Question

How many electrons and orbitals are associated with ${\text{n}} = 4$?
A. \[32,64\]
B. \[16,32\]
C. \[4,16\]
D. \[8,16\]

Answer 1

Hint: Principal, azimuthal and magnetic quantum numbers are interconnected. Thereby we can calculate the total number of electrons and orbitals. These quantum numbers can be used to explain the energy of the electron in an atom.

Given data:
Principal quantum number, \[{\text{n}} = 4\]

Complete step by step solution:
Atomic orbital is specified by four quantum numbers. They are used to describe the movement of electrons in atoms.
Principal quantum number \[{\text{n}}\] explains about the shell of the atom where \[{\text{n}}\]can be 1,2,3 etc.
Azimuthal quantum number \[{\text{l}}\] , also called orbital angular momentum quantum number explains about the shape of the orbital. Also it depends upon the value of \[{\text{n}}\] . The value of \[{\text{l}}\]lies between \[0\]and \[{\text{n}} - 1\]. It indicates the s, p, d, f subshell. If \[{\text{n}} = 3\], \[{\text{l}}\] can have values \[0,1,2\] i.e., it has s, p, d subshells.
Magnetic quantum number \[{{\text{m}}_{\text{l}}}\] defines the total number of orbitals in a subshell and their orientation. It depends upon the \[{\text{l}}\] value. \[{{\text{m}}_{\text{l}}}\]values range from \[ - {\text{l}}\]to \[ + {\text{l}}\] values. Total number of orbitals in a given subshell is given by \[2{\text{l}} + 1\]. Moreover, each orbital has two electrons.
Spin quantum number \[{{\text{m}}_{\text{s}}}\] tells about the direction in which the electron spins and is independent of other quantum numbers.
Here \[{\text{n}} = 4\].
Therefore \[{\text{l}}\] can have values \[0,1,2,3\] i.e., s, p, d, f subshells.
For \[{\text{l}} = 0,{{\text{m}}_{\text{l}}} = 2{\text{l}} + 1 = 2 \times 0 + 1 = 1\]values\[ \Rightarrow 0\]
For \[{\text{l}} = 1,{{\text{m}}_{\text{l}}} = 2l + 1 = 2 \times 1 + 1 = 3\]values\[ \Rightarrow - 1,0, + 1\]
For \[{\text{l}} = 2,{{\text{m}}_{\text{l}}} = 2{\text{l}} + 1 = 2 \times 2 + 1 = 5\]values\[ \Rightarrow - 2, - 1,0, + 1, + 2\]
For \[{\text{l}} = 3,{{\text{m}}_{\text{l}}} = 2{\text{l}} + 1 = 2 \times 3 + 1 = 7\]values\[ \Rightarrow - 3, - 2, - 1,0, + 1, + 2, + 3\]
Total number of orbitals is the total number of \[{{\text{m}}_{\text{l}}}\]values.
i.e., \[1 + 3 + 5 + 7 = 16\]orbitals.
Two electrons can be occupied in an orbital.
Therefore total number of electrons\[ = 2 \times 16 = 32\] electrons.
Hence option B is correct.

Additional information:
\[{\text{n}} = 1\]denotes the innermost shell or the first shell which is near to the nucleus. When the value of \[{\text{n}}\]increases, there will be a larger distance from the nucleus. Spin quantum number also describes whether a magnetic field can be produced by an atom or not.

Note: Magnetic quantum number depends on \[{\text{l}}\] value thereby can connect to \[{\text{n}}\]value. Two electrons of the same atom cannot have the same quantum state. This is Hund’s rule. Or it does not have the same quantum numbers.