n = 5 , ` m _ s = + ( 1 ) /(2 ) ` How many orbitals are possible :

Options

A. 25
B. 30
C. 50
D. 35

Answer

To determine how many orbitals are possible when \( n = 5 \) and \( m_s = +\frac{1}{2} \), we can follow these steps:

### Step 1: Identify the Principal Quantum Number
The principal quantum number \( n \) indicates the shell number. Here, \( n = 5 \) means we are dealing with the fifth shell.

### Step 2: Calculate the Azimuthal Quantum Number \( l \)
The azimuthal quantum number \( l \) can take values from \( 0 \) to \( n-1 \). Therefore, for \( n = 5 \):
- \( l \) can be \( 0, 1, 2, 3, 4 \).

### Step 3: Determine the Number of Orbitals for Each \( l \)
The number of orbitals corresponding to each \( l \) value can be calculated using the formula:
\[ \text{Number of orbitals} = 2l + 1 \]
Now, we will calculate this for each value of \( l \):
- For \( l = 0 \) (s orbital):
\[ 2(0) + 1 = 1 \text{ orbital} \]
- For \( l = 1 \) (p orbital):
\[ 2(1) + 1 = 3 \text{ orbitals} \]
- For \( l = 2 \) (d orbital):
\[ 2(2) + 1 = 5 \text{ orbitals} \]
- For \( l = 3 \) (f orbital):
\[ 2(3) + 1 = 7 \text{ orbitals} \]
- For \( l = 4 \) (g orbital):
\[ 2(4) + 1 = 9 \text{ orbitals} \]

### Step 4: Sum the Total Number of Orbitals
Now, we will add the number of orbitals from all \( l \) values:
\[ 1 \text{ (s)} + 3 \text{ (p)} + 5 \text{ (d)} + 7 \text{ (f)} + 9 \text{ (g)} = 25 \text{ orbitals} \]

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

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