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For the beta^(+) (positron) emission from a nucleus, there is another competing process known as electron capture (electron from an inner orbit, say, the K - shell, is captured by the nucleus and a neutrino is emitted). e^(+) ._(Z)^(A)X to ._(Z-1)^(A)Y+v Show that if beta^(+) emission is energetically allowed, electron capture is necessarily allowed but not vice-versa. |
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Answer» Solution :The `beta^(+)` emission from a nucleus `._(Z)X^(A)` may be represented as `._(Z)X^(A)= ._(Z-1)Y^(A)+ ._(1)e^(0)+ V + Q_(1)` ______ (i) The other competing process of electron CAPTURE may be represented as `._(-1)e^(0)+ ._(Z)X^(A)= ._(Z-1)Y^(A)+ v + Q_(2)` _______ (ii) The energy released `Q_(1)` in 1. is given by `Q_(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)m_(e)-2m_(e)]c^(2)` `Q_(1)=[m(._(Z)X^(A))-m(._(Z-1)Y^(A))-2m_(e)]c^(2)` _______ (iii) Note that `m_(N)` here denotes mass ofnucleus and m denotes the mass of atom similarly from (ii) `Q_(2)=[m_(N)(._(Z)X^(A))+m_(e)-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)-m_(e)]c^(2)` `Q_(2)=[m(._(X)Z^(A))+m-(._(Z-1)Y^(A))]c^(2)` `= [m_(N)(._(Z)X^(A))+Zm_(e)+m_(e)-m_(N) (._(Z-1)Y^(A))-(Z-1)m_(e)-m_(e)]c^(2)` `Q_(2)=[m(._(Z)X^(A))-m-(._(Z-1)Y^(A))]c^(2)` Ir `Q_(1)gt 0` then `Q_(2) gt 0`. i.e., If positron emission is energetically allowed electron capture is necessarily allowed. But `Q_(2)gt 0` does not necessarily mean `Q_(1)gt 0`. Hence the REVERSE is not TRUE. |
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