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

Question

Accepted Answer

The total number of orbitals associated with a given principal quantum number $ n $ can be determined using the formula $ n^2 $. This formula arises from the quantum mechanical model of the atom, which describes the distribution of electrons in an atom.

For the principal quantum number $ n = 3 $:

1. **Calculation of Total Orbitals:**
   - Using the formula $ n^2 $, we calculate the total number of orbitals as $ 3^2 = 9 $.
   - Therefore, there are 9 orbitals associated with the principal quantum number $ n = 3 $.

2. **Explanation of the Correct Option (Option 3):**
   - Option 3 correctly states that there are 9 orbitals for $ n = 3 $. This is derived from the formula $ n^2 $, which is a fundamental concept in quantum mechanics. Each principal quantum number $ n $ corresponds to a set of orbitals, and the number of these orbitals is given by squaring the principal quantum number.

3. **Explanation of Why Other Options are Incorrect:**
   - **Option 1 (3):** This option is incorrect because it underestimates the number of orbitals. For $ n = 3 $, the calculation $ 3^2 = 9 $ clearly shows that there are more than 3 orbitals.
   - **Option 2 (6):** This option is incorrect because it also underestimates the number of orbitals. The correct calculation $ 3^2 = 9 $ indicates that there are more than 6 orbitals.
   - **Option 4 (12):** This option is incorrect because it overestimates the number of orbitals. The calculation $ 3^2 = 9 $ confirms that there are fewer than 12 orbitals.

In conclusion, Option 3 is the correct answer because it accurately reflects the total number of orbitals associated with the principal quantum number $ n = 3 $, as determined by the formula $ n^2 $.