Question: Which of the following nuclei is most likely to decay by positron emission? Explain your choice.

(a) chromium\({\rm{ - 53}}\)

(b) manganese\({\rm{ - 51}}\)

(c) iron\({\rm{ - 59}}\)

Positron emission, also known as beta plus decay or\({{\rm{\beta }}^{\rm{ + }}}\)decay, is a kind of radioactive decay in which a proton inside a radionuclide nucleus is converted to a neutron while simultaneously emitting a positron and an electron neutrino\({\rm{(}}{{\rm{v}}_{\rm{e}}}{\rm{)}}\). The weak force is involved in positron emission.

(a)

Through the periodic table, it is known that the atomic number of\(Cr\)is\(24\), which means the number of protons is\(24\).

Now, calculate the number of neutrons –

\({\rm{A = N}}\left( {{{\rm{p}}^{\rm{ + }}}} \right){\rm{ + N}}\left( {{{\rm{n}}^{\rm{0}}}} \right)\)

On rearranging the equation, substituting the values, and solving:

\(\begin{array}{c}{\rm{N}}\left( {{{\rm{n}}^{\rm{0}}}} \right){\rm{ = A - N}}\left( {{{\rm{p}}^{\rm{ + }}}} \right)\\{\rm{N}}\left( {{{\rm{n}}^{\rm{0}}}} \right){\rm{ = 53 - 24 = 29 }}\end{array}\)

The ratio is calculated as :

\(\begin{array}{c}{\rm{ratio = }}\frac{{{\rm{N}}\left( {{{\rm{n}}^{\rm{0}}}} \right)}}{{{\rm{N}}\left( {{{\rm{p}}^{\rm{ + }}}} \right)}}\\{\rm{ratio = }}\frac{{{\rm{29}}}}{{{\rm{24}}}}{\rm{ = 1}}{\rm{.21}}\end{array}\)

Therefore, the value of the ratio is obtained as \({\rm{1}}{\rm{.21}}\), which is not the lowest ratio among chromium\({\rm{ - 53}}\), magnesium\({\rm{ - 51}}\) , and iron\({\rm{ - 59}}\).

(b)

Through the periodic table, it is known that the atomic number of\(Cr\)is\(25\), which means the number of protons is\(25\).

Now, calculate the number of neutrons:

\({\rm{A = N}}\left( {{{\rm{p}}^{\rm{ + }}}} \right){\rm{ + N}}\left( {{{\rm{n}}^{\rm{0}}}} \right)\)

On rearranging the equation, substituting the values, and solving:

\(\begin{array}{c}{\rm{N}}\left( {{{\rm{n}}^{\rm{0}}}} \right){\rm{ = A - N}}\left( {{{\rm{p}}^{\rm{ + }}}} \right)\\{\rm{N}}\left( {{{\rm{n}}^{\rm{0}}}} \right){\rm{ = 51 - 25 = 26 }}\end{array}\)

The ratio is calculated as:

\(\begin{array}{c}{\rm{ratio = }}\frac{{{\rm{N}}\left( {{{\rm{n}}^{\rm{0}}}} \right)}}{{{\rm{N}}\left( {{{\rm{p}}^{\rm{ + }}}} \right)}}\\{\rm{ratio = }}\frac{{{\rm{26}}}}{{{\rm{25}}}}{\rm{ = 1}}{\rm{.04}}\end{array}\)

Therefore, the value of the ratio is obtained as \({\rm{1}}{\rm{.04}}\), the lowest ratio among chromium\({\rm{ - 53}}\), magnesium\({\rm{ - 51}}\) , and iron\({\rm{ - 59}}\).

(c)

Through the periodic table, it is known that the atomic number of\(Cr\)is\(26\), which means the number of protons is\(26\).

Now, calculate the number of neutrons:

\({\rm{A = N}}\left( {{{\rm{p}}^{\rm{ + }}}} \right){\rm{ + N}}\left( {{{\rm{n}}^{\rm{0}}}} \right)\)

On rearranging the equation, substituting the values, and solving:

\(\begin{array}{c}{\rm{N}}\left( {{{\rm{n}}^{\rm{0}}}} \right){\rm{ = A - N}}\left( {{{\rm{p}}^{\rm{ + }}}} \right)\\{\rm{N}}\left( {{{\rm{n}}^{\rm{0}}}} \right){\rm{ = 59 - 26 = 33 }}\end{array}\)

The ratio is calculated as:

\(\begin{array}{c}{\rm{ratio = }}\frac{{{\rm{N}}\left( {{{\rm{n}}^{\rm{0}}}} \right)}}{{{\rm{N}}\left( {{{\rm{p}}^{\rm{ + }}}} \right)}}\\{\rm{ratio = }}\frac{{33}}{{{\rm{26}}}}{\rm{ = 1}}{\rm{.27}}\end{array}\)

Therefore, the value of the ratio is obtained as \({\rm{1}}{\rm{.27}}\), which is the highest ratio among chromium\({\rm{ - 53}}\), magnesium\({\rm{ - 51}}\) , and iron\({\rm{ - 59}}\).