The maximum number of orbitals which can be identified with n = 4 and m_l = 0 is_____

The correct answer is 4.

To determine the maximum number of orbitals that can be identified with the quantum numbers $n=4$ and $m_l=0$, we analyze the permissible values of the electronic quantum numbers step-by-step:

1. Principal Quantum Number (n):
We are given the principal quantum number:
$n=4$
This corresponds to the fourth energy shell.

2. Azimuthal Quantum Number (l):
For a given principal quantum number $n$, the azimuthal quantum number $l$ can take any integer value ranging from $0$ to $n-1$.
For $n=4$, the possible values for $l$ are:
$l=0,1,2,3$
These correspond to the 4s, 4p, 4d, and 4f subshells, respectively.

3. Magnetic Quantum Number (m_l):
For each value of the azimuthal quantum number $l$, the magnetic quantum number $m_l$ can have integer values ranging from $-l$ to $+l$ (including zero). Each distinct value of $m_l$ represents a unique orbital within that subshell. Let us check the availability of $m_l=0$ for each subshell:
- For the 4s subshell ($l=0$): $m_l$ can only be $0$ (1 orbital).
- For the 4p subshell ($l=1$): $m_l$ can be $-1,0,+1$ (1 orbital has $m_l=0$, which is the 4p_z orbital).
- For the 4d subshell ($l=2$): $m_l$ can be $-2,-1,0,+1,+2$ (1 orbital has $m_l=0$, which is the 4d_z² orbital).
- For the 4f subshell ($l=3$): $m_l$ can be $-3,-2,-1,0,+1,+2,+3$ (1 orbital has $m_l=0$, which is the 4f_z³ orbital).

4. Total Count of Orbitals:
To find the maximum number of orbitals with $n=4$ and $m_l=0$, we sum the orbitals with $m_l=0$ from all possible subshells in the fourth shell:
Total orbitals = 1 (from 4s) + 1 (from 4p) + 1 (from 4d) + 1 (from 4f) = 4 orbitals.