Problem 1

^{131}\ce{I}

is a radioactive isotope of iodine (a

\beta^{-}

emitter) used in nuclear medicine for scintigraphy of thyroid disorders. Its decay rate constant is

k = 9.93\times10^{-7}\ \mathrm{s^{-1}}

. (1.1) Write the decay reaction of

^{131}\ce{I}

. (1.2) Calculate the half-life of

^{131}\ce{I}

in days. (1.3) Calculate the time (in days) needed for a sample of

^{131}\ce{I}

to fall to 30 % of its initial activity. (1.4) A Geiger counter detects activities of order

10^{-4}\ \mu

Ci; calculate the minimum mass (in grams) of

^{131}\ce{I}

detectable by this counter. (1 curie is the amount of a radioisotope producing

3.7\times10^{10}

disintegrations per second.)

Step 3 of 4: Time to 30 % activity

t = \frac{1}{k}\ln\frac{A_0}{A} = \frac{\ln(100/30)}{9.93\times10^{-7}} = 1.21\times10^{6}\ \mathrm{s} = 14.0\ \mathrm{d}

Analysis

Activity is proportional to N, so $A/A_0 = e^{-kt}$ ; solving $t = \ln(1/0.30)/k$ gives $1.21\times10^{6}$ s ≈ 14.0 days.