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.

Explanation:

All the radioactive reactions follow first order kinetics.

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

$t_{1/2}=\frac{0.693}{k}$

We are given:

$t_{1/2}=10000yrs$

Putting values in above equation, we get:

$k=\frac{0.693}{10000}=6.93\times 10^{-5}yr^{-1}$

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:

$N=N_o\times e^{-k\times t}$

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 = $6.93\times 10^{-5}yr^{-1}$

Putting values in above equation, we get:

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

Hence, the rock is 20004.2 years old.