Question

Half life of radium is $1620$ years. How many radium nuclei decay in $5$ hours in $5gm$ radium? (Atomic weight of radium $=223)$.

• A. $9.1\times {10}^{12}$
• B. $3.15\times {10}^{15}$
• C. $1.72\times {10}^{20}$
• D. $3.3\times {10}^{17}$

Answer 1

The correct option is B $3.15\times {10}^{15}$

Given,

Half-life, $T_{{}^{\frac{1}{2}}}=\frac{0.693}{\lambda }$

$\Rightarrow \lambda =\frac{0.693}{1620\times 365\times 24}h{r}^{-1}$

In time $(t=5hours)$, $\lambda t=\frac{0.693\times 5}{1620\times 365\times 24}=2.44\times {10}^{-7}$

$N_o=\frac{m{N}_A}{M_o}=\frac{5\times (6.023\times {10}^{23})}{233}=1.292\times {10}^{22}$nuclei

Number of radium nuclei decay in 5 hours is

$(N_o-N)=N_0(1-e^{-\lambda t})$

$(N_o-N)=1.292\times {10}^{22}(1-e^{-2.44\times {10}^{-7}})$

$(N_o-N)=3.15\times {10}^{15}$

Number of radium nuclei decay in 5 hours is $3.15\times {10}^{15}nuclie$