The number of orbitals associated with quantum numbers n =5, ` m _ s = + ( 1 ) /(2) ` is :

Options

A. 15
B. 50
C. 25
D. 11

Answer

To find the number of orbitals associated with the principal quantum number \( n = 5 \), we can follow these steps:

### Step 1: Understand Quantum Numbers
Quantum numbers describe the properties of atomic orbitals and the electrons in those orbitals. The principal quantum number \( n \) indicates the energy level of an electron in an atom.

### Step 2: Identify the Orbitals for \( n = 5 \)
For a given principal quantum number \( n \), the possible azimuthal quantum numbers \( l \) can take values from \( 0 \) to \( n-1 \). Thus, for \( n = 5 \):
- \( l = 0 \) corresponds to the s orbital (1 orbital)
- \( l = 1 \) corresponds to the p orbital (3 orbitals)
- \( l = 2 \) corresponds to the d orbital (5 orbitals)
- \( l = 3 \) corresponds to the f orbital (7 orbitals)
- \( l = 4 \) corresponds to the g orbital (9 orbitals)

### Step 3: Count the Total Number of Orbitals
Now, we can sum the number of orbitals for each type:
- s: 1 orbital
- p: 3 orbitals
- d: 5 orbitals
- f: 7 orbitals
- g: 9 orbitals

Total number of orbitals = \( 1 + 3 + 5 + 7 + 9 = 25 \)

### Step 4: Verify Using the Formula
Alternatively, we can use the formula for the total number of orbitals associated with a principal quantum number \( n \):
\[ \text{Number of orbitals} = n^2 \]
For \( n = 5 \):
\[ \text{Number of orbitals} = 5^2 = 25 \]

### Conclusion
Thus, the total number of orbitals associated with \( n = 5 \) is **25**.

### Final Answer
The number of orbitals associated with quantum numbers \( n = 5 \) and \( m_s = +\frac{1}{2} \) is **25**.

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