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 2 of 4: Half-life

t_{1/2} = \frac{\ln 2}{k} = \frac{0.693}{9.93\times10^{-7}\ \mathrm{s^{-1}}} = 6.98\times10^{5}\ \mathrm{s} = 8.08\ \mathrm{d}

Analysis

For a first-order process $t_{1/2} = \ln 2/k$ ; converting seconds to days ( $86400$ s per day) gives 8.08 days.