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

Question

Accepted Answer

To determine the total number of orbitals associated with the principal quantum number $ n = 3 $, we need to understand the relationship between the principal quantum number and the types of orbitals it can accommodate.

1. **Principal Quantum Number $ n $:** The principal quantum number $ n $ indicates the main energy level or shell of an electron in an atom. For $ n = 3 $, the possible values of the azimuthal quantum number $ l $ are 0, 1, and 2, corresponding to the s, p, and d subshells, respectively.

2. **Subshells and Orbitals:**
   - **s Subshell ($ l = 0 $):** Contains 1 orbital.
   - **p Subshell ($ l = 1 $):** Contains 3 orbitals.
   - **d Subshell ($ l = 2 $):** Contains 5 orbitals.

3. **Total Number of Orbitals:**
   - For $ n = 3 $, the total number of orbitals is the sum of the orbitals in each subshell:
     $$
     1 (\text{s}) + 3 (\text{p}) + 5 (\text{d}) = 9 \text{ orbitals}
     $$

Therefore, the total number of orbitals associated with the principal quantum number $ n = 3 $ is 9, which corresponds to Option 3.

**Explanation of Incorrect Options:**

- **Option 1 (3 orbitals):** This option is incorrect because it only accounts for the orbitals in the p subshell, ignoring the s and d subshells.

- **Option 2 (6 orbitals):** This option is incorrect because it might consider only the s and p subshells, ignoring the d subshell, which also contributes to the total number of orbitals.

- **Option 4 (18 orbitals):** This option is incorrect because it overestimates the number of orbitals. It might be confused with the total number of electrons that can be accommodated in the $ n = 3 $ shell, which is indeed 18 (since each orbital can hold 2 electrons), but the question specifically asks for the number of orbitals, not electrons.

In conclusion, the correct answer is Option 3, as it accurately reflects the total number of orbitals for $ n = 3 $.