Repeated halving

Problem

A sample of iodine-131 ($\tau_{1/2} = 8\ \text{days}$) has an initial activity $A_0 = 400\ \text{MBq}$. What is the activity after (a) 8 days, (b) 24 days, (c) 40 days?

Activity of iodine-131 as a function of time: the red dots mark the requested instants (8, 24 and 40 days).

Solution

Every $8$-day interval corresponds exactly to one halving. It is enough to count how many halvings have elapsed and divide $A_0$ by $2$ raised to that number, using $A(t) = A_0\, 2^{-t/\tau_{1/2}}$.

(a) $t = 8$ days $\to$ 1 halving:

$A = \frac{400}{2} = 200\ \text{MBq}$

(b) $t = 24$ days $\to$ 3 halvings:

$A = \frac{400}{2^3} = \frac{400}{8} = 50\ \text{MBq}$

(c) $t = 40$ days $\to$ 5 halvings:

$A = \frac{400}{2^5} = \frac{400}{32} = 12{,}5\ \text{MBq}$

$\ev{200,\ 50,\ 12{,}5\ \text{MBq}}$

Connections

Topics: Nuclear physics Concepts: Radioactive decay Skills: Reading graphs