Understanding Electron Energy Order Based on Quantum Numbers

To determine the order of increasing energy for electrons with given quantum numbers, we use the (n + ℓ) rule:

Step 1: Calculate (n + ℓ) for each electron

• For (a) n=4, ℓ=1: $n+\ell =4+1=5$
• For (b) n=4, ℓ=0: $n+\ell =4+0=4$
• For (c) n=3, ℓ=2: $n+\ell =3+2=5$
• For (d) n=3, ℓ=1: $n+\ell =3+1=4$

Step 2: Compare (n + ℓ) values

Lower (n + ℓ) means lower energy. If (n + ℓ) is equal, the electron with lower n has lower energy.

• (b) and (d) both have (n + ℓ)=4. Since (b) has n=4 and (d) has n=3, (d) has lower energy than (b).
• (a) and (c) both have (n + ℓ)=5. Since (a) has n=4 and (c) has n=3, (c) has lower energy than (a).

Step 3: Arrange all in increasing energy order

From lowest to highest energy: (d) < (b) < (c) < (a)

Final Answer: The correct order is (d) < (b) < (c) < (a), which matches the second option.

Related Topics

• Quantum Numbers: These describe the properties of atomic orbitals and the electrons in them. The principal quantum number (n) determines the energy level/shell, and the azimuthal quantum number (ℓ) determines the subshell shape (s, p, d, f).
• Aufbau Principle: Electrons fill orbitals in order of increasing energy. The (n + ℓ) rule helps predict this order.
• Orbital Energy: For multi-electron atoms, orbital energy depends on both n and ℓ due to electron-electron interactions and shielding.

Key Formulae and Theory

• (n + ℓ) Rule: To compare energies of orbitals:
  • Orbital with lower (n + ℓ) value has lower energy.
  • If two orbitals have the same (n + ℓ) value, the orbital with lower n has lower energy.
• Energy Order Formula: The energy of an orbital is approximately proportional to (n + ℓ).
• Standard Order: The increasing energy order of orbitals is: 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s...