Question

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

Time t (minute):	0	25	50	75	100
Ctount rate R (109 s−1):	30	16	8.0	3.8	2.0

(a) Plot In (R₀/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.

Answer 1

(a) For t = 0,
$\ln \left(\frac{R_{\mathit{0}}}{R}\right)=\mathrm{In}\left(\frac{30\times {10}^9}{30\times {10}^9}\right)=0$
For t = 25 s,
$\ln \left(\frac{R_{\mathit{0}}}{R_{\mathit{2}}}\right)=\mathrm{In}\left(\frac{30\times {10}^9}{16\times {10}^9}\right)=0.63$
For t = 50 s,
$\mathrm{In}\left(\frac{{\mathit{R}}_{\mathit{0}}}{{\mathit{R}}_{\mathit{3}}}\right)=\mathrm{In}\left(\frac{30\times {10}^9}{8\times {10}^9}\right)=1.35$
For t = 75 s,
$\ln \left(\frac{R_0}{R_4}\right)=\mathrm{In}\left(\frac{30\times {10}^9}{3.8\times {10}^9}\right)=2.06$
For t = 100 s,
$\mathrm{In}\left(\frac{R_0}{R_5}\right)=\mathrm{In}\left(\frac{30\times {10}^9}{2\times {10}^9}\right)=2.7$
The required graph is shown below.


(b) Slope of the graph = 0.028
∴ Decay constant, $\lambda$ = 0.028 min^$-1$
The half-life period$\left(T_{\frac{1}{2}}\right)$ is given by
$$\begin{gathered} T_{\frac{1}{2}}=\frac{0.693}{\lambda } \\ =\frac{0.693}{0.028}=25\min \end{gathered}$$