Effective Nuclear Charge and Shielding
A valence electron in sodium does not feel the full +11 pull of the nucleus — the ten inner electrons get in the way. What it actually feels is called the effective nuclear charge (Zeff), and the most widely taught way to estimate it is Slater's rules. Every periodic trend you have memorised — atomic radius, ionisation energy, electronegativity — ultimately traces back to how Zeff changes across the table.
The formula and Slater's grouping rule
where Z = actual atomic number and S = the shielding (screening) constant
To calculate S by Slater's rules, first write the electron configuration in these groups, in order: (1s)(2s,2p)(3s,3p)(3d)(4s,4p)(4d)(4f)(5s,5p)… Note that ns and np are grouped together, but nd and nf are each their own separate group.
| For an electron in an ns or np group | Contribution per electron |
|---|---|
| Other electrons in the same group | 0.35 each (0.30 if the group is 1s) |
| Electrons in the (n−1) shell | 0.85 each |
| Electrons in (n−2) or lower shells | 1.00 each |
For an electron in an nd or nf group, the rule is simpler: other electrons in the same group contribute 0.35 each, and every electron in a lower-listed group (regardless of shell) contributes 1.00 each. The electron whose Zeff you are calculating is never counted towards its own shielding.
Worked example 1 — the 3s valence electron in Na (Z = 11)
Configuration: (1s²)(2s²2p⁶)(3s¹). For the 3s electron:
Same group (3s,3p), other electrons: 0 → 0 × 0.35 = 0
(n−1) shell, i.e. n = 2 (2s²2p⁶ = 8 electrons): 8 × 0.85 = 6.80
(n−2) shell, i.e. n = 1 (1s² = 2 electrons): 2 × 1.00 = 2.00
S = 0 + 6.80 + 2.00 = 8.80
Zeff = 11 − 8.80 = 2.20
Worked example 2 — a 2p electron in F (Z = 9)
Configuration: (1s²)(2s²2p⁵). For one 2p electron:
Same group (2s,2p), other electrons: 2 + 5 − 1 = 6 → 6 × 0.35 = 2.10
(n−1) shell, i.e. n = 1 (1s² = 2 electrons): 2 × 0.85 = 1.70
S = 2.10 + 1.70 = 3.80
Zeff = 9 − 3.80 = 5.20
Worked example 3 — the trend across Period 2 (Li vs Be)
Li (Z = 3): (1s²)(2s¹). Same-group others = 0. (n−1) shell (1s²) =
2 × 0.85 = 1.70. S = 1.70, so Zeff = 3 − 1.70 = 1.30.
Be (Z = 4): (1s²)(2s²). Same-group others = 1 × 0.35 = 0.35. (n−1) shell
(1s²) = 2 × 0.85 = 1.70. S = 0.35 + 1.70 = 2.05, so Zeff = 4 − 2.05 =
1.95.
Adding one proton (+1.00 to Z) only adds 0.35 to the shielding from the new same-shell
electron, so Zeff rises by about 0.65 each step across the period — this is
exactly why atomic radius shrinks steadily left to right.
Worked example 4 — down a group: Na vs K
K (Z = 19): (1s²)(2s²2p⁶)(3s²3p⁶)(4s¹). For the 4s electron:
Same group (4s,4p), other electrons: 0.
(n−1) shell, i.e. n = 3 (3s²3p⁶ = 8 electrons): 8 × 0.85 = 6.80.
(n−2) or lower, i.e. n = 1,2 (2 + 8 = 10 electrons): 10 × 1.00 = 10.00.
S = 6.80 + 10.00 = 16.80, so Zeff = 19 − 16.80 = 2.20 — almost
identical to sodium's 2.20 calculated above. Slater's simplified rule gives alkali metals a
nearly constant valence-electron Zeff down the group; more accurate self-consistent
field calculations (Clementi–Raimondi values) actually show Zeff rising slightly
down a group. Either way, atomic radius still increases down the group, because each new
shell adds a larger principal quantum number n, which outweighs the small change in
Zeff.
Common mistakes that cost marks
- Mixing up the 0.85 and 1.00 contributions: the (n−1) shell always contributes 0.85 per electron for an s/p electron being calculated — only shells two or more below contribute the full 1.00.
- Grouping s and p electrons separately, or lumping d/f in with s/p: Slater's rule groups (ns, np) together as one unit but treats nd and nf as their own separate groups — this is the single most common setup error.
- Using 0.30 outside the 1s group: the reduced same-group value of 0.30 applies only within the 1s group itself; every other same-group interaction uses 0.35.
- Counting the electron itself in its own shielding: always subtract one from the group's total electron count before multiplying by 0.35.
- Treating Slater's rules as exact: they are a hand-calculable approximation, not a precise quantum-mechanical result — real trends can deviate slightly, as shown in worked example 4.
Where effective nuclear charge appears in exams
| Exam | Typical use |
|---|---|
| CBSE Class 11 | Structure of Atom / Periodicity chapters — shielding effect concept |
| JEE/NEET | Conceptual questions linking Zeff to trends |
| IIT-JAM / CUET-PG | Direct Slater's-rules numerical calculations |
| GATE / CSIR-NET | Slater's rules problems are a recurring, high-frequency question type |
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