Question

In an $\alpha -\text{decay}$, the kinetic energy of $\alpha -\text{particle}$ is $48\text{MeV}$ and $Q$ value of the reaction is $50\text{MeV}$. The mass number of the mother nucleus is (assume that daughter nucleus is in ground state)

• A. $96$
• B. $100$
• C. $104$
• D. none of these

Answer 1

The correct option is B $100$

The $\alpha -\text{decay}$ is represented as,

${}_Z^{A}X\to {}_{A-2}^{A-4}Y+{}_2^4\text{He}$

We know that,

$Q=K.E_{\alpha }+K.E_Y$

$\Rightarrow K.E_Y=Q-K.E_{\alpha }$

$\Rightarrow K.E_Y=50-48=2\text{MeV}$

According to conservation of momentum,

$\Rightarrow \left|P_Y\right|=\left|P_{\alpha }\right|$

We know that,

$K.E=\frac{P^2}{2m}$

$\Rightarrow \frac{K.E_{\alpha }}{K.E_Y}=\frac{m_Y}{m_{\alpha }}$

We know that, mass of the nucleus is proportional to mass number,

$\Rightarrow \frac{K.E_{\alpha }}{K.E_Y}=\frac{A-4}{4}$

$\Rightarrow \frac{48\text{MeV}}{2\text{MeV}}=\frac{A-4}{4}$

$\Rightarrow A=100$

Therefore, the mass number of the daughter nucleus is $A=100$.

Hence, option $\left(B\right)$ is correct.