`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) " 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)) c^(2) " and " Q_(2) = (M_(x) -M_(y) +2c_(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

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