Half-life of radium is 1620 years. How many r | vedclass

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(B) Given: Half-life $T_{1/2} = 1620 	ext{ years}$.Decay constant $\lambda = \frac{0.693}{T_{1/2}} = \frac{0.693}{1620 	imes 365 	imes 24} 	ext{ h

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$3.23 	imes 10^{15}$

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(B) Given: Half-life $T_{1/2} = 1620 	ext{ years}$.Decay constant $\lambda = \frac{0.693}{T_{1/2}} = \frac{0.693}{1620 	imes 365 	imes 24} 	ext{ hr}^{-1}$.In time $t = 5 	ext{ hours}$, the product $\lambda t = \frac{0.693 	imes 5}{1620 	imes 365 	imes 24} \approx 2.44 	imes 10^{-7}$.Initial number of nuclei $N_0 = \frac{m}{M} 	imes N_A = \frac{5}{223} 	imes 6.023 	imes 10^{23} \approx 1.35 	imes 10^{22} 	ext{ nuclei}$.Number of decayed nuclei $\Delta N = N_0(1 - e^{-\lambda t})$.Since $\lambda t$ is very small, $e^{-\lambda t} \approx 1 - \lambda t$.Therefore, $\Delta N \approx N_0(1 - (1 - \lambda t)) = N_0 \lambda t$.$\Delta N = (1.35 	imes 10^{22}) 	imes (2.44 	imes 10^{-7}) \approx 3.29 	imes 10^{15}$.Comparing with options, the closest value is $3.23 	imes 10^{15}$.

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

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