A nucleus with mass number 184 initially at r | vedclass

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Question

(D) Let $M = 184$ be the mass number of the parent nucleus. The daughter nucleus has mass number $M' = 180$ and the $\alpha$-particle has mass number

Vedclass · Accepted Answer

$5.38$

Vedclass · Answer

(D) Let $M = 184$ be the mass number of the parent nucleus. The daughter nucleus has mass number $M' = 180$ and the $\alpha$-particle has mass number $m_{\alpha} = 4$.By conservation of linear momentum,$p_{\alpha} = p_{d}$,where $p_{\alpha}$ is the momentum of the $\alpha$-particle and $p_{d}$ is the momentum of the daughter nucleus.Since $K = \frac{p^2}{2m}$,the kinetic energy $K_{\alpha}$ of the $\alpha$-particle and $K_{d}$ of the daughter nucleus are related by $K_{d} = K_{\alpha} \cdot \frac{m_{\alpha}}{M'} = K_{\alpha} \cdot \frac{4}{180} = \frac{K_{\alpha}}{45}$.The total $Q$ value is the sum of the kinetic energies: $Q = K_{\alpha} + K_{d} = K_{\alpha} + \frac{K_{\alpha}}{45} = K_{\alpha} \left(1 + \frac{1}{45}ight) = K_{\alpha} \left(\frac{46}{45}ight)$.Given $Q = 5.5\, 	ext{MeV}$,we have $5.5 = K_{\alpha} \cdot \frac{46}{45}$.Therefore,$K_{\alpha} = 5.5 \cdot \frac{45}{46} \approx 5.38\, 	ext{MeV}$.

$A$ nucleus with mass number $184$ initially at rest emits an $\alpha$-particle. If the $Q$ value of the reaction is $5.5\, \text{MeV}$,calculate the kinetic energy of the $\alpha$-particle in $\text{MeV}$.

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