An electron is in one of the `3d` orbitals. Give the possible values of `n, l,` and `m` for this electron.

Options

A. n = 3, l = 0, `m_(1)` = 0
B. n = 3, l = 1, `m_(1)` = -1, 0, +1
C. n = 3, l = 2, `m_(1)` = -2, -1, 0, +1, +2
D. n = 3, l = 3, `m_(1)` = -3, -2, -1, 0, +1, +2, +3

Answer

To solve the question regarding the possible values of the quantum numbers \( n \), \( l \), and \( m \) for an electron in one of the \( 3d \) orbitals, 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 and is determined by the shell in which the electron resides. For \( 3d \) orbitals, the principal quantum number is:
\[ n = 3 \]

### Step 2: Determine the Azimuthal Quantum Number (\( l \))
The azimuthal quantum number \( l \) defines the shape of the orbital. The value of \( l \) is calculated using the formula:
\[ l = n - 1 \]
For \( n = 3 \):
\[ l = 3 - 1 = 2 \]

### Step 3: Calculate the Magnetic Quantum Number (\( m \))
The magnetic quantum number \( m \) describes the orientation of the orbital in space. The possible values of \( m \) range from \( -l \) to \( +l \). Therefore, for \( l = 2 \):
\[ m = -2, -1, 0, +1, +2 \]
This gives us a total of five possible values for \( m \).

### Summary of Values
- The possible values for the quantum numbers are:
- \( n = 3 \)
- \( l = 2 \)
- \( m = -2, -1, 0, +1, +2 \)

### Final Answer
Thus, the possible values of \( n \), \( l \), and \( m \) for an electron in one of the \( 3d \) orbitals are:
- \( n = 3 \)
- \( l = 2 \)
- \( m = -2, -1, 0, +1, +2 \)

---