1.

Find the energy `Q` liberated in `beta^(-_(-))` and `beta^(+)-` decays and in `K`- caputure if the masses of the parent atom `m_(p)`, the daughter atom `M_(d)` and an electron `m` are known.

Answer» In `beta^(-)` decay
`Z^(X^(A))rarr_(Z+1)Y^(A)+e^(_)+Q`
`Q=(M_(x)-M_(y)-m_(e ))c^(2)`
`=[(M_(x)+ZM_(e )c^(2))-(M_(y)+Zm_(e )+m_(e ))]c^(2)`
`=(M_(p)-M_(d))c^(2)`
since `M_(p),M_(d)` are the masses of the atoms. The binding energy of the electrons in ignored. In `K` capture
`e_(k)^(-)+._(Z)X^(A) rarr_(z-1)Y^(A)+Q`
`Q=(M_(X)-M_(Y))c^(2)+m_(e )c^(2)`
`=(M_(x)^(c^(2))+Zm_(e )c^(2))-(M_(Y)c^(2)+(Z-1)m_(e )c^(2))`
`=c^(2)(M_(p)-M_(d))`
In `beta^(+) decay _(Z)X^(A) rarr_(Z-1)Y^(A)+e^(+)+Q`
Then `Q= (M_(x)-M_(y)-m_(e ))c^(2)`
`[M_(x)+Zm_(e )]c^(2)-[M_(y)+(Z-1)m_(e )]c^(2)-2m_(e )c^(2)`
`=(M_(p)-M_(d)-2m_(e ))c^(2)`


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