Answer
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Hint: We will have to calculate the radioactive decay constant first. Then after finding the no. of atoms in 1 gm of $^{24} Na$ ,we can easily solve this problem.
Formula used:
$N=N_0 .e^{-\lambda t}$
$\lambda =\dfrac{\ln 2}{{{t}_{\dfrac{1}{2}}}}$
Complete step-by-step solution:
Half life is given as ${{t}_{\dfrac{1}{2}}}=15hrs$ . If $\lambda$ be the radioactive decay constant, we have,
$\begin{align}
& \lambda =\dfrac{\ln 2}{{{t}_{\dfrac{1}{2}}}} \\
& \Rightarrow \lambda =\dfrac{\ln 2}{15}=0.0462hr{{s}^{-1}} \\
\end{align}$
Now, 24 gm $^{24}Na$ contains $6.023\times {{10}^{23}}$ number of atoms.
Or, 1 gm $^{24}Na$ contains
${{N}_{0}}=\dfrac{6.023\times {{10}^{23}}}{24}=2.51\times {{10}^{22}}$ number of atoms.
The time of decay is given to be t = 7.5 hrs.
So, putting the values of $N_0$ , $\lambda$ and t , we obtain,
$N={{N}_{0}}.{{e}^{-\lambda t}}=1.77\times {{10}^{22}}$
This is the number of atoms that has remained after 7.5 hrs of decay. Hence, the number of atoms that had been decayed is,
${{N}_{\beta }}={{N}_{0}}-N=(2.51\times {{10}^{22}})-(1.77\times {{10}^{22}})=7.4\times {{10}^{21}}$
This is the number of $\beta$ particles that has been emitted.
So, option C is the correct answer.
Additional information:
A radioactive decay of a certain amount of radioactive substance takes infinite time to come to an end. It means, the process never completes and some radioactivity is always present in the substance.
Note: While calculating $\lambda$, know that (ln 2) is the natural logarithm and not the usual log (10-base) we use. Their relation is $\ln 2=2.303\times \log 2$. Another point to remember to avoid making mistakes is that N in the formula is not the number of atoms that has decayed, but the number that has remained.
Formula used:
$N=N_0 .e^{-\lambda t}$
$\lambda =\dfrac{\ln 2}{{{t}_{\dfrac{1}{2}}}}$
Complete step-by-step solution:
Half life is given as ${{t}_{\dfrac{1}{2}}}=15hrs$ . If $\lambda$ be the radioactive decay constant, we have,
$\begin{align}
& \lambda =\dfrac{\ln 2}{{{t}_{\dfrac{1}{2}}}} \\
& \Rightarrow \lambda =\dfrac{\ln 2}{15}=0.0462hr{{s}^{-1}} \\
\end{align}$
Now, 24 gm $^{24}Na$ contains $6.023\times {{10}^{23}}$ number of atoms.
Or, 1 gm $^{24}Na$ contains
${{N}_{0}}=\dfrac{6.023\times {{10}^{23}}}{24}=2.51\times {{10}^{22}}$ number of atoms.
The time of decay is given to be t = 7.5 hrs.
So, putting the values of $N_0$ , $\lambda$ and t , we obtain,
$N={{N}_{0}}.{{e}^{-\lambda t}}=1.77\times {{10}^{22}}$
This is the number of atoms that has remained after 7.5 hrs of decay. Hence, the number of atoms that had been decayed is,
${{N}_{\beta }}={{N}_{0}}-N=(2.51\times {{10}^{22}})-(1.77\times {{10}^{22}})=7.4\times {{10}^{21}}$
This is the number of $\beta$ particles that has been emitted.
So, option C is the correct answer.
Additional information:
A radioactive decay of a certain amount of radioactive substance takes infinite time to come to an end. It means, the process never completes and some radioactivity is always present in the substance.
Note: While calculating $\lambda$, know that (ln 2) is the natural logarithm and not the usual log (10-base) we use. Their relation is $\ln 2=2.303\times \log 2$. Another point to remember to avoid making mistakes is that N in the formula is not the number of atoms that has decayed, but the number that has remained.
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