`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)andQ_(2)=[M_(x)-M_(y)-2m_(e)]c^(2)`
B. `Q_(1)=(M_(x)-M_(y))c^(2)andQ_(2)=(M_(x)-M_(y))c^(2)`
C. `Q_(1)=(M_(x)-M_(y)-2m_(e))c^(2)andQ_(2)=(M_(x)-M_(y)+2c_(e))c^(2)`
D. `Q_(1)=(M_(x)-M_(y)+2m_(e))c^(2)andQ_(2)=(M_(x)-M_(y)+2m_(e))c^(2)`

Answer

A

Explanation

Let the nucleus is `""_(z)X^(A).beta^(+)` decay is represented as
`""_(Z)X^(A)to_(Z-1)^(A)+_(+1)e^(0)+n+Q_(2)`
`Q_(2)=[(m_(n)(""_(z)X^(A))+zm_(e)-m_(n)(""z_(1)Y^A)-m_(e)]c^(2)`
`=[m_(n)(""_(z)X^(A))+zm_(e)(""_(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)-2m_(e))c^(2)`
`beta^(-)`-decay is represented as
`=""_(z)X^(A)to_(z+1)A^(Y)+_(-1)e^(0)+bar(v)+alpha_(1)`
`Q_(1)=[m_(n)(""_(z)X^(A))-m_(n)(""_(z+1)Y^(A))-(z+1)me]c^(2)`
`=[m_(n)(""_(z)X^(A))+zm_(e)-m_(n)(""_(z+1)Y^(A))-(z+1)me]c^(e)`
`=[m(""_(z)X^(A))-m(""_(z-1)Y^(A))]c^(2)=(m_(x)-M_(y))c^(2)`