to stable daughter isotope is 1:3

Answer 1

Answer: The rock is 20004.2 years old.

Step-by-step explanation:

All the radioactive reactions follow first order kinetics.

The equation used to calculate rate constant from given half life for first order kinetics:

`t_(1/2)=(0.693)/(k)`

We are given:

`t_(1/2)=10000yrs`

Putting values in above equation, we get:

`k=(0.693)/(10000)=6.93* 10^(-5)yr^(-1)`

We are given:

Ratio of parent isotope : daughter isotope = 1 : 3

Let us assume that the amount of parent isotope is 100 g

So, amount of daughter isotope formed is 75 g

Hence, the amount of parent isotope left = 100 - 75 = 25 g

The equation used to calculate time period follows:

`N=N_o* e^(-k* t)`

where,

`N_o` = initial mass of isotope = 100 g

N = mass of the parent isotope left after the time = 25 g

t = time = ? years

k = rate constant =
`6.93* 10^(-5)yr^(-1)`

Putting values in above equation, we get:

`25=100* e^{-(6.93* 10^(-5)yr^(-1))* t}\\\\t=20004.2\text{ years}`

Hence, the rock is 20004.2 years old.