Question

The half-life of radium is $1620$ year and its atomic weight is $226$. The number of atoms that will decay from its $1\, g$ sample per second will be :

• A. $ 3.6\times {{10}^{10}} $
• B. $ 3.6\times {{10}^{12}} $
• C. $ 3.1\times {{10}^{15}} $
• D. $ 31.1\times {{10}^{15}} $

Answer 1

Answer: (A)

According to Avogadro's hypothesis.
$N_{0}=\frac{6.02 \times 10^{23}}{226}=2.66 \times 10^{21}$
Half-life $=T=\frac{0.693}{\lambda}=1620 $ year
$\therefore \lambda=\frac{0.693}{1620 \times 3.16 \times 10^{7}}$
$=1.35 \times 10^{-11} s^{-1}$
Because half-life is very much large as compared to its times interval,
hence $N=N_{0}$
Now, $\frac{d N}{d t}=\lambda N=\lambda N_{0}$
or $d N=\lambda N_{0} d t$
$=\left(1.35 \times 10^{-11}\right)\left(2.66 \times 10^{21}\right) \times 1$
$=1.35 \times 10^{-11} s^{-1}$