Find whether alpha decay or any of the beta decays are allowed for { }_{89}^{226} Ac.
Strategy For each of the possible four reactions, we first write the reaction, list the disintegration energy Q equation, look up the appropriate masses, and calculate the disintegration energy Q. If Q > 0, the decay is allowed.
We outline the solution, but leave looking up the masses and doing the math to the student.
Alpha decay: { }_{89}^{226} Ac \rightarrow{ }_{87}^{222} Fr +\alpha
\begin{aligned}Q &=\left[M\left({ }_{89}^{226} Ac \right)-M\left({ }^{222} Fr \right)-M\left({ }^{4} He \right)\right] c^{2} \\&=5.54 MeV \quad \text { Alpha decay is allowed }\end{aligned}
\beta ^{-} decay: { }_{89}^{226} Ac \rightarrow{ }_{90}^{226} Th +\beta^{-}+\bar{\nu}
\begin{aligned}Q &=\left[M\left({ }_{89}^{226} Ac \right)-M\left({ }_{90}^{226} Th \right)\right] c^{2} \\&=1.12 MeV \quad \beta^{-} \text {decay is allowed }\end{aligned}
\beta ^{+} decay: { }_{89}^{226} Ac \rightarrow{ }_{88}^{226} Ra +\beta^{+}+\nu
\begin{aligned}Q &=\left[M\left({ }_{89}^{226} Ac \right)-M\left({ }_{88}^{226} Ra \right)-2 m_{e}\right] c^{2} \\&=-0.38 MeV \quad \beta^{+} \text {decay is not allowed }\end{aligned}
Electron capture: { }_{89}^{226} Ac +e^{-} \rightarrow{ }_{88}^{226} Ra +\nu<br />
\begin{aligned}Q &=\left[M\left({ }_{89}^{226} Ac \right)-M\left({ }_{88}^{226} Ra \right)\right] c^{2} \\&=0.64 MeV \quad \text { Electron capture is allowed }\end{aligned}
We find that α decay, \beta^{-} decay, and electron capture are all possible from the same nucleus. Experiment shows that alpha decay occurs only 0.006% of the time for { }^{226} Ac , \beta^{-} decay 83%, and electron capture 17%.