`M_(x) and M_(y)` denote the atomic masses of the parent and the dougther 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) and Q_(2)=(M_(x)-M_(y)-2m_(e))c^(2)`
B. `Q_(1)=(M_(x)-M_(y))c^(2) and Q_(2)=(M_(x)-M_(y))c^(2)`
C. `Q_(1)=(M_(x)-M_(y)-2m_(e))c^(2) and Q_(2)=(M_(x)-M_(y)+2m_(e))c^(2)`
D. `Q_(1)=(M_(x)-M_(y)+2m_(e))c^(2) and Q_(2)=(M_(x)-M_(y)+2m_(e))c^(2)`

Answer

A

Explanation

`beta^(-)` decay is represented as
`._(Z)X^(A)to._(Z+1)A^(Y)+._(-1)e^(0)+barv+Q_(1)`
`:. Q_(1)=[m_(N)(._(Z)X^(A))-m_(N)(._(Z+1)Y^(A))-m_(e)]c^(2)=[m_(N)(._(Z)X^(A))+Zm_(e)-m_(N)(._(Z+1)Y^(A))-(Z+1)m_(e)]c^(2)`
`=[m(._(Z)X^(A))-m(._(Z+1)Y^(A))]c^(2)=(M_(x)-M_(y))c^(2)`
`beta^(+)` decay is represented as
`._(Z)X^(A)=._(Z-1)Y^(A)+._(1)e^(0)+v+Q_(2)`
`:. Q_(2)=[m_(N)(._(Z)X^(A))-m_(N)(._(Z-1)Y^(A))-m_(e)]c^(2)=[m_(N)(._(Z)X^(A))+Zm_(e)-m_(N)(._(Z-1)Y^(A))-(Z-1)m_(e)-2m_(e)] c^(2)`
`=[m(._(Z)X^(A))-m(._(Z-1)Y^(A))-2m_(e) ]c^(2)=(M_(x)-M_(y)-2m_(e))c^(2)`
`:.` Choice (a) is correct.