Effective Nuclear Charge Calculator Formula

Understand the math behind the effective nuclear charge calculator. Each variable explained with a worked example.

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Formulas Used

Effective Nuclear Charge (Z_eff)

z_eff = z - sigma

Variables

VariableDescriptionDefault
zAtomic Number (Z)11
sigmaShielding Constant (σ)8.8

How It Works

Effective Nuclear Charge

Formula

Z_eff = Z − σ

Where:

  • Z_eff = effective nuclear charge felt by valence electron
  • Z = atomic number (total protons)
  • σ = shielding constant (from Slater's rules)

    • Inner electrons shield outer electrons from the full nuclear charge. Z_eff increases across a period (more protons, similar shielding) and stays roughly constant down a group.

    Worked Example

    Sodium (Z = 11) with shielding constant σ = 8.8 for the 3s electron.

    z = 11sigma = 8.8
    1. 01Z_eff = Z - σ
    2. 02Z_eff = 11 - 8.8
    3. 03Z_eff = 2.2
    4. 04The 3s electron feels an effective charge of +2.2

    Frequently Asked Questions

    What is effective nuclear charge?

    Z_eff is the net positive charge experienced by an electron after accounting for shielding by inner electrons. It determines how tightly the electron is held.

    What are Slater's rules?

    Slater's rules provide a systematic way to estimate the shielding constant σ. Different electron groups contribute different amounts of shielding.

    Why does Z_eff matter?

    Z_eff explains periodic trends in atomic radius, ionization energy, and electronegativity. Higher Z_eff pulls electrons closer and makes them harder to remove.

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