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 determine the number of orbitals associated with the quantum number \( n = 5 \) and \( m_s = +\frac{1}{2} \), we can follow these steps:

### Step 1: Understand the Quantum Numbers
- The principal quantum number \( n \) indicates the energy level of the electron.
- The azimuthal quantum number \( l \) can take values from \( 0 \) to \( n-1 \). Thus, for \( n = 5 \), \( l \) can be \( 0, 1, 2, 3, \) or \( 4 \).

### Step 2: Identify the Possible Values of \( l \)
- For \( l = 0 \): This corresponds to the **s** orbital (1 orbital).
- For \( l = 1 \): This corresponds to the **p** orbitals (3 orbitals).
- For \( l = 2 \): This corresponds to the **d** orbitals (5 orbitals).
- For \( l = 3 \): This corresponds to the **f** orbitals (7 orbitals).
- For \( l = 4 \): This corresponds to the **g** orbitals (9 orbitals).

### Step 3: Calculate the Total Number of Orbitals
Now, we add the number of orbitals for each value of \( l \):
- For \( l = 0 \): 1 orbital (s)
- For \( l = 1 \): 3 orbitals (p)
- For \( l = 2 \): 5 orbitals (d)
- For \( l = 3 \): 7 orbitals (f)
- For \( l = 4 \): 9 orbitals (g)

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

### Step 4: Consider the Spin Quantum Number
The spin quantum number \( m_s \) indicates the spin of the electron, which can be either \( +\frac{1}{2} \) or \( -\frac{1}{2} \). However, this does not affect the count of orbitals; each orbital can hold two electrons with opposite spins.

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

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