Penetrating Power of Orbitals

The penetrating power of an orbital refers to how close its electron density can get to the nucleus. Orbitals with higher penetration experience less shielding and lower effective nuclear charge.

Step 1: Understanding Orbital Shapes

Different orbitals have different shapes and radial distributions:

• s-orbitals: spherical, highest electron density near nucleus
• p-orbitals: dumbbell-shaped, nodal plane through nucleus
• d-orbitals: cloverleaf-shaped, more complex nodal structure
• f-orbitals: even more complex shapes with multiple nodal planes

Step 2: Radial Distribution Analysis

The radial probability function shows that for a given principal quantum number n:

$P_{\mathrm{ns}}>P_{\mathrm{np}}>P_{\mathrm{nd}}>P_{\mathrm{nf}}$

where P represents the probability of finding electrons near the nucleus.

Step 3: Penetration Order Determination

Since s-orbitals have the highest electron density near the nucleus, they have the greatest penetrating power. The order decreases as we move to orbitals with more nodal planes:

$s>p>d>f$

Final Answer

The correct order of penetrating power for orbitals with the same principal quantum number is: $s>p>d>f$

Related Topics

• Effective Nuclear Charge (Z_eff)
• Shielding Effect
• Radial Distribution Functions
• Orbital Energies in Multi-electron Atoms
• Atomic Structure and Periodicity

Key Formulae and Theory

Effective Nuclear Charge: $Z_{\text{eff}}=Z-\sigma$

where Z is the atomic number and σ is the shielding constant

Radial Probability Function: $P\left(r\right)=r^2{R}^2\left(r\right)$

where R(r) is the radial wavefunction

Energy Order: Due to penetration effects, within the same shell: $E_s<E_p<E_d<E_f$