`M_(x) and M_(y)` denote the atomic masses of the parent and the daughter nuclei 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 electrons, then which of the following statements is correct?

Options

A. `Q_(1)=(M_(X)-M_(Y))c^(2)" &"`
`Q_(2)=(M_(X)-M_(Y)-2m_(e))c^(2)`
B. `Q_(1)=(M_(X)-M_(Y))c^(2)" &"`
`Q_(2)=(M_(X)-M_(Y))c^(2)`
C. `Q_(1)=(M_(X)-M_(Y)-2m_(e))c^(2)" &"`
`Q_(2)=(M_(X)-M_(Y)+2m_(e))c^(2)`
D. `Q_(1)=(M_(X)-M_(Y)+2m_(e))c^(2)" &"`
`Q_(2)=(M_(X)-M_(Y)+2m_(e))c^(2)`

Answer

A

Explanation

`beta^(-)` decay: As `""_(Z)^(A)Xto""_(Z+1)^(" "A)Y+""_(-1)^(" "0)e+Q_(1)`
`Q_(1)=[m_(N)(""_(Z)^(A)X)-m_(N)(""_(Z+1)^(" "A)Y)-m_(e)]c^(2)`
`=[{m(""_(Z)^(A)X)-Zme}-{m(""_(Z+1)^(" "A)Y)-(Z+1)m_e)}-m_(e)]c^(2)`
`=[M_(X)-M_(Y)-Zm_(e)+(Z+1)m_(e)-m_(e)]c^(2)`
`=(M_(X)-M_(Y))c^(2)`
`B^(+)" decay:"`
`""_(Z)^(A)Xto""_(Z+1)^(" "A)Y+""_(-1)^(" "0)e+Q_(2)`
`Q_(2)=[m_(N)(""_(Z)^(A)X)-m_(N)(""_(Z+1)^(" "A)Y)-m_(e)]c^(2)`
`=[{m(""_(Z)^(A)X)-Zm_(e)}-{m(""_(Z)^(A)Y)-(Z-1)m_(e)}-m_(e)]c^(2)`
`=(M_(X)-M_(Y)-2m_(e))c^(2)`