Effective Nuclear Charge
Every periodic trend you have studied in this chapter - atomic radius, ionization energy, electron affinity, electronegativity, metallic character - can be explained by a single underlying concept. If you understand effective nuclear charge, you do not need to memorize five separate trends. You can derive all of them from first principles, right there on test day.
Effective nuclear charge (Zeff) is the net positive charge actually felt by a valence electron. The nucleus contains Z protons, each pulling the electron inward. But inner-shell electrons sit between the nucleus and the valence electrons, repelling them outward and partially canceling the nuclear pull. This partial cancellation is called shielding (or screening).
The Formula
The key insight is that S approximately equals the number of core electrons. Electrons in the same shell as the valence electron provide very little shielding because they are at roughly the same distance from the nucleus. Only the electrons in lower energy levels act as effective shields.
Worked Example: Sodium
Sodium (Na) has an atomic number of 11, so Z = 11. Its electron configuration is 1s² 2s² 2p⁶ 3s¹. The single valence electron sits in the 3s orbital, and there are 10 core electrons (the 1s², 2s², and 2p⁶ electrons) between it and the nucleus.
Zeff = Z - S = 11 - 10 = +1
That lone 3s electron feels an effective pull of roughly +1, despite there being 11 protons in the nucleus. The 10 inner electrons shield almost all of the nuclear charge.
Worked Example: Chlorine
Chlorine (Cl) has an atomic number of 17. Its electron configuration is 1s² 2s² 2p⁶ 3s² 3p⁵. The valence electrons are in the n = 3 shell, and there are 10 core electrons (same 1s² 2s² 2p⁶ as sodium).
Zeff = Z - S = 17 - 10 = +7
Chlorine’s valence electrons feel a much stronger pull (+7) than sodium’s (+1), even though both elements are in the same period. This single number explains why chlorine has a smaller atomic radius, higher ionization energy, higher electron affinity, and higher electronegativity than sodium.
Zeff Across a Period
As you move left to right across a period, each element has one more proton (Z increases by 1) and one more electron. But the added electron goes into the same shell - it does not add a new layer of shielding. The shielding constant S stays approximately the same because same-shell electrons are poor shielders.
The result: Zeff increases by roughly +1 for each step across a period. From Na to Cl, Zeff climbs from about +1 to +7. Every periodic trend across a period follows directly from this steady climb.
| Element | Z | Core Electrons (S) | Approximate Zeff | Consequence |
|---------|---|-------------------|------------------|-------------|
| Na | 11 | 10 | +1 | Large radius, low IE, low EN |
| Mg | 12 | 10 | +2 | Slightly smaller, slightly higher IE |
| Al | 13 | 10 | +3 | Smaller still, higher IE |
| Si | 14 | 10 | +4 | Continuing the trend |
| P | 15 | 10 | +5 | Continuing the trend |
| S | 16 | 10 | +6 | Continuing the trend |
| Cl | 17 | 10 | +7 | Small radius, high IE, high EN |
Zeff Down a Group
Moving down a group, each new period adds an entirely new electron shell. Z increases (more protons), but S also increases dramatically because each new inner shell adds a full layer of effective shielding. The net effect is that Zeff felt by valence electrons increases only slightly - or stays roughly flat - as you go down a group.
However, even though Zeff does not change much, the valence electrons are now much farther from the nucleus. They occupy higher energy levels with larger orbital radii. The combination of roughly constant Zeff and increasing distance means the nucleus has a weaker grip on valence electrons as you go down a group.
Consider the Group 1 elements:
| Element | Z | Core Electrons (S) | Approximate Zeff | Valence Shell |
|---------|---|-------------------|------------------|---------------|
| Li | 3 | 2 | +1 | n = 2 |
| Na | 11 | 10 | +1 | n = 3 |
| K | 19 | 18 | +1 | n = 4 |
| Rb | 37 | 36 | +1 | n = 5 |
| Cs | 55 | 54 | +1 | n = 6 |
The Zeff is about +1 for all of them. But lithium’s valence electron is in the second shell, while cesium’s is in the sixth. That enormous distance difference is why cesium has a much larger atomic radius, much lower ionization energy, and much higher metallic character than lithium.
How Zeff Explains Every Periodic Trend
Here is the complete picture. Every trend reduces to two variables: how strongly the nucleus pulls (Zeff) and how far away the valence electrons are (shell number).
Periodic trends, and the single cause behind them
Scroll sideways to see the whole map.
The one cause underneathAcross a period, protons are added but the new electrons go into the same shell, so they shield each other poorly and effective nuclear charge climbs. Down a group, a whole new shell is added and the outer electrons are both further out and screened by everything beneath. Every trend on this page is one of those two sentences.
The two ionisation-energy dipsBe to B falls because boron's outermost electron is in a higher-energy 2p rather than 2s. N to O falls because oxygen is the first period-2 element forced to put two electrons in one 2p orbital, and their mutual repulsion makes one easier to remove. Both dips are on the plot, not smoothed away.
Ionic radiusCations are always smaller than their parent atom, often dramatically so when a whole shell is lost: Na is 186 pm, Na⁺ is 102 pm. Anions are always larger, because added electrons increase repulsion without adding any protons.
| Trend | Across a Period (L to R) | Explanation | Down a Group | Explanation |
|-------|--------------------------|-------------|-------------|-------------|
| Atomic radius | Decreases | Zeff increases, pulling electrons closer | Increases | New shells place electrons farther out |
| Ionization energy | Increases | Higher Zeff holds electrons more tightly | Decreases | Greater distance weakens the grip |
| Electron affinity | Increases (more exothermic) | Higher Zeff attracts incoming electrons more | Decreases | Distance reduces attraction for new electrons |
| Electronegativity | Increases | Higher Zeff pulls bonding electrons harder | Decreases | Distance weakens the pull on bonding electrons |
| Metallic character | Decreases | Higher Zeff makes electron loss harder | Increases | Greater distance makes electron loss easier |
A Note on Slater’s Rules
The approximation Zeff = Z - S (where S equals the number of core electrons) is a simplification. The more precise method, called Slater’s rules, assigns different shielding values to electrons depending on their orbital type and relative position. For example, electrons in the same shell provide about 0.35 units of shielding, while electrons one shell below provide about 0.85 units.
The MCAT does not require you to use Slater’s rules. The simplified formula is sufficient for all periodic trend questions. But understanding that same-shell electrons provide some shielding (just much less than inner-shell electrons) helps explain why the Zeff increase across a period is slightly less than +1 per element in reality.
Zeff = +5. Phosphorus has the electron configuration 1s² 2s² 2p⁶ 3s² 3p³. There are 10 core electrons (in shells 1 and 2). Using Zeff = Z - S: Zeff = 15 - 10 = +5. The five valence electrons each feel an effective pull of roughly +5 from the nucleus.
Across a period, each added electron enters the same shell (poor shielding) while Z increases by 1, so Zeff rises steadily. Down a group, each new element adds a full inner shell of core electrons that effectively shield the additional nuclear charge. Z increases, but S increases by nearly the same amount, keeping Zeff approximately constant.
Chlorine’s valence electrons feel a much higher Zeff (+7) than sodium’s (+1). Both elements have 10 core electrons providing shielding, but chlorine has 17 protons versus sodium’s 11. The stronger net pull in chlorine draws the electron cloud inward, producing a smaller radius despite having more total electrons. More Zeff means a tighter, more compact atom.