Question

Iodine-131, t_1/2 = 8.0 days, is used in diagnosis and treatment off thyroid gland diseases. If a laboratory sample of iodine-131 initially emits 9.95 times 10^18 beta particles per day, how long will it take for the activity to drop to 6.22 times 10^17 beta particles per day?

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 of I-131, t_1/2= 8.0 days

time t = ?

Initial activity of I-131 [A]o = 9.95 x 10¹⁸

Final activity of I-131   [A]_t = 6.22 x 10¹⁷

Substitute all the values in Eq (3),

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

[0.693/8] hrs = (1/t )  ln {9.95 x 10¹⁸/ 6.22 x 10¹⁷}

Then,

t = [ 8/0.693] x ln {9.95 x 10¹⁸/ 6.22 x 10¹⁷}

= 32 days

t = 32 days

Therefore,

required time = 32 days