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Under certain circumstances, a nucleus can decay by emitting a particle more massive than an $\alpha$-particle. Consider the following decay processes:
$$
{ }_{88}^{223} \mathrm{Ra} \rightarrow{ }_{82}^{209} \mathrm{~Pb}+{ }_6^{14} \mathrm{C},{ }_{88}^{223} \mathrm{Ra} \rightarrow{ }_{86}^{219} \mathrm{Rn}+{ }_2^4 \mathrm{He}
$$
Calculate the $Q$-values for these decays and determine that both are energetically allowed.
$$
{ }_{88}^{223} \mathrm{Ra} \rightarrow{ }_{82}^{209} \mathrm{~Pb}+{ }_6^{14} \mathrm{C},{ }_{88}^{223} \mathrm{Ra} \rightarrow{ }_{86}^{219} \mathrm{Rn}+{ }_2^4 \mathrm{He}
$$
Calculate the $Q$-values for these decays and determine that both are energetically allowed.
Solution:
1458 Upvotes
Verified Answer
Let us calculate $Q$ value for the given decay process.
For first decay process,
$$
\begin{aligned}
Q &=m\left({ }_{88}^{223} \mathrm{Ra}\right)-m\left({ }_{82}^{209} \mathrm{~Pb}\right)-m\left({ }_6^{14} \mathrm{C}\right) \\
Q &=[223.01850-208.981107-14.000324] \mathrm{u}\left(\mathrm{C}^2\right) \\
&=[0.034109] \times 931.5 \mathrm{MeV}=31.85 \mathrm{MeV}
\end{aligned}
$$
For the second decay process
$Q=m\left({ }_{88}^{223} R a\right)-m\left({ }_{86}^{219} R n\right)-m\left({ }_2^4 H e\right)$
$Q=[223.01850-219.00948-4.00260] \mathrm{u} c^2$.
$Q=0.00642 \times 931.5 \mathrm{MeV}=5.98 \mathrm{MeV}$
Since, $Q$ value is positive in both the cases, hence decay process in both ways are possible.
For first decay process,
$$
\begin{aligned}
Q &=m\left({ }_{88}^{223} \mathrm{Ra}\right)-m\left({ }_{82}^{209} \mathrm{~Pb}\right)-m\left({ }_6^{14} \mathrm{C}\right) \\
Q &=[223.01850-208.981107-14.000324] \mathrm{u}\left(\mathrm{C}^2\right) \\
&=[0.034109] \times 931.5 \mathrm{MeV}=31.85 \mathrm{MeV}
\end{aligned}
$$
For the second decay process
$Q=m\left({ }_{88}^{223} R a\right)-m\left({ }_{86}^{219} R n\right)-m\left({ }_2^4 H e\right)$
$Q=[223.01850-219.00948-4.00260] \mathrm{u} c^2$.
$Q=0.00642 \times 931.5 \mathrm{MeV}=5.98 \mathrm{MeV}$
Since, $Q$ value is positive in both the cases, hence decay process in both ways are possible.
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