`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)- 2m_(e)) 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) rarr ""_(z-1)y^(4) + ""(+1) e^(0) + v+ Q_(2)`
`therefore 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)`
`beta^(-1)` decay is represented as :
`""_(z)X^(A) rarr ""_(z+1)A^(y) + ""_(-1) e^(0)+ vec(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))- (x+1) me ]c^(2)`
`= [m (""_(z)X^(A))- m (""_(z-1)Y^(A))] c^(2)`
`= (M_(x)- M_(y))c^(2)`