Quantum Mechanical Model Of Atom

To determine which set of quantum numbers is not allowed, let's review the rules for quantum numbers:

1. Principal quantum number ( ): It must be a positive integer (1, 2, 3, ...).
2. Azimuthal quantum number ( ): It defines the shape of the orbital and ranges from 0 to .
3. Magnetic quantum number ( ): Its values range from to , including zero.
4. Spin quantum number ( ): It can either be or .

Now, let's analyze each option:

• Option 1:
  • All quantum numbers are valid: is a positive integer, is within [0, 2], is valid under , and .
• Option 2:
  • This is valid: lies in the range , i.e., [-2, 2] for .
• Option 3:
  • This set is not allowed: For , the maximum value of is 2 (since must be ). Here is beyond the permissible range.
• Option 4:
  • This is valid: All values conform to the constraints ( allows only ).

Thus, the correct answer is: , as it does not conform to the quantum number rules, particularly violating the rule for .

In a multi-electron atom, in the absence of electric and magnetic fields, the energy of an orbital depends only on the principal quantum number and the azimuthal quantum number . The magnetic quantum number has no effect on energy under these conditions (only affects energy in presence of magnetic field — Zeeman effect).

So, we need to find pairs of orbitals that have the same and values:

• A. → 1s orbital
• B. → 2s orbital
• C. → 2p orbital
• D. → 3d orbital
• E. → 3d orbital

D and E have the same and , so they belong to the same energy level in the absence of external fields.

Final Answer:
The final answer is .

In the absence of electric and magnetic fields, the energy of orbitals depends only on the principal quantum number for a hydrogen-like atom.
Step 1: For multielectron atoms, the energy levels are further split due to electron-electron interactions and quantum numbers and .
Step 2: Orbitals with the same but different and have the same energy due to the degeneracy of the level.

Final Conclusion: The correct answer is Option (1), where D and E have the same energy.