Question
The radionuclide $^{11}C$ decays according to
$^{11}_{6}\text{C}\rightarrow^{11}_{5}\text{B}+\text{e}^{+}+\text{v}:\ \text{T}_{1/2}=20.3 \text{ min}$
The maximum energy of the emitted positron is 0.960 MeV.
Given the mass values:
$\text{m}(^{11}_{6})=10=11.011434\text{ u and m}(^{11}_{6}\text{B})=11.009305\text {u}.$
calculate Q and compare it with the maximum energy of the positron emitted.

Answer

For the given reaction, mass defect is,
$\Delta\text{m}=[\text{m}(^{11}_{6}-6\text{m}_\text{e})]-[\text{m}(^{11}_{5}\text{B})-5\text{ m}_\text{e}+\text{m}_\text{e}]$
$=\text{m}(^{11}_{6}\text{C})-\text{m}(^{11}_{5}\text{B})-2\text{m}_\text{e}$
$= 11.011434\ u -11.009305\ u - 2 × 0.000548\ u$
$= 0.001033$ u
Now, Q-value is,
$Q = 0.001033 × 931.5$ MeV
$= 0.962$ MeV
which, is the maximum energy of the positron.
We have,
$\text{Q}=\text{E}_\text{d}+\text{E}_\text{e}+\text{E}_\text{v}$
The daughter nucleus is too heavy compared to $e^+$ and v. So, it carries negligible energy $(\text{E}_\text{d}\approx0).$ If the kinetic energy $(E_v)$ carried by the neutrino is minimum (i.e., zero), the positron carries maximum energy, and this is practically all energy Q.
Hence, maximum $\text{E}_\text{e}\approx\text{Q}.$

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