Question

The following quantum numbers are possible for how many orbital $n = 3,{\rm{ }}l = 2,{\rm{ }}m = + 2$?
A. $1$
B. $2$
C. $3$
D. $4$

Answer 1

In the question $n$ denotes principal quantum number; $l$ is known as azimuthal quantum number while $m$ is known as magnetic quantum number. For the quantum number we have to first find that quantum number which satisfies all other conditions i.e. $n = 3,{\rm{ }}l = 2,{\rm{ }}m = + 2$ .

Complete step by step answer:
The quantum numbers are the identification numbers for an individual electron in an atom, since these fully describe the position and energy of an electron in an atom.
The principal quantum number ‘$n$’ is a positive integer with value of n = 1,2,3.......The principal quantum number determines the size and to large extent the energy of the orbital. For hydrogen atom and hydrogen like species (He+, Li2+, .... etc.) energy and size of the orbital depends only on ‘$n$’. Azimuthal quantum number. ‘$l$’ is also known as orbital angular momentum or subsidiary quantum number. It defines the three-dimensional shape of the orbital. For a given value of $n$, $l$ can have $n$values ranging from 0 to n – 1, that is, for a given value of $n$, the possible value of $l$is
$:{\rm{ }}l{\rm{ }} = {\rm{ }}0,{\rm{ }}1,{\rm{ }}2,{\rm{ }}..........\left( {n-1} \right)$.
Magnetic orbital quantum number. ‘$m$’ gives information about the spatial orientation of the orbital with respect to standard set of co-ordinate axis. For any sub-shell (defined by ‘l’ value) $2l + 1$ values of ${m_l}$ are possible and these values are given by:
${m_l} = {\rm{ }}-{\rm{ }}l,{\rm{ }}-{\rm{ }}\left( {l{\rm{ }}-1} \right),{\rm{ }}-{\rm{ }}\left( {l-2} \right)...{\rm{ }}0,1...{\rm{ }}\left( {l{\rm{ }}-2} \right),{\rm{ }}\left( {l-1} \right),{\rm{ }}l$.
$\begin{array}{l} n = 3,l = 2\\\ m = - 2, - 1,0,1,2 \end{array}$
So, for $n = 3$ and $m = + 2$ we have only one orbital.

The correct option is A.

Note:
The principle quantum number 3 denotes 3rd orbit while magnetic quantum number 3 denotes we are taking about 3d subshell. And we know that d subshell contains 5 atomic orbitals and can hold a maximum of 10 electrons.