Question

A nuclide has a half life of 2.41 x104 years. How long will it take a sample to reach a level of radioactivity that is 2.5% the level it had when it was first made?

Answer 1

Radio active decay is a first order reaction.

For first order recation,

half life t_1/2 = 0.693 /k where k is rate constant

k = 0.693/ t_{1/2 --- Eq (1)}

k = 1/t ln { [A]_o/[A]_t} -----Eq (2)

From Eqs (1) and (2),

0.693/ t_1/2 = (1/t) ln {[A]_o/ [A]_t} ------Eq (3)

Given that

half life t_1/2= 2.41 x10⁴ yrs = 24100 yrs

time t = ? hrs

Initial radioactivity  [A]_o = 100 %

Final radioactivity   [A]_t = 2.5 %

Substitute all the values in Eq (3),

0.693/ t_1/2 = (1/t) ln {[A]_o/ [A]_t}

[(0.693)/ 24100 yrs)] = (1/t )  ln{100/2.5}

t = ln {100/2.5} x ( 24100 yrs/ 0.693)

= 128285 yrs

t = 128285 yrs

Therefore, 128285 yrs required to reach a level of radioactivity that is 2.5% the level it had when it was first made.