Question

A stationary Thorium nucleus $(A=220,Z=90)$ emits an alpha particle with kinetic energy $E_{\alpha }$. What is the kinetic energy of the recoiling nucleus?

• A. $\frac{E_{\alpha }}{108}$
• B. $\frac{E_{\alpha }}{110}$
• C. $\frac{E_{\alpha }}{55}$
• D. $\frac{E_{\alpha }}{54}$

Answer 1

The correct option is D $\frac{E_{\alpha }}{54}$

The given $\alpha -$decay can be represented by

${}_{90}^{220}\text{Th}\to {}_{66}^{216}\text{Ra}+{}_2^4\text{He}$

As the Thorium nucleus is at rest,

By conservation of momentum,

$0=P_{\text{Ra}}+P_{\alpha }$

$\Rightarrow \left|P_{\text{Ra}}\right|=\left|P_{\alpha }\right|$

We know that, $K.E=\frac{P^2}{2m}$

$\Rightarrow \frac{E_{\text{Ra}}}{E_{\alpha }}=\frac{m_{\alpha }}{m_{\text{Ra}}}$

We know that,

$\text{mass of the nucleus}\propto \text{mass number}$

$\Rightarrow \frac{E_{\text{Ra}}}{E_{\alpha }}=\frac{4}{216}$

$\Rightarrow E_{\text{Ra}}=\frac{E_{\alpha }}{54}$

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

Why this Question? Key concepts: $\alpha -decay$ decreases mass number by 4 and reduces atomic number by 2. We can apply conservation of momentum when the net external force on a system is zero.