For a sub-shell with azimuthal quantum number 'l', the total values of magnetic quantum number m can be related to l as

Options

A. `m = (l+2)`
B. `m = (2l^(2)+2)`
C. `l = ((m-1))/(2)`
D. `l = 2m + 1`

Explanation

To solve the question regarding the relationship between the azimuthal quantum number (l) and the magnetic quantum number (m), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Quantum Numbers**:
- The azimuthal quantum number (l) determines the shape of the orbital and can take values from 0 to (n-1), where n is the principal quantum number.
- The magnetic quantum number (m) describes the orientation of the orbital in space and can take values ranging from -l to +l.

2. **Determine the Range of m**:
- For a given value of l, the possible values of m are:
- m = -l, -l + 1, ..., 0, ..., l - 1, l.
- This means that the total number of possible values for m is given by the formula:
\[ \text{Total values of } m = 2l + 1 \]

3. **Example Calculation**:
- For example, if l = 0 (s orbital), then:
\[ m = 0 \quad \text{(only one value)} \]
Hence, the total values of m = 2(0) + 1 = 1.
- If l = 1 (p orbital), then:
\[ m = -1, 0, +1 \quad \text{(three values)} \]
Hence, the total values of m = 2(1) + 1 = 3.
- If l = 2 (d orbital), then:
\[ m = -2, -1, 0, +1, +2 \quad \text{(five values)} \]
Hence, the total values of m = 2(2) + 1 = 5.
- If l = 3 (f orbital), then:
\[ m = -3, -2, -1, 0, +1, +2, +3 \quad \text{(seven values)} \]
Hence, the total values of m = 2(3) + 1 = 7.

4. **Conclusion**:
- The relationship between the azimuthal quantum number (l) and the total values of the magnetic quantum number (m) can be summarized as:
\[ \text{Total values of } m = 2l + 1 \]