How much nuclear charge does an outer electron feel?
A sodium nucleus carries a charge of +11. But between it and the outermost 3s electron sit 10 inner electrons. Their negative charge cancels much of the nucleus’s pull, so the outer electron feels far less than +11. In lesson 1.4.5 you explained atomic size with two factors, the number of shells and the nuclear charge. This lesson puts both into one physical law and adds a more useful idea: effective nuclear charge.
Coulomb’s law
The force between charged particles follows Coulomb’s law库仑定律 (you’ll meet it in more detail in physics):
Here q₁ and q₂ are the two charges, r is the distance between them, and k is a constant. Applied to a nucleus and an electron, it says two things:
- The greater the nuclear charge, the stronger the attraction.
- The farther the electron is from the nucleus, the weaker the attraction, and it weakens with the square of the distance: double the distance and the force drops to 1/4.
Shielding and effective nuclear charge
Inner electrons sit between the nucleus and the outer electrons and weaken the nucleus’s pull on them. This is called shielding屏蔽效应 (or screening). The nuclear charge an outer electron actually feels is the effective nuclear charge有效核电荷, Zeff:
Z is the nuclear charge and S is the shielding constant, the total charge cancelled by the other electrons. For a rough estimate, assume each core (inner) electron cancels exactly one unit of nuclear charge and that electrons in the same level don’t shield each other at all:
For a main-group element, this estimate equals the number of valence electrons: 11 − 10 = +1 for sodium and 17 − 10 = +7 for chlorine.
- Across a period: the number of core electrons stays the same while the nuclear charge goes up by 1 at each step, so Zeff increases. With the same number of energy levels, the outer electrons are pulled closer and held more tightly, so atoms get smaller and electrons get harder to remove.
- Down a group: Zeff stays about the same (about +1 for every alkali metal), but each atom adds an energy level, so the outer electrons are farther from the nucleus and attracted less. Atoms get larger and electrons get easier to remove.
A better estimate: Slater’s rules (optional)
In reality, core electrons don’t block all of the nuclear charge, and electrons in the same level shield each other a little. In 1930, J. C. Slater summed this up in a set of rules of thumb. For an electron in the outermost s or p sublevel:
- each other electron in the same ns and np sublevels shields 0.35 (the other electron in 1s shields 0.30);
- each electron in level n − 1 shields 0.85;
- each electron in deeper levels shields 1.00.
For sodium’s 3s electron, S = 8 × 0.85 + 2 × 1.00 = 8.80, so Zeff = 11 − 8.80 = 2.20. For chlorine’s 3p electron, S = 6 × 0.35 + 8 × 0.85 + 2 × 1.00 = 10.90, so Zeff = 17 − 10.90 = 6.10. The numbers differ from the simple estimate, but the pattern is the same: Zeff rises clearly across a period. Neither AP nor most Honors courses require Slater’s rules; they’re here to show that shielding is never perfect.
Compare two elements side by side: start with Na and Cl in the same period, then Li and Na in the same group, and check your reasoning against the radius and ionization-energy data:
Example 1: Why is a magnesium atom smaller than a sodium atom?
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Find what they share.
Arrangement Inner electrons Na 2, 8, 1 10 Mg 2, 8, 2 10 Both have 3 shells with 10 inner electrons.
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Estimate Zeff. Zeff ≈ nuclear charge − inner electrons:
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Reason with Coulomb’s law. With the same number of shells, the outer electrons are at similar distances; the larger Zeff pulls harder and draws the electrons in, so the magnesium atom is smaller.
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Check with data. Atomic radii: Na 166 pm, Mg 141 pm. First ionization energies (the energy to remove one electron, lesson 2.3.4): Na 496 kJ/mol, Mg 738 kJ/mol, so magnesium holds its electrons more tightly, as predicted.
Example 2: Which loses an electron more easily, fluorine or chlorine?
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Estimate Zeff.
They’re the same, so something else must decide.
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Compare distance. Fluorine has 2 shells and chlorine 3, so chlorine’s outer electrons are farther from the nucleus (atomic radii: F 57 pm, Cl 102 pm).
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Conclusion. With equal Zeff, the more distant electron is held more weakly, so chlorine loses an electron more easily. Data: first ionization energies F 1681 kJ/mol, Cl 1251 kJ/mol.
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What Slater’s rules say (optional). A more careful calculation gives Zeff = 5.20 for F and 6.10 for Cl, so chlorine’s is actually larger; but its electron is so much farther away that it is still held more weakly. Down a group, distance outweighs the change in Zeff.
- Na (Z = 11)
2, 8, 1: there are 10 core electrons, so 11 − 10 = +1.
- Mg (Z = 12)
2, 8, 2: 12 − 10 = +2.
- B (Z = 5)
2, 3: the only core electrons are the 2 in 1s, so 5 − 2 = +3.
- C (Z = 6)
2, 4: 6 − 2 = +4.
- P (Z = 15)
2, 8, 5: 15 − 10 = +5.
- O (Z = 8)
2, 6: 8 − 2 = +6.
- Br (Z = 35)
[Ar]4s²3d¹⁰4p⁵: the core holds 2 + 8 + 18 = 28 electrons, including the full 3d¹⁰, so 35 − 28 = +7.
- Ne (Z = 10)
2, 8: 10 − 2 = +8. That large Zeff makes neon's electrons very hard to remove.
- K (Z = 19)
2, 8, 8, 1: 19 − 18 = +1, the same as sodium. But potassium has one more energy level, so its outer electron is farther away.
Common mistakes
Mistake 1: “The bigger the nuclear charge, the more strongly the outer electrons are attracted.” Potassium’s nuclear charge (+19) is much larger than sodium’s (+11), but its 8 extra protons are cancelled by 8 extra core electrons, so Zeff is about the same. Potassium’s outer electron is farther away, so it is attracted less, not more.
Mistake 2: “Core electrons block the nuclear charge completely, and electrons in the same level don’t block it at all.” That is only a rough estimate. Shielding is never complete, and electrons in the same level do shield each other a little (0.35 each in Slater’s rules), so an accurate Zeff is usually not a whole number.
Mistake 3: “A larger effective nuclear charge always means a smaller atom.” That comparison only works for atoms with the same number of energy levels. Down a group, Slater’s rules give a slightly larger Zeff (F 5.20, Cl 6.10), yet the atom gets clearly bigger because it has an extra energy level.
Quick check
Estimating Zeff as the nuclear charge minus the core electrons, what effective nuclear charge do sulfur's outer electrons feel?
Sulfur is 2, 8, 6, with 10 core electrons: 16 − 10 = +6, which equals its number of valence electrons.
Atomic radius decreases from Na to Cl. What is the main reason?
Every period 3 atom has 3 energy levels and the same number of core electrons. As the nuclear charge rises, Zeff rises with it and pulls the outer electrons closer.
Summary
- Coulomb’s law: the larger the charges, the stronger the attraction; the greater the distance, the weaker the attraction (it falls with the square of the distance).
- Core electrons shield the nuclear charge. Zeff = Z − S; as a rough estimate, Zeff ≈ Z − core electrons, which equals the number of valence electrons for a main-group element. Slater’s rules give a more accurate value.
- Across a period, the number of energy levels stays the same and Zeff increases, so the attraction gets stronger. Down a group, Zeff stays about the same but the electrons are in higher levels, farther away, so the attraction gets weaker. The next lessons use these two ideas to explain atomic radius, ionization energy, and electronegativity.
On to AP (for later)
AP U1.5 and U1.7 ask you to explain periodic trends with Coulomb’s law. A full-credit answer names both factors: charge (“more protons, with the same core-electron shielding”) and distance (“the electron is in a higher energy level, farther from the nucleus”). Writing only “greater Zeff” or only “bigger atom” often loses points. AP doesn’t require Slater’s rules.