The number of orbitals associated with quantum number n = 4,`m_s=+1/2` is :

Options

A. 15
B. 16
C. 25
D. 9

Explanation

To determine the number of orbitals associated with the principal quantum number \( n = 4 \) and the spin quantum number \( m_s = +\frac{1}{2} \), we can follow these steps:

### Step 1: Identify the Principal Quantum Number
The principal quantum number \( n \) indicates the main energy level or shell of an electron in an atom. Here, \( n = 4 \).

### Step 2: Determine the Possible Values of the Azimuthal Quantum Number \( l \)
The azimuthal quantum number \( l \) can take values from \( 0 \) to \( n-1 \). For \( n = 4 \), the possible values of \( l \) are:
- \( l = 0 \) (s subshell)
- \( l = 1 \) (p subshell)
- \( l = 2 \) (d subshell)
- \( l = 3 \) (f subshell)

### Step 3: Identify the Number of Orbitals for Each Subshell
Each value of \( l \) corresponds to a specific number of orbitals:
- For \( l = 0 \) (s subshell): 1 orbital
- For \( l = 1 \) (p subshell): 3 orbitals
- For \( l = 2 \) (d subshell): 5 orbitals
- For \( l = 3 \) (f subshell): 7 orbitals

### Step 4: Calculate the Total Number of Orbitals
Now, we can sum the number of orbitals from each subshell:
\[ \text{Total orbitals} = 1 \, (\text{s}) + 3 \, (\text{p}) + 5 \, (\text{d}) + 7 \, (\text{f}) = 1 + 3 + 5 + 7 = 16 \]

### Step 5: Consider the Spin Quantum Number
The spin quantum number \( m_s = +\frac{1}{2} \) indicates the spin orientation of the electrons. However, this does not affect the total number of orbitals; it only indicates the possible spin states of the electrons within those orbitals.

### Final Answer
Thus, the total number of orbitals associated with the quantum number \( n = 4 \) is **16**.

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