What is the half-life of iron-59, a radioisotope used medically in the diagnosis of anemia, if a sample with an initial decay rate of 16,800 disintegrations/min decays at a rate of 10,860 disintegrations/min after 28.0 days?

1) Data:

initial decay rate, No =16,800 disintegrations/min

final decay rate, Nf = 10,860 disintegrations/min

t = 28.0 days

t half-life =?

2) Formulas

Radioactive disintegration =>

Nt = No * e ^ (-kt)

t half-life = ln (2) / k

3) Solution

From Nt = No * e^ (-kt) =>

Nt / No = e ^ (-kt)

=> -kt = ln (Nt / No)

=> kt = ln (No/Nt)

=> k = ln (No / Nt) / t

=> k = ln (16,800 / 10,860) / 28 days = 0.01558 days^ -1

From t half-life = ln(2) / k

t half-life = ln(2) / (0.01558 days^-1) = 44.5 days.

44.5 days.