A) \[{{Q}_{1}}=\left( {{M}_{x}}-{{M}_{y}} \right){{c}^{2}}\And \]\[{{Q}_{2}}=\left( {{M}_{x}}-{{M}_{Y}}-2{{m}_{e}} \right){{c}^{2}}\]
B) \[{{Q}_{1}}=\left( {{M}_{x}}-{{M}_{Y}} \right){{c}^{2}}\And \]\[{{Q}_{2}}=\left( {{M}_{x}}-{{M}_{y}} \right){{C}^{2}}\]
C) \[{{Q}_{1}}=\left( {{M}_{x}}-{{M}_{Y}}-2{{m}_{e}} \right){{c}^{2}}\And \]\[{{Q}_{2}}=\left( {{M}_{x}}-{{M}_{y}}+2{{m}_{e}} \right){{c}^{2}}\]
D) \[{{Q}_{1}}=\left( {{M}_{x}}-{{M}_{y}}+2{{m}_{e}} \right){{c}^{2}}\And \]\[{{Q}_{2}}=\left( {{M}_{x}}-{{M}_{Y}}+2{{m}_{e}} \right){{c}^{2}}\]
Correct Answer: A
Solution :
[a] \[{{\beta }^{-}}\] decay: As \[_{Z}^{A}X\xrightarrow{{}}_{Z+1}^{A}Y+_{-1}^{0}e+{{Q}_{1}}\] \[{{Q}_{1}}=\left[ {{m}_{N}}\left( _{Z}^{A}X \right)-{{m}_{N}}\left( _{Z+1}^{A}Y \right)-{{m}_{e}} \right]{{c}^{2}}\] \[=\left[ \left\{ m\left( _{Z}^{A}X \right)-Zme \right\}-\left\{ m\left( _{Z+1}^{A}Y \right) \right\} \right]\] \[-\left( Z+1 \right){{m}_{e}}\}-{{m}_{e}}]{{c}^{2}}\] \[=\left[ {{M}_{x}}-{{M}_{Y}}-Z{{m}_{e}}+\left( Z+1 \right){{m}_{e}}-{{m}_{e}} \right]{{c}^{2}}\] \[=\left( {{M}_{x}}-{{M}_{y}} \right){{c}^{2}}.\] \[{{B}^{+}}\] decay: \[_{Z}^{A}X\xrightarrow{{}}_{Z-1}^{A}Y+_{-1}^{0}e+{{Q}_{2}}\] \[{{Q}_{2}}=[{{m}_{N}}\left( _{Z}^{A}X \right)-{{m}_{N}}\left( _{Z-1}^{A}Y \right)-{{m}_{e}}]{{c}^{2}}\] \[=\left[ \left\{ m\left( _{Z}^{A}X \right)-Z{{m}_{e}} \right\}-\left\{ m\left( _{Z}^{A}Y \right)-(Z-1){{m}_{e}} \right\}-{{m}_{e}} \right]{{c}^{2}}\]\[=\left( {{M}_{x}}-{{M}_{Y}}-2{{m}_{e}} \right){{c}^{2}}\]
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