Question

The magnetic quantum number ${m_l}$ is fixed by the azimuthal quantum number, $l$ , if$l = 2$ , the type and number of the orbitals indicated by,
$(A)f,7$
$(B)d,5$
$(C)p,3$
$(D)s,1$

Answer 1

The magnetic quantum number represents the orbitals available inside a subshell and is used to find the azimuthal component of the orientation of the orbital in space. Electrons inside a particular subshell are denoted by values of azimuthal quantum numbers. Use the relation between the number of orbitals and the azimuthal quantum numbers. After getting the orbital number the type of the orbital can also be found.
Formula used:
The number of the orbitals is $= 2l + 1$ , where $l$ is the azimuthal quantum number.

Complete answer:
Magnetic Quantum Number ${m_l}$ $( - l,....0,....l)$ denotes the orientation in space of an orbital of given energy and shape$\left( l \right)$ . This number classifies the subshell into individual orbitals which keep the electrons; there are $(2l + 1)$ orbitals in every subshell.
The number of the orbitals is $= 2l + 1$ , where $l$ is the azimuthal quantum number.
$\therefore$ The number of the orbitals$= 2 \times 2 + 1 = 5$
Electrons inside a particular subshell (such as $s,p,d,f$ ) are denoted by values of azimuthal quantum numbers $l(0,1,2,3....)$.
The type of orbital depends on the azimuthal quantum number, such as,
$l = 0 \Rightarrow s{\text{ orbital}}$
$l = 1 \Rightarrow p{\text{ orbital}}$
$l = 2 \Rightarrow d{\text{ orbital}}$
$l = 3 \Rightarrow f{\text{ orbital}}$
So, the type $d{\text{ orbital}}$ and the number of the orbitals are $5$.

Hence, the right answer is in option $(B) \Rightarrow d,5$.

Note:
To define an electron in an atom, four quantum numbers are necessary: energy$\left( n \right)$ , angular momentum $(l)$ , magnetic moment $({m_l})$and, spin$\left( {{m_s}} \right)$ .
The first quantum number defines the shell of an electron i.e. energy level of an atom. The value of $n$ranges from one to the shell having the outermost electron of that atom.
The dynamics of the quantum system are defined by a quantum Hamiltonian $\left( H \right)$