Effective nuclear chargeCourse level: High School / HonorsCourse level: AP Chemistry

Shielding and Effective Nuclear Charge

What you'll be able to do: Estimate effective nuclear charge and use it to explain why every periodic trend behaves the way it does.

Best after: Electron Configurations and Orbital Filling

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Introduction

Every trend in this unit reduces to one competition: how strongly the nucleus pulls on an outer electron, and how much the inner electrons get in the way. Effective nuclear charge is the number that captures that competition.

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Learning objectives

  • Define shielding and effective nuclear charge
  • Estimate Zeff using Z minus the number of core electrons
  • Explain why effective nuclear charge generally increases across a period and why shell size and shielding dominate many down-group trends
  • Use Zeff to justify a trend rather than restate it

Shielding is an electron-interaction effect

An electron is attracted to the positively charged nucleus and repelled by other electrons. Effective nuclear charge summarizes the net nuclear attraction in a simplified one-electron model. Shielding is not a solid wall that switches the nucleus off: an orbital samples a range of distances, and other electrons alter the average attraction it experiences.

Core electrons usually shield valence electrons more effectively than electrons in the same shell do. Adding a proton and a same-shell electron therefore does not simply cancel the proton's effect. The nuclear charge Z is the actual proton count; Zeff is a model for the attraction experienced by a particular electron and can differ for different orbitals of the same atom.

Same-shell electrons are at roughly the same distance, so they are poor shields for one another.

Effective nuclear charge

Zeff = Z - S, where Z is the number of protons and S is the shielding constant.

A quick classroom estimate takes S as the number of core electrons, so sodium has Zeff approximately 11 - 10 = 1, while chlorine has 17 - 10 = 7.

Zeff = Z - S

Across a period: Zeff climbs

Moving left to right, each step adds one proton and one electron, but the new electron enters the same shell and shields poorly. Z rises while S stays nearly constant, so Zeff climbs sharply.

From Na to Cl the estimate goes from 1 to 7, a large increase.

Compare Period 3 atoms with the same [Ne] core. Shell circles and the selected electron marker are schematic bookkeeping, not trajectories.
This single fact explains why radius shrinks and ionization energy and electronegativity rise across a period.

Down a group: shell size and shielding matter

The crude Z − core-electron estimate gives about +1 for every alkali-metal valence electron, because added core electrons offset the additional protons in that model. Do not conclude that real effective nuclear charge is exactly constant: shielding is incomplete, and more detailed models can predict increases down a group.

The useful chemical conclusion is that an additional occupied principal shell places the valence electron farther out and adds shielding. Those effects usually dominate the alkali-metal radius and first-ionization-energy comparisons: K is larger than Na and its valence electron is easier to remove. Saying that potassium has more protons is true but incomplete; it also has an outer 4s electron rather than a 3s electron.

Use Z − core electrons as a trend model

The estimate Zeff ≈ Z − (core electron count) is a quick model, not an exact experimental value. Electrons in the same shell still repel and shield one another, and orbital penetration affects how strongly an electron samples the nucleus. A 3s electron can reach closer to the nucleus than a 3p electron, so two electrons with the same n need not feel identical attraction. The model is most useful for a comparison: Na and Mg both have a [Ne] core, but Mg has one more proton, so its valence electrons generally feel a stronger pull. State the shared shell and the shielding assumption when using it.

Penetration explains why equal n does not mean equal attraction. An s orbital has some electron density close to the nucleus and typically penetrates the core more than a p orbital of the same shell. Its electrons can be held more tightly. The simple core-count subtraction is therefore suitable for qualitative trends, not precise orbital energies, ionization-energy values, or numerical force ratios. Coulomb force also depends on distance, which changes as the electron cloud contracts.

Compare atoms while holding a shell constant

Across a period, added protons increase the pull on valence electrons, while added electrons enter roughly the same principal shell and do little to shield one another. Down a group, a new occupied shell places valence electrons farther from the nucleus and adds inner-electron shielding.

Say which comparison you are making before deciding which effect dominates.

Across: stronger nuclear pull in the same shell. Down: distance and shielding rise.

Key ideas

Definition
Shielding

Repulsion from inner electrons that reduces the nuclear attraction felt by an outer electron.

Definition
Effective nuclear charge

The net positive charge experienced by a specified electron, Zeff = Z - S.

Rule
Period behaviour

Across a period Z increases and S is nearly constant, so Zeff increases.

Rule
Group behaviour

Down a group, occupied shell size and shielding increase. The crude core-count model can give constant Zeff, but actual effective nuclear charge need not be constant.

Equation
Effective nuclear charge

Zeff = Z - S

  • Zeff = net nuclear charge felt by a valence electron
  • Z = atomic number, the number of protons
  • S = shielding constant, estimated as the number of core electrons

Worked examples

Worked example 1

Estimate Zeff for a valence electron in sodium (Z = 11) and in chlorine (Z = 17), and say what the comparison predicts.

Try it first: Count core electrons for each before doing any subtraction.

    0 of 3 steps revealed.

    Worked example 2

    Explain, using Zeff, why first ionization energy decreases from Li to Cs.

      0 of 3 steps revealed.

      Worked example 3

      Use the core-electron estimate to compare a valence electron in Na (Z = 11) and Mg (Z = 12). What does this predict about their atomic radii?

      Try it first: Both atoms have ten neon-core electrons and outer electrons in n = 3.

        0 of 2 steps revealed.

        Worked example 4

        Why is phosphorus generally smaller than silicon despite both using the third valence shell?

        Try it first: Compare proton counts and inner-shell shielding.

          0 of 3 steps revealed.

          Common mistakes

          Common mistakes and better checks
          MistakeWhy it's wrongCheck instead
          Saying atoms get smaller across a period because electrons are added.Adding electrons alone would enlarge the atom. The shrinking comes from the added protons raising Zeff on a shell that does not change.Attribute the trend to increasing Zeff at constant principal quantum number.
          Treating Z minus core-electron count as an exact Zeff measurement down a group.Same-shell shielding and penetration are omitted, so actual effective nuclear charge need not remain constant.Use the crude estimate for qualitative comparisons and explain down-group trends with shell size and shielding.
          Treating shielding by same-shell electrons as complete.Electrons at the same distance screen each other poorly, which is why Zeff climbs across a period at all.Count only core electrons in the quick S estimate.

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          What you should now know

          Zeff = Z − S models the net nuclear attraction experienced by a specified electron. Core electrons shield strongly and same-shell shielding is incomplete, explaining a general increase across a period. Down a group, greater shell size and shielding often dominate radius and ionization trends; exact Zeff is not required to remain constant.

          • Zeff is an electron-specific model, not the bare proton count.
          • Z minus core-electron count is a qualitative estimate, not an exact force or energy.
          • Across a period, more protons with similar core shielding generally increase attraction.
          • Down a group, a larger occupied shell and more shielding usually make valence electrons easier to remove.
          • Orbital penetration explains why electrons with the same n can feel different attraction.
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          Sources and further reading

          • Chemistry 2e, Section 6.5: Periodic Variations in Element Properties
            Flowers, Theopold, Langley, Robinson · OpenStax, Rice University · Chapter 6.5
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          • General Chemistry bookshelf (LibreTexts)
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