Question

Which of the following nuclei are doubly magic?
A) ${}_{{\text{92}}}{{\text{U}}^{{\text{238}}}}$
B) ${}_{\text{2}}{\text{H}}{{\text{e}}^{\text{4}}}$
C) ${}_{\text{8}}{{\text{O}}^{{\text{16}}}}$
D) ${}_{{\text{82}}}{\text{P}}{{\text{b}}^{{\text{208}}}}$

Answer 1

To solve this we must know that the nuclei which contain magic numbers of protons and neutrons are known as doubly magic nuclei. For each of the given nuclei calculate the number of protons and neutrons. If both the number of protons and neutrons belong to the list of magic numbers then the nuclei are doubly magic.

Complete step-by-step answer:
We know that the nuclei which contain a magic number of protons and neutrons are known as doubly magic nuclei.
The list of magic numbers is 2, 8, 20, 28, 50, 82, 126 and 184.
We know that the general representation of elements is $_{\text{Z}}{{\text{X}}^{\text{A}}}$. Where ${\text{Z}}$ is the atomic number of the element, ${\text{A}}$ is the mass number of the element and ${\text{X}}$ is the atomic symbol of the element.
The number of protons in the nucleus of the atom or the number of electrons surrounding the nucleus of the atom of any element is known as the atomic number of the element.
The sum of the number of protons and the number of neutrons in the nucleus of an atom of an element is known as the mass number of the element. Thus,
${\text{Mass number}} = {\text{Number of protons}} + {\text{Number of neutrons}}$
${\text{Number of neutrons}} = {\text{Mass number}} - {\text{Number of protons}}$
We are given four nuclei ${}_{{\text{92}}}{{\text{U}}^{{\text{238}}}}$, ${}_{\text{2}}{\text{H}}{{\text{e}}^{\text{4}}}$, ${}_{\text{8}}{{\text{O}}^{{\text{16}}}}$ and ${}_{{\text{82}}}{\text{P}}{{\text{b}}^{{\text{208}}}}$.
For ${}_{{\text{92}}}{{\text{U}}^{{\text{238}}}}$:
The atomic number is 92. Thus, the number of protons is 92.
The mass number is 238. The number of neutrons is,
${\text{Number of neutrons}} = {\text{238}} - {\text{92}} = {\text{146}}$
The numbers 92 and 146 do not belong to the list of magic numbers. Thus, ${}_{{\text{92}}}{{\text{U}}^{{\text{238}}}}$ is not a doubly magic nuclei.

For ${}_{\text{2}}{\text{H}}{{\text{e}}^{\text{4}}}$:
The atomic number is 2. Thus, the number of protons is 2.
The mass number is 4. The number of neutrons is,
${\text{Number of neutrons}} = 4 - 2 = {\text{2}}$
The number 2 belongs to the list of magic numbers. Thus, ${}_{\text{2}}{\text{H}}{{\text{e}}^{\text{4}}}$ is a doubly magic nuclei.

For ${}_{\text{8}}{{\text{O}}^{{\text{16}}}}$:
The atomic number is 8. Thus, the number of protons is 8.
The mass number is 16. The number of neutrons is,
${\text{Number of neutrons}} = 16 - 8 = {\text{8}}$
The number 8 belongs to the list of magic numbers. Thus, ${}_{\text{8}}{{\text{O}}^{{\text{16}}}}$ is a doubly magic nuclei.
For ${}_{{\text{82}}}{\text{P}}{{\text{b}}^{{\text{208}}}}$:
The atomic number is 82. Thus, the number of protons is 82.
The mass number is 208. The number of neutrons is,
${\text{Number of neutrons}} = {\text{208}} - {\text{82}} = {\text{126}}$
The numbers 82 and 126 belong to the list of magic numbers. Thus, ${}_{{\text{82}}}{\text{P}}{{\text{b}}^{{\text{208}}}}$ is a doubly magic nuclei.
Thus, the doubly magic nuclei are ${}_{\text{2}}{\text{H}}{{\text{e}}^{\text{4}}}$, ${}_{\text{8}}{{\text{O}}^{{\text{16}}}}$ and ${}_{{\text{82}}}{\text{P}}{{\text{b}}^{{\text{208}}}}$.

Thus, the correct options are (B), (C) and (D).

Note: Remember that the list of magic numbers is 2, 8, 20, 28, 50, 82, 126 and 184. The magic number is the number of nucleons such that they are arranged into complete shells within the atomic nucleus. The nucleons are the protons or neutrons or both.