Question

Calculate the mass defect for the ${\mathit{\beta }}^{\mathbf{+}}$decay of ${}_{\mathbf{6}}{}^{\mathbf{11}}\mathbf{}$C. Is this decay energetically possible? Why or why not? The atomic mass of ${}_{\mathbf{6}}{}^{\mathbf{11}}\mathbf{}$C is 11.011434 u.

Short answer

This decay is energetically possible.

Step-by-step solution

Definition of β+ decay

${\mathit{\beta }}^{\mathbf{+}}$decay can occur whenever the mass of the original neutral atom is at least two electron masses larger than that of the final atom.

Determine the daughter nucleus

The decay equation is given as:

${}_{\mathbf{6}}{}^{\mathbf{11}}\mathit{C}\mathbf{\to }{}_{\mathbf{Z}}{}^{\mathbf{A}}\mathit{X}\mathbf{+}{}_{\mathbf{1}}{}^{\mathbf{0}}\mathit{e}$

From the conservation of atomic number and the number of nucleons,

$$\begin{gathered} \mathbf{}\mathbf{6}\mathbf{=}\mathit{Z}\mathbf{+}\mathbf{1}\mathbf{\Rightarrow }\mathit{Z}\mathbf{=}\mathbf{5} \\ \mathbf{11}\mathbf{=}\mathit{A}\mathbf{+}\mathbf{0}\mathbf{\Rightarrow }\mathit{A}\mathbf{=}\mathbf{11} \end{gathered}$$

So, the daughter nucleus is ${}_{\mathbf{5}}{}^{\mathbf{11}}\mathit{B}$ with atomic mass of 11.009305u.

Determine if the decay is energetically possible or not

${\mathit{\beta }}^{\mathbf{+}}$decay can occur whenever the mass of the original neutral atom is at least two electron masses larger than that of the final atom.

Mass defect can be given as:

$\mathbf{∆}\mathit{M}\mathbf{=}\mathit{M}\mathbf{\left(}{}_{\mathbf{6}}{}^{\mathbf{11}}\mathit{C}\mathbf{\right)}\mathbf{-}\mathbf{2}{\mathit{m}}_{\mathbf{e}}\mathbf{-}\mathit{M}\mathbf{\left(}{}_{\mathbf{5}}{}^{\mathbf{11}}\mathit{B}\mathbf{\right)}$

Plug the values,

$\mathbf{∆}\mathit{M}\mathbf{=}\mathbf{11}\mathbf{.}\mathbf{011434}\mathbf{-}\mathbf{2}\mathbf{*}\mathbf{0}\mathbf{.}\mathbf{000548580}\mathbf{-}\mathbf{11}\mathbf{.}\mathbf{009305}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{032}\mathbf{*}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}\mathbf{}\mathit{u}$

Since the mass defect is positive, the decay is energetically possible.

Thus, this decay is energetically possible.