Question

$M_x$ and $M_y$ denote the atomic masses of the parent and the daughter atom respectively in a radioactive decay. The $Q$-value for a $\beta^-$ decay is $Q_1$ and that for a $\beta^+$ decay is $Q_2$ . If $m_e$ denotes the mass of an electron, then which of the following statements is correct?

• A. $Q_{1} = \left(M_{x}-M_{y}\right)c^{2}$ and $Q_{2} = \left(M_{x} -M_{y} -2m_{e}\right)c^{2}$
• B. $Q_{1} = \left(M_{x}-M_{y}\right)c^{2}$ and $Q_{2} = \left(M_{x} -M_{y} \right)c^{2}$
• C. $Q_{1} = \left(M_{x}-M_{y} -2m_{e}\right)c^{2}$ and $Q_{2} = \left(M_{x} -M_{y} + 2m_{e}\right)c^{2}$
• D. $Q_{1} = \left(M_{x}-M_{y} +2m_{e}\right)c^{2}$ and $Q_{2} = \left(M_{x} +M_{y} + 2m_{e}\right)c^{2}$

Answer 1

Answer: (A)

$Q_{1} = \left(M_{x}-M_{y}\right)c^{2}$ and $Q_{2} = \left(M_{x} -M_{y} -2m_{e}\right)c^{2}$

Answer 2

$\beta^-$ decay is represented as $_{z}X^{A} \rightarrow _{z+1}Y^{A} +_{-1}e^{0} + \bar{\upsilon} +Q_{1}$ $\therefore Q_{1} = \left[m_{N}\left(_{Z}X^{A}\right)-m_{N}\left(_{Z+1}Y^{A}\right) -m_{e}\right]c^{2}$ $= \left[m_{N} \left(_{Z}X^{A}\right) + Zm_{e} -m_{N}\left(_{Z+1}Y^{A}\right)-\left(Z+1\right)m_{e}\right]c^{2}$ $= \left[m \left(_{Z}X^{A}\right) -m\left(_{Z+1}Y^{A}\right)\right] c^{2}$ $= \left(M_{x} - M_{y}\right)c^{2}$ $\beta^+$ decay is represented as $_{Z}X^{A} = _{Z-1}Y^{A} +_{1}e^{0}+\upsilon +Q_{2}$ $\therefore Q_{2} = \left[m_{N}\left(_{Z}X^{A}\right) -m_{N}\left(_{Z-1}Y^{A}\right)-m_{e}\right]c^{2}$ $= \left[m_{N}\left(_{Z}X^{A}\right) +Zm_{e}-m_{N}\left(_{Z-1}Y^{A}\right)-\left(Z-1\right)m_{e}-2m_{e}\right]c^{2}$ $=\left[m\left(_{Z}X^{A}\right) -m\left(_{Z-1}Y^{A}\right) -2m_{e}\right]c^{2}$ $=\left(M_{x}-M_{y}-2m_{e}\right)c^{2}$