Question

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

• A. 3
• B. 6
• C. 9
• D. 10
• E. 14

Hint:

Total orbitals $= n^2$; Total electrons $= 2n^2$.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
The principal quantum number, \(n\), specifies the main energy shell of an atom. Each shell contains a set of sub-shells, and each sub-shell contains a set of orbitals. We need to find the total number of orbitals in the third shell (\(n=3\)).
Step 2: Key Formula or Approach
There are two common methods to determine the total number of orbitals in a shell.
Method 1: The total number of orbitals in a shell with principal quantum number \(n\) is given by the formula \(n^2\).
Method 2: Sum the number of orbitals in each sub-shell. For a given \(n\), the possible values of \(l\) are \(0, 1, \dots, n-1\). The number of orbitals for each \(l\) is \(2l+1\).
Step 3: Detailed Explanation
Using Method 1 (Direct Formula):
Given the principal quantum number \(n=3\).
The total number of orbitals is \(n^2\).
\[ \text{Total Orbitals} = 3^2 = 9 \] Using Method 2 (Summing Sub-shells):
For \(n=3\), the possible values for the azimuthal quantum number \(l\) are 0, 1, and 2.
- For \(l=0\) (the 3s sub-shell), the number of orbitals = \(2(0)+1 = 1\).
- For \(l=1\) (the 3p sub-shell), the number of orbitals = \(2(1)+1 = 3\).
- For \(l=2\) (the 3d sub-shell), the number of orbitals = \(2(2)+1 = 5\).
The total number of orbitals is the sum of the orbitals in all these sub-shells:
\[ \text{Total Orbitals} = 1 (\text{from 3s}) + 3 (\text{from 3p}) + 5 (\text{from 3d}) = 9 \] Both methods yield the same result.
Step 4: Final Answer
The total number of orbitals associated with the principal quantum number n=3 is 9.