$M_{x}and{M}_y$  denote the atomic masses of the parent and daughter nuclei respectively in a radio active decay. The Q value of a ${\beta }^{-}$ decay is $Q_1$  and for ${\beta }^{+}$  decay is $Q_2$. If $m_e$ denotes the mass of electron, then which of following statements is correct?

detailed solution

Correct option is A

Case 1: Emission of β -10 This increases atomic numberMN(x),MN(y) mass of nucleus of X and YMx=MN(x)+Zme MN(x)=Mx-Zme My=MN(y)+(Z+1)meMN(y)=My-(Z+1)meX ZA→-βY Z+1A+β -10+ν¯+Q1Q1=MN(x)-MN(y)-mec2=Mx-Zme-My-(Z+1)me-mec2=Mx-Zme-My+Zme+me-mec2=Mx-Myc2 Case 2: Emission of β +10   This decreases atomic numberMx=MN(x)+ZmeMN(x)=Mx-ZmeMy=MN(y)+(Z-1)meMN(y)=My-(Z-1)meX ZA→βY Z-1A+β 10+ν +Q2Q2=MN(x)-MN(y)-mec2=Mx-Zme-My-(Z-1)me-mec2=Mx-Zme-My+Zme-me-mec2=Mx-My-2mec2