Gujarati
Hindi
13.Nuclei
hard

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

A

$9.1 \times 10^{12}$

B

$3.23 \times 10^{15}$

C

$1.72 \times 10^{20}$

D

$3.3 \times 10^{17}$

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

Standard 12
Physics

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