In a particular decay process \(^{41}_{20}\text{Ca}\) converts to \(^{41}_{19}\text{K}\) via a beta-decay process. From the given information, conclude which type of decay mode is energetically possible. [Given: \(M(^{41}_{20}\text{Ca}) = 40.962278\) u, \(M(^{41}_{19}\text{K}) = 40.961825\) u, \(M(^{1}_{1}\text{H}) = 1.007825\) u, \(M_n = 1.008665\) u, \(m_e = 0.00055\) u, \(1\ \text{u} = 931.5\ \text{MeV}/c^{2}\).]
Electron capture
Since \(Z\) decreases from 20 to 19, only \(\beta^{+}\) decay or electron capture (EC) can convert \(^{41}\text{Ca}\) into \(^{41}\text{K}\); \(\beta^{-}\) decay is ruled out.
Using atomic masses, \(Q_{EC} = [M(^{41}\text{Ca}) - M(^{41}\text{K})]\,c^{2} = (40.962278 - 40.961825) \times 931.5 = 0.000453 \times 931.5 \approx 0.422\) MeV. Positive, so EC is allowed.
For \(\beta^{+}\): \(Q_{\beta^{+}} = [M(^{41}\text{Ca}) - M(^{41}\text{K}) - 2 m_e]\,c^{2} = (0.000453 - 0.00110) \times 931.5 \approx -0.60\) MeV. Negative, so \(\beta^{+}\) is energetically forbidden.
Hence, only electron capture is energetically possible.
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