Question

A rock is radiometrically dated to determine its age. The laboratory doing the dating discovers that the rock currently has 200 atoms of radioactive parent isotope and 600 atoms of stable daughter product.
Assuming that the rock had 800 atoms of radioactive parent isotope and 0 atoms of daughter product at the time it formed, how old is the rock, if the half-life of the radioactive isotope is 1,000,000 years?

Answer 1

Answer: The rock is 2000785 years old.

Explanation:

Expression for rate law for first order kinetics is given by:

$t=\frac{2.303}{k}\log\frac{a}{a-x}$

where,

k = rate constant

t = age of sample

a = initial amount of the reactant  = 800

x = amount of daughter isotope = 600

a - x = amount left after decay process = amount of parent isotope = (800-600) = 200

Half life is the amount of time taken by a radioactive material to decay to half of its original value.

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

$k=\frac{0.693}{1000000}=0.000000693years^{-1}$

$t=\frac{2.303}{0.000000693}\log\frac{800}{200}$

$t=2000785years$

The rock is 2000785 years old.