For how many orbitals are the quantum numbers `n = 3, l = 2, m = +2` possible?

Options

A. 1
B. 5
C. 3
D. 7

Answer

To determine how many orbitals correspond to the quantum numbers \( n = 3 \), \( l = 2 \), and \( m = +2 \), we can follow these steps:

### Step 1: Identify the Principal Quantum Number (n)
The principal quantum number \( n \) indicates the energy level of the electron. Here, \( n = 3 \) means we are dealing with the third energy level.

**Hint:** The principal quantum number \( n \) defines the shell where the electron is located.

### Step 2: Identify the Azimuthal Quantum Number (l)
The azimuthal quantum number \( l \) defines the subshell and its shape. The value of \( l \) can range from \( 0 \) to \( n-1 \). For \( n = 3 \):
- \( l = 0 \) corresponds to the s subshell
- \( l = 1 \) corresponds to the p subshell
- \( l = 2 \) corresponds to the d subshell

Since \( l = 2 \), we are dealing with the d subshell.

**Hint:** The azimuthal quantum number \( l \) tells you the type of orbital (s, p, d, f).

### Step 3: Identify the Magnetic Quantum Number (m)
The magnetic quantum number \( m \) can take values from \( -l \) to \( +l \). For \( l = 2 \), the possible values of \( m \) are:
- \( m = -2 \)
- \( m = -1 \)
- \( m = 0 \)
- \( m = +1 \)
- \( m = +2 \)

This means there are five possible orbitals in the d subshell.

**Hint:** The magnetic quantum number \( m \) specifies the orientation of the orbital in space.

### Step 4: Determine the Specific Orbital
In this case, we are given \( m = +2 \). This means we are specifically looking for the orbital that corresponds to this orientation. Since \( m = +2 \) is one of the possible values for \( l = 2 \), it corresponds to one specific orbital in the d subshell.

**Hint:** Each value of \( m \) corresponds to a unique orbital within the subshell.

### Conclusion
Thus, for the quantum numbers \( n = 3 \), \( l = 2 \), and \( m = +2 \), there is only **one orbital** possible.

**Final Answer:** 1 orbital