Question

The half-life of radium is $1620$ year and its atomic weight is $226$. The number of atoms that will decay from its $1g$ 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, $\frac{dN}{dt}=\lambda N=\lambda N_0$
or $dN=\lambda N_{0}dt$
$=\left(1.35\times 10^{-11}\right)\left(2.66\times 10^{21}\right)\times 1$
$=1.35\times 10^{-11}{s}^{-1}$