Write the decay equations and expression for the disintegration energy` Q` of the following decay: `beta-` decay,electron capture.

Explanation

In all the cases discussed here, we neglect any neutrino mass.
`beta-` decay: `M_(nucl) (._(Z)^(X) X) =M_(nucl) (._(Z+1)^(A) D) + m_(e) +Q//c^(2)`
Where `M_(nucl)` indicates the nuclear mass. In order to convert it to atomic mass, we add `Zm_(e)` on both sided:
`M._(nucl) (._Z^(AX)+Zm._(e) =M._(nucl) (._(Z+1)^(A)D)+(Z+1) m._(e)+Q//c^(2)`
Since atomic binding energies are less than nuclear binding energies, they are neglected on the two sides of the equation.
`[M((._Z)^(A)X)=M(._(Z+1)^(A)D)]c^(2)`
`Q` for this equation `=[M(._Z^AX)-M(._(Z+1)^(A)D)]c^(2)`
`beta+` decay: `M_(nucl) (._Z^AX)=M_(nucl) (._(Z-1)^(A)D) +m._(e) +Q//c^(2)`
In this case, only `Z-1 m` is needed for the daughter atomic mass which gives us a remaining mass of `2 m_(e)`
`[M(._Z)^(A)X)-M (._(Z-1)^(A)D)-2m_(e)]c^(2)`
`Q` for this equation` =[M(._Z)^AX)-M (._(Z-1)^AD)-2m_(e)]c^(2)`
We add `(Z-1) m_(e)` to each side above and obtain
`M._(nucl)(._Z^AX)-M (._(Z-1)^AD)+Q//c^(2)`
`Q` for this equtation`=[M(._Z)^AX)-M (._(Z-1)^AD)-2m_(e)]c^(2)`
Electron capture`.