The count rate of nuclear radiation coming from a radiation coming from a radioactive sample containing ^{128}I varies with time as follows.

Time t (minute): | 0 | 25 | 50 | 75 | 100 |

Ctount rate R (10^{9} s^{−1}): |
30 | 16 | 8.0 | 3.8 | 2.0 |

(a) Plot In (R_{0}/R) against t. (b) From the slope of the best straight line through the points, find the decay constant λ. (c) Calculate the half-life t_{1}_{/2}.

#### Solution

(a) For *t* = 0,

`"In" (R_0/R) = "In" ((30 xx 10^9)/(30 xx 10^9)) = 0`

For *t* = 25 s,

`"In" (R_0/R_2) = "In" ((30 xx 10^9)/(16 xx 10^9)) = 0.63`

For *t *= 50 s,

`"In" (R_0/R_3) = "In" ((30 xx 10^9)/(8 xx 10^9)) = 1.35`

For *t* = 75 s,

`"In" (R_0/R_4) = "In" ((30 xx 10^9)/(3.8 xx 10^9)) = 2.06`

For *t* = 100 s,

`"In" (R_0/R_5) = "In" ((30 xx 10^9)/(2 xx 10^9)) = 2.7`

The required graph is shown below.

(b) Slope of the graph = 0.028

∴ Decay constant, `lambda` = 0.028 `"min"^-1`

The half-life period (`T_"1/2"`) is given by

`T_"1/2" = 0.693/lambda`

= `0.693/0.028 = 25 "min"`