The quantum numbers 'm' of a free gaseous atom is associated with :

Options

A. The effective volume of the orbital
B. The shape of the orbital
C. The spatial orientation of the orbital
D. The energy of the orbital in the absence of a magnetic field.

Explanation

To solve the question regarding the quantum number 'm' of a free gaseous atom, we can follow these steps:

### Step-by-Step Solution

1. **Understanding Quantum Numbers**:
- Quantum numbers are used to describe the properties of atomic orbitals and the electrons in those orbitals. There are four types of quantum numbers: principal (n), azimuthal (l), magnetic (m), and spin (s).

2. **Identifying the Magnetic Quantum Number (m)**:
- The magnetic quantum number (m) specifically relates to the orientation of the orbital in space. It can take on integer values ranging from -l to +l, where l is the azimuthal quantum number.

3. **Analyzing the Options**:
- The question provides four options regarding what the magnetic quantum number (m) is associated with:
1. Effective volume of the orbital
2. Shape of the orbital
3. Spatial orientation of the orbital
4. Energy of the orbital in absence of magnetic field

4. **Evaluating Each Option**:
- **Option 1**: Effective volume of the orbital - This is incorrect. The effective volume is not determined by the magnetic quantum number.
- **Option 2**: Shape of the orbital - This is also incorrect. The shape of the orbital is determined by the azimuthal quantum number (l).
- **Option 3**: Spatial orientation of the orbital - This is correct. The magnetic quantum number (m) defines the orientation of the orbital in space.
- **Option 4**: Energy of the orbital in absence of magnetic field - This is incorrect. The energy levels are primarily determined by the principal quantum number (n) and azimuthal quantum number (l).

5. **Conclusion**:
- Based on the evaluation, the correct answer is that the quantum number 'm' is associated with the spatial orientation of the orbital.

### Final Answer
The quantum number 'm' of a free gaseous atom is associated with the spatial orientation of the orbital.

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