Question

The number of orbitals with \[{\mathbf{n}} = {\text{ }}{\mathbf{5}}\],\[{{\mathbf{m}}_{\mathbf{1}}}\; = {\text{ }} + {\mathbf{2}}\] is ___________. (Round off to the nearest integer).

Answer 1

Hint: The quantum numbers are principal quantum number (n), azimuthal quantum number (l), magnetic quantum number (ml) and spin quantum number (ms). These quantum numbers are used to describe an electron in an orbital. Principle quantum number (n) signifies shell and azimuthal quantum number (l) signifies sub-shell of the orbital. Magnetic quantum number represents the orientation of orbitals in the subshell. Spin quantum number represents the angular momentum of the electron.

Complete Step by Step Solution:
Given principal quantum number, \[{\mathbf{n}} = {\text{ }}{\mathbf{5}}\]
Therefore, azimuthal quantum number is given by formula \[{\mathbf{l}} = {\mathbf{n}} - {\mathbf{1}} \ldots \ldots {\mathbf{1}} = {\text{ }}{\mathbf{4}},{\mathbf{3}},{\mathbf{2}},{\mathbf{1}}\]. Also, magnetic quantum number is given \[{\mathbf{m}} = - {\mathbf{l}}{\text{ }}{\mathbf{to}} + {\mathbf{l}}\]

Thus, \[{\mathbf{m}} = - {\mathbf{4}}, - {\mathbf{3}}, - {\mathbf{2}}, - {\mathbf{1}},{\mathbf{0}},{\mathbf{1}},{\mathbf{2}},{\mathbf{3}},{\mathbf{4}}\]for \[{\mathbf{l}} = {\mathbf{4}}\]
\[{\mathbf{m}} = - {\mathbf{3}}, - {\mathbf{2}}, - {\mathbf{1}},{\mathbf{0}},{\mathbf{1}},{\mathbf{2}},{\mathbf{3}}\]for \[{\mathbf{l}} = {\mathbf{3}}\]
\[{\mathbf{m}} = - {\mathbf{2}}, - {\mathbf{1}},{\mathbf{0}},{\mathbf{1}},{\mathbf{2}}\]for \[{\mathbf{l}} = {\mathbf{2}}\]
\[{\mathbf{m}} = - {\mathbf{1}},{\mathbf{0}},{\mathbf{1}}\]for \[{\mathbf{l}} = {\mathbf{1}}\]
Thus, \[{\mathbf{m}} = + {\mathbf{2}}\] appears in\[{\mathbf{l}} = {\text{ }}{\mathbf{4}},{\mathbf{3}},{\mathbf{2}}\]. So, the number of orbitals having value of m as \[ + {\mathbf{2}}\] is\[{\mathbf{3}}\].

Note: The value of l cannot exceed the value of n and also, the value of m cannot exceed the value of l. Principal and azimuthal quantum numbers cannot have a negative value. Spin quantum numbers have a value \[ + \frac{1}{2}\]and $ - \frac{1}{2}$. There are only two values of spin quantum number because a single orbital can accommodate only two electrons in an orbital. It is not necessary that \[ + \frac{1}{2}\]represents clockwise direction and $ - \frac{1}{2}$represents negative. Positive and negative signs just represent that the spin direction is the reverse of one another.