Calculate effective nuclear charge (Z‑eff) using Slater’s rules for any element or custom electron configuration. Get the shielding constant, Z‑eff value, and a complete step-by-step breakdown of how each electron group contributes to shielding.
Effective Nuclear Charge and Slater's Rules Explained
Effective nuclear charge (Z‑eff) is the net positive charge experienced by a specific electron in a multi-electron atom. While the actual nuclear charge Z equals the number of protons, the inner electrons partially shield the valence electrons from the full nuclear attraction. The shielding constant σ quantifies this screening effect, and Z‑eff = Z − σ.
Z‑eff is the single most important concept for explaining periodic table trends. It explains why atomic radius decreases across a period, why ionization energy increases across a period, why electronegativity trends exist, and why electron affinity varies — all because of how strongly the nucleus holds onto its outermost electrons.
Z-eff Values Across Periods 2 and 3
Element
Z
Valence
σ (Slater)
Z-eff
Li
3
2s¹
2(0.85)=1.70
1.30
C
6
2s²2p²
3(0.35)+2(0.85)=2.75
3.25
N
7
2s²2p³
4(0.35)+2(0.85)=3.10
3.90
F
9
2s²2p⁵
6(0.35)+2(0.85)=3.80
5.20
Ne
10
2s²2p⁶
7(0.35)+2(0.85)=4.15
5.85
Na
11
3s¹
8(0.85)+2(1.00)=8.80
2.20
Cl
17
3s²3p⁵
6(0.35)+8(0.85)+2(1.00)=10.90
6.10
Ar
18
3s²3p⁶
7(0.35)+8(0.85)+2(1.00)=11.25
6.75
💡 Periodic Table Trends from Z-eff:
Atomic radius decreases → across period: Z-eff rises, pulling electrons closer
Ionization energy increases → across period: higher Z-eff = harder to remove electron
Electronegativity increases → across period: stronger pull on bonding electrons
All trends reverse going ↓ down group: new shells placed farther from nucleus
Why Z-eff Drops at the Start of Each Period
Notice that Z-eff drops sharply from Ne (5.85) to Na (2.20) despite Z increasing from 10 to 11. The new 3s electron in Na is shielded by all 10 inner electrons, 8 of which contribute 0.85 each and 2 contribute 1.00 each, giving massive shielding of 8.80. This makes alkali metals hold their valence electron very loosely — exactly why they are so reactive and have low ionization energies.
Frequently Asked Questions
Z-eff = net positive charge felt by a valence electron = Z − σ. Inner electrons partially shield valence electrons from the full nuclear charge. Z-eff ≤ Z for all multi-electron atoms.
Empirical rules to calculate shielding constant σ. Group electrons as [1s][2s,2p][3s,3p][3d] etc. For s/p valence: same group = 0.35, one shell inside = 0.85, two+ shells inside = 1.00. For d/f valence: all inner = 1.00.
Z-eff increases left to right. Each added electron enters the same shell (only 0.35 shielding each other) while Z increases by 1. Net gain ≈ 0.65 per element across a period.
Na (Z=11): [Ne]3s¹. For 3s electron: σ = 8(0.85) + 2(1.00) = 8.80. Z-eff = 11 − 8.80 = 2.20.
Higher Z-eff = nucleus attracts electrons more strongly = more energy needed to remove them. IE increases across a period (rising Z-eff) and decreases down a group (farther from nucleus despite slight Z-eff increase).
Z = actual proton count. Z-eff = charge felt by a specific electron after shielding. Na: Z=11, but 3s electron feels only Z-eff=2.20 because 10 inner electrons shield 8.80 units of charge.
Total screening by all other electrons on one target electron, via Slater's rules. Same-group: 0.35; one shell inside (s/p): 0.85; two+ shells inside: 1.00. For d/f targets: inner = 1.00 always.
Higher Z-eff pulls electrons closer, shrinking the atom. Atomic radius decreases across a period (Z-eff rises) and increases down a group (new shells much farther from nucleus).
d electrons have poor radial penetration and spend most time in outer regions. They don't shield each other well. In Slater's rules, d electrons in the same shell contribute 1.00 to each other (not 0.35 like s/p electrons).