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

$ZX^{A} \rightarrow_{Z + 1}Y^{A} +_{- 1}e^{0} + \bar{\upsilon} + Q_{1}$

$\therefore Q_{1} = \left\lbrack m_{N}(_{Z}X^{A}) - m_{N}(_{Z + 1}Y^{A}) - m_{e} \right\rbrack c^{2}$

$= \left\lbrack m_{N}(_{Z}X^{A}) + Zm_{e} - m_{N}(_{Z + 1}Y^{A}) - (Z + 1)m_{e} \right\rbrack c^{2}$

$= \lbrack m(_{Z}X^{A}) - m(_{Z + 1}Y^{A})\rbrack c^{2} = (M_{x} - M_{y})c^{2}$

$\beta^{+}$decay is represented as

$ZX^{A} \rightarrow_{Z - 1}Y^{A} +_{1}e^{0} + \upsilon + Q_{2}\therefore Q_{2} = \left\lbrack m_{N}(_{Z}X^{A}) - m_{N}(_{Z - 1}Y^{A}) - m_{e} \right\rbrack c^{2}$

$= \left\lbrack m_{N}(_{Z}X^{A}) + Zm_{e} - m_{N}(_{Z - 1}Y^{A}) - (Z - 1)m_{e} - 2m_{e} \right\rbrack c^{2}$

$= \lbrack m(_{Z}X^{A}) - m(_{Z - 1}Y^{A}) - 2m_{e}\rbrack c^{2} = (M_{x} - M_{y} - 2m_{e})c^{2}$