What is the total number of orbitals associated with the principal qua

What is the total number of orbitals associated with the principal quantum number 3?

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UPSC Geoscientist – 2023

The principal quantum number (n) defines the energy level or shell. For a given principal quantum number n, the number of possible subshells is equal to n. These subshells are characterized by the azimuthal quantum number (l), which can take integer values from 0 to n-1.
For n = 3, the possible values of l are 0, 1, and 2.
l = 0 corresponds to the s subshell. The number of orbitals in an s subshell is 1 (m_l = 0).
l = 1 corresponds to the p subshell. The number of orbitals in a p subshell is 3 (m_l = -1, 0, +1).
l = 2 corresponds to the d subshell. The number of orbitals in a d subshell is 5 (m_l = -2, -1, 0, +1, +2).
The total number of orbitals associated with a principal quantum number n is the sum of the number of orbitals in each subshell:
Total orbitals for n=3 = (number of orbitals for l=0) + (number of orbitals for l=1) + (number of orbitals for l=2)
Total orbitals for n=3 = 1 (3s) + 3 (3p) + 5 (3d) = 9.
Alternatively, the total number of orbitals in a shell with principal quantum number n is given by n².
For n = 3, total orbitals = 3² = 9.

For a given principal quantum number n, the total number of atomic orbitals in that shell is n².

Each atomic orbital can hold a maximum of 2 electrons with opposite spins (Pauli Exclusion Principle). Thus, the maximum number of electrons in a shell with principal quantum number n is 2n². For n=3, the maximum number of electrons is 2 * 3² = 18.

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