Question

If 25% of a radioactive element is left after 40 days, then its half life period is

• A. 80 days
• B. 20 days
• C. 4 days
• D. 10 days.

Answer 1

Answer: (B)

20 days

Answer 2

Radioactive disintegration follows the first order kinetics.
Disintegration constant $(k) = \frac{2.303}{t}\log \frac{N_0}{N}$
$k = \frac{2.303}{12}\log \frac{100}{25}$
$k = \frac{2.303}{12}\log (2)^2 = \frac{2 \times 2.303}{40} \log \, 2$
$k = \frac{2 \times 2.303}{40} \times 0.3010 = 0.035 \, {days^{-1}}$
Now, $k = \frac{2.303}{t_{1/2}} \log \frac{100}{50}$
$0.035 = \frac{2.303}{t_{1/2}} \log \, 2$
$t_{1/2} = \frac{2.303}{0.035} \times 0.3010 = 19.80 \, {days \approx 20 \, days}$
So, the correct option is (B): 20 days.