Periodic trends

Result

Periodic trends

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Statement

Define the effective nuclear charge felt by an electron as \(Z_{\text{eff}}=Z-S\), where \(S\) (the shielding constant) is computed from Slater's rules: for an electron in an \(ns\) or \(np\) orbital, sum \(0.35\) for every other electron in the same \(n s,np\) group, \(0.85\) for every electron in the \(n{-}1\) shell, and \(1.00\) for every electron in shells \(n{-}2\) and below. Then, moving left to right across a period, \(Z_{\text{eff}}\) for the valence electron increases (atomic radius decreases, first ionisation energy increases); moving down a group, \(Z_{\text{eff}}\) stays roughly constant while the valence principal quantum number \(n\) increases (atomic radius increases, first ionisation energy decreases).

Why it matters

This single quantity, effective nuclear charge, is the mechanism behind essentially every periodic trend taught in introductory chemistry — atomic radius, ionic radius, ionisation energy, electron affinity, electronegativity — rather than each trend needing its own separate explanation. Once \(Z_{\text{eff}}\) is understood, the periodic table stops being a list of memorised up/down and left/right arrows and becomes the readout of one physical cause, screening, playing out differently along the table's two axes.

It is also the concept that makes sense of the exceptions students otherwise memorise by rote — why oxygen's first ionisation energy is slightly lower than nitrogen's, against the naive left-to-right trend (see Fails Without), and why post-transition-metal elements have systematically smaller radii than a naive extrapolation down their group would predict (the "d-block contraction," a direct consequence of the poor shielding provided by \(d\) electrons).

Hypotheses

Slater's rules are an empirical approximation to the true, many-electron shielding, not an exact quantum-mechanical result.Genuine shielding depends on the detailed radial probability distribution of every other electron, which differs orbital by orbital and element by element; Slater's rules replace this with fixed, universal constants (\(0.35\), \(0.85\), \(1.00\)) chosen to fit atomic spectra reasonably well across the table. They reproduce correct trends reliably (the qualifying comparisons in Discussion) but should not be read as giving exact ionisation energies to high precision — a genuinely accurate calculation needs a full self-consistent-field (Hartree–Fock or better) treatment. Comparing atoms with broadly similar electron configurations (main-group, same block).The clean left-to-right and top-to-bottom trend statements are calibrated on the \(s\)- and \(p\)-blocks. Transition metals and lanthanides/actinides show additional, more subtle trend irregularities (arising from \(d\)- and \(f\)-electron shielding being systematically weaker than \(s\)/\(p\) shielding of the same shell), covered separately once this unit reaches transition-metal chemistry.

Proof

Z_{\text{eff}} = Z - S, \qquad S = 0.35\,n_{\text{same}} + 0.85\,n_{\,n-1} + 1.00\,n_{\le n-2}

Slater's rules (cited, empirical, not re-derived): the shielding constant is a weighted count of every other electron in the atom, weighted by how effectively that electron's shell shields the electron in question — electrons in the same shell shield weakly (\(0.35\)), electrons one shell in shield substantially (\(0.85\)), and electrons two or more shells in shield almost completely (\(1.00\)). B

\text{Moving left to right across a period: same-shell electron count } n_{\text{same}}\text{ rises by 1 per element; } Z \text{ also rises by 1; inner shells unchanged.}

Adding one proton and one valence electron per step along a period increases \(Z\) by exactly \(1\), but only adds \(0.35\) to \(S\) (the new electron is in the same, weakly-shielding group as the others already there). Net change in \(Z_{\text{eff}}\) per step: \(+1-0.35=+0.65\) — \(Z_{\text{eff}}\) climbs steadily across a period. B

\text{Moving down a group: valence shell } n \text{ increases by 1; new complete inner shell added, contributing } \approx1.00\text{ per inner electron; } Z_{\text{eff}} \text{ changes only slightly.}

Descending a group replaces the valence electron's \(n{-}1\)-shell neighbours (previously weighted \(0.85\)) with a newly completed shell now sitting at \(n{-}1\) again relative to the NEW valence shell, while the OLD valence shell (now fully inside) contributes at the stronger \(1.00\) weight; the net effect very nearly cancels the extra nuclear charge picked up, leaving \(Z_{\text{eff}}\) for the valence electron roughly constant down a group (see Worked Example 2, Na vs K) — even as the electron itself sits in a higher, larger-radius shell. B

r \sim \frac{n^2}{Z_{\text{eff}}}a_0, \qquad I_1 \sim \frac{Z_{\text{eff}}^2}{n^2}\cdot13.6\,\text{eV}

Apply the Bohr-model radius and energy scaling (this unit's Bohr-model result) with \(Z\) replaced by \(Z_{\text{eff}}\) and the true valence \(n\) — a standard, qualitatively reliable extension of the one-electron formulas to the valence electron of a many-electron atom. Increasing \(Z_{\text{eff}}\) at fixed \(n\) shrinks \(r\) and raises \(I_1\) (the period trend, Step 2); increasing \(n\) at roughly fixed \(Z_{\text{eff}}\) grows \(r\) and lowers \(I_1\) (the group trend, Step 3). B

Result

\text{left}\to\text{right: } Z_{\text{eff}}\uparrow,\ r\downarrow,\ I_1\uparrow \qquad\qquad \text{top}\to\text{bottom: } Z_{\text{eff}}\approx\text{const},\ n\uparrow,\ r\uparrow,\ I_1\downarrow

Reading. Two different mechanisms drive the table's two axes: across a period, the nucleus wins a tug-of-war against weak same-shell shielding; down a group, a fresh, more effective inner shell almost exactly cancels the extra nuclear charge, leaving shell number (\(n\), hence size) as the dominant remaining effect.

Scope. Reliable, qualitative-to-semi-quantitative trend prediction across the \(s\)- and \(p\)-blocks. Not intended as a precision tool for absolute ionisation energies (see Hypotheses); several well-known exceptions to the strict period trend exist and are discussed below.

Corollaries & converses

Electronegativity and electron affinity broadly track the same \(Z_{\text{eff}}\) trend as ionisation energy (higher \(Z_{\text{eff}}\) pulls harder on both an atom's own electrons and any electron it might gain), which is why all of these properties are typically taught together as "the periodic trends."
Ionic radii follow the same logic applied to ions rather than neutral atoms: removing an electron (forming a cation) reduces electron–electron repulsion and effectively raises \(Z_{\text{eff}}\) per remaining electron, shrinking the ion; adding an electron (forming an anion) does the reverse.
Converse: given two elements' measured first ionisation energies, one can infer which has the larger \(Z_{\text{eff}}\) (at comparable \(n\)) without doing the Slater calculation explicitly — the Result runs in both directions as a diagnostic.

Fails without

The strict period-trend prediction, O vs N: naively, oxygen (\(Z=8\)) should have a higher first ionisation energy than nitrogen (\(Z=7\)), following Step 2. Experimentally it is slightly lower (\(I_1(\text N)=14.53\,\text{eV}\) vs \(I_1(\text O)=13.62\,\text{eV}\)). The reason lies outside Slater's rules entirely: nitrogen's \(2p^3\) configuration is half-filled (extra exchange stabilisation, this unit's building-up-principle result), so removing an electron from oxygen's \(2p^4\) (breaking a paired orbital, relieving electron–electron repulsion within that orbital) is easier than the naive \(Z_{\text{eff}}\)-only picture predicts.
Drop the "roughly constant" qualifier in the group trend (\(d\)-block shielding): gallium (just after the first row of \(d\)-block elements) is almost the same atomic radius as aluminium directly above it, rather than noticeably larger as the naive group trend of Step 3 would suggest, because the intervening \(3d\) electrons shield the nuclear charge unusually poorly (Slater's rules with \(d\)-electrons behave differently, weighting same-group \(d\) electrons only at \(0.35\) but crucially not counting toward shielding the \(4p\) valence electron as generously as a full \(s/p\) shell would) — the well-known "\(d\)-block contraction."

Common errors

Assuming ionisation energy increases strictly monotonically across every period with no exceptions — the Be/B and N/O pairs are the two standard counter-examples every introductory course covers, both traceable to subshell-filling effects outside the bare \(Z_{\text{eff}}\) picture.
Assuming \(Z_{\text{eff}}\) itself increases significantly down a group, rather than staying roughly constant (Step 3) — the actual driver of the group-trend decrease in ionisation energy is the growing \(n\), not a shrinking \(Z_{\text{eff}}\).
Applying Slater's simplified weighting scheme to \(d\)- or \(f\)-block valence electrons with the same confidence as \(s\)/\(p\)-block elements, without the caveat noted in Fails Without.

Discussion

John C. Slater published his shielding rules in 1930 as a practical, hand-calculable approximation, at a time when solving the many-electron Schrödinger equation numerically for every atom was not yet feasible. The rules were deliberately built to be usable with pencil and paper — a small, fixed set of weighting constants rather than a full variational calculation — and their continued use in introductory chemistry today reflects exactly that original design goal: they get the qualitative and semi-quantitative physics right with an amount of arithmetic a student can actually do.

More accurate effective nuclear charges are available from later, more sophisticated treatments (notably Clementi and Raimondi's 1963 values, fit directly to self-consistent-field wavefunctions rather than a fixed rule), which refine but do not overturn the qualitative period/group story told here.

The deeper reason Slater's rules work as well as they do is that they are, in effect, a crude discretisation of the true radial shielding function \(\langle 1/r\rangle\) integrated over each other electron's charge density — electrons genuinely closer to the nucleus on average (smaller \(n\), or smaller \(l\) at fixed \(n\), owing to greater orbital penetration) shield more effectively, and Slater's three-tier weighting (\(0.35\), \(0.85\), \(1.00\)) is a coarse but serviceable stand-in for a continuous, orbital-by-orbital calculation.

Common misconception: that \(Z_{\text{eff}}\) is a single, fixed property of an element, like atomic number. It is specific to a particular electron in a particular orbital of a particular atom (or ion) — a \(1s\) electron and a valence-shell electron in the same atom have very different \(Z_{\text{eff}}\) values, precisely because they are shielded differently (compare the near-full \(Z_{\text{eff}}\approx Z\) felt by a \(1s\) electron, essentially unshielded from the nucleus, against the much-reduced valence-electron values computed throughout this page).

Worked examples

\text{Period 2 trend: Li (}Z{=}3\text{) vs F (}Z{=}9\text{), both valence electrons in }n{=}2

Li \(2s\) valence electron (config \(1s^22s^1\)): same-group electrons besides itself \(=0\); \(n{-}1\) shell (\(1s\)) \(=2\) electrons \(\times0.85=1.7\); \(S=1.7\), \(Z_{\text{eff}}=3-1.7=1.3\). A

\text{F } 2p \text{ electron (config } 1s^22s^22p^5\text{): same-group}=6\times0.35=2.1;\ \ n{-}1\ (1s)=2\times0.85=1.7;\quad S=3.8,\ \ Z_{\text{eff}}=9-3.8=5.2

Fluorine's \(2p\) electron has six other electrons in the same \((2s,2p)\) group (\(2s^22p^5\) has \(7\) electrons total, minus itself). \(Z_{\text{eff}}\) has climbed from \(1.3\) (Li) to \(5.2\) (F) across the same period — a factor of \(4\), matching the sharp rise in first ionisation energy from lithium (\(5.39\,\text{eV}\)) to fluorine (\(17.42\,\text{eV}\)). A

\text{Group 1 trend: Na (}Z{=}11,\ n{=}3\text{) vs K (}Z{=}19,\ n{=}4\text{), both valence }ns^1

Na \(3s\): same-group\(=0\); \(n{-}1\) (\(2s,2p\)) \(=8\times0.85=6.8\); \(n{-}2\) (\(1s\)) \(=2\times1.00=2.0\); \(S=8.8\), \(Z_{\text{eff}}=11-8.8=2.2\). K \(4s\): same-group\(=0\); \(n{-}1\) (\(3s,3p\)) \(=8\times0.85=6.8\); \(n{-}2\) and lower (\(1s,2s,2p\)) \(=10\times1.00=10.0\); \(S=16.8\), \(Z_{\text{eff}}=19-16.8=2.2\). A

Z_{\text{eff}}(\text{Na, }3s)=Z_{\text{eff}}(\text{K, }4s)=2.2 \quad\text{exactly, despite }\Delta Z=8

Reading. Slater's rule predicts, essentially exactly, that sodium and potassium's valence electrons feel the same effective nuclear charge — direct confirmation of Step 3's claim that the group trend is driven by \(n\), not by \(Z_{\text{eff}}\). Both elements' measured first ionisation energies (Na: \(5.14\,\text{eV}\); K: \(4.34\,\text{eV}\)) are lower than the corresponding Bohr-scaling estimate would give for hydrogen at the same \(Z_{\text{eff}}\) and \(n\) (a reminder of the Hypotheses' precision caveat), but the two values' ordering, and the fact that they are close to each other rather than differing by a large factor, both match the near-equal \(Z_{\text{eff}}\) found here.

Scope. The same calculation, repeated down any main group, shows the same near-constant-\(Z_{\text{eff}}\) pattern — the general mechanism behind every group's decreasing ionisation-energy trend.

Problems

Compute \(Z_{\text{eff}}\) for the \(2p\) valence electron of nitrogen (\(Z=7\), configuration \(1s^22s^22p^3\)) using Slater's rules, and compare with the fluorine value found in Worked Example 2.
Solution
Same-group electrons besides itself: \(2s^22p^3\) has \(5\) total, minus itself \(=4\), \(\times0.35=1.4\). \(n{-}1\) shell (\(1s\)): \(2\times0.85=1.7\). \(S=1.4+1.7=3.1\), \(Z_{\text{eff}}=7-3.1=3.9\). This is lower than fluorine's \(5.2\), consistent with the general period trend (Step 2), and lower than oxygen would give by the same method — the "exception" in Fails Without concerns the actual ionisation energy ordering of N and O, not the \(Z_{\text{eff}}\) ordering itself, which still increases monotonically with \(Z\) here.
A student argues that since caesium (\(Z=55\)) has a far larger nuclear charge than lithium (\(Z=3\)), it must have a much larger \(Z_{\text{eff}}\) for its valence electron, and therefore should be harder to ionise. Explain the flaw, referencing the Result.
Solution
The flaw is treating \(Z_{\text{eff}}\) as tracking \(Z\) directly. By the Result (and confirmed numerically for Na/K in Worked Example 3), \(Z_{\text{eff}}\) for the valence electron stays roughly constant down a group, regardless of how large \(Z\) itself grows — caesium's much larger \(Z\) is almost entirely cancelled by its much larger shielding \(S\) from many more complete inner shells. What changes down the group is \(n\) (Cs valence \(n=6\)), which lowers ionisation energy via the \(1/n^2\) factor in Step 4 — caesium is in fact one of the most easily ionised elements (\(I_1=3.89\,\text{eV}\)), the opposite of the student's prediction.
Using Slater's rules, find \(Z_{\text{eff}}\) for a \(1s\) electron in helium (\(Z=2\), configuration \(1s^2\)), and explain why it is so much closer to the bare \(Z\) than any valence-electron example on this page.
Solution
Same-group (\(1s\)) electrons besides itself: \(1\), weighted at \(0.30\) for the special \(1s\)-group case of Slater's rules (not \(0.35\), a documented exception for the innermost shell), giving \(S=0.30\), \(Z_{\text{eff}}=2-0.30=1.70\). This is much closer to the bare \(Z=2\) than any of the valence \(Z_{\text{eff}}\) values computed above, because a \(1s\) electron has no inner shell to be shielded by at all — only its one same-shell partner contributes, at the weakest available weighting.
Explain qualitatively, without recomputing, why the first ionisation energy of neon (\(Z=10\), end of period 2) is higher than that of sodium (\(Z=11\), start of period 3), even though sodium has the larger \(Z\).
Solution
Neon and sodium are not in the same period, so the period-trend comparison of Step 2 does not apply directly between them; sodium's extra electron starts a brand-new shell (\(n=3\)), which the Result's group-trend reasoning (Step 3) says should have a much lower \(Z_{\text{eff}}\)-to-\(n\) ratio than neon's tightly-shielded, complete \(n=2\) shell. In effect, comparing across a period boundary combines both trends: sodium's new, poorly-shielded \(3s\) electron is easier to remove than any electron in neon's complete, tightly-bound \(2p^6\) shell, despite sodium's higher \(Z\).