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Half life of radium is $1620$ years. How many radium nuclei decay in $5$ hours in $5\, gm$ radium? ( Atomic weight of radium $= 223$)
$9.1 \times 10^{12}$
$3.23 \times 10^{15}$
$1.72 \times 10^{20}$
$3.3 \times 10^{17}$
Solution
Given, Half-life, $T_{\frac{1}{2}}=\frac{0.693}{\lambda}$
$\overline{2}$
$\Rightarrow \lambda=\frac{0.693}{1620 \times 365 \times 24} \mathrm{hr}^{-1}$
In time $(t=5$ hours $), \lambda t=\frac{0.693 \times 5}{1620 \times 365 \times 24}=2.44 \times 10^{-7}$
$N_{o}=\frac{m N_{A}}{M_{o}}=\frac{5 \times\left(6.023 \times 10^{23}\right)}{233}=1.292 \times 10^{22}$ nuclei
Number of radium nuclei decay in 5 hours is $\left(N_{o}-N\right)=N_{0}\left(1-e^{-\lambda t}\right)$
$\left(N_{o}-N\right)=1.292 \times 10^{22}\left(1-e^{-2.44 \times 10^{-7}}\right)$
$\left(N_{o}-N\right)=3.23 \times 10^{15}$
Number of radium nuclei decay in 5 hours is $3.23 \times 10^{15}$ nuclie
Similar Questions
The activity $R$ of an unknown radioactive nuclide is measured at hourly intervals. The results found are tabulated as follows:
$t(h)$ | $0$ | $1$ | $2$ | $3$ | $4$ |
$R(MBq)$ | $100$ | $35.36$ | $12.51$ | $4.42$ | $1.56$ |
$(i)$ Plot the graph of $R$ versus $t$ and calculate half-life from the graph.
$(ii)$ Plot the graph of $\ln \left( {\frac{R}{{{R_0}}}} \right) \to t$ versus $t$ and obtain the value of half-life from the graph.