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(B) For a nuclear decay to be spontaneous,the $Q$-value of the reaction must be positive,which implies that the mass of the parent nucleus must be gre

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$M(A, Z) > M(A, Z+1)$

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(B) For a nuclear decay to be spontaneous,the $Q$-value of the reaction must be positive,which implies that the mass of the parent nucleus must be greater than the sum of the masses of the daughter nucleus and the emitted particles.The decay process is: $X_Z^A ightarrow Y_{Z+1}^A + e^{-} + \bar{
u}_e$.Let $M_n(A, Z)$ be the mass of the nucleus. The condition is: $M_n(A, Z) > M_n(A, Z+1) + m_e$ (neglecting neutrino mass).Since $M(A, Z)$ represents the mass of the neutral atom,we have $M_n(A, Z) = M(A, Z) - Z m_e$.Substituting this into the condition:$(M(A, Z) - Z m_e) > (M(A, Z+1) - (Z+1) m_e) + m_e$.Simplifying the right side:$M(A, Z) - Z m_e > M(A, Z+1) - Z m_e - m_e + m_e$.$M(A, Z) - Z m_e > M(A, Z+1) - Z m_e$.Therefore,$M(A, Z) > M(A, Z+1)$.

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