Question

`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?
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 1

Correct Answer - a
`beta^(-)` decay is represented as `._ZX^A to ._(Z+1)Y^A + ._(-1)e^0 + barupsilon+Q_1`
`therefore Q_1=[m_N(._ZX^A)-m_N(._(Z+1)Y^A)-m_e]c^2`
`=[m_N(._ZX^A)+Zm_e-m_N(._(Z+1)Y^A) - (Z+1)m_e]c^2`
`=[m(._ZX^A)-m(._(Z+1)Y^A)]c^2=(M_x- M_y)c^2`
`beta^+` decay is represented as `._ZX^A=._(Z-1)Y^A + ._1e^0 + upsilon + Q_2`
`therefore Q_2=[m_N(._ZX^A)-m_N(._(Z-1)Y^A)-m_e]c^2`
`=[m_N(._ZX^A)+Zm_e-m_N(._(Z-1)Y^A) - (Z-1)m_e-2m_e]c^2`
`=[m(._ZX^A)-m(._(Z-1)Y^A)-2m_e] c^2=(M_x-M_y-2m_e) c^2`