Appearance
9.5 — Periodic Trends, With the Reasoning
Chapter 9.4 stated the trends and gave hand-waving reasons. This chapter computes them.
Two quantities do all the work, and everything else follows from how they compete.
Z_{\text{eff}} — the effective nuclear charge felt by an outer electron.
n — the principal quantum number of the outer shell, which sets how far out it is.
Since Chapter 7.6 gives the orbital size as r \sim n^2a_0/Z_{\text{eff}} and the binding energy as E \sim -13.6Z_{\text{eff}}^2/n^2, those two numbers determine size, ionisation energy, electron affinity and electronegativity together.
Computing Z_{\text{eff}} across a period
Use Slater's rules from Chapter 9.3 for the second period, for an outer 2s or 2p electron.
Lithium, 1s^22s^1:
S = 2(0.85) = 1.70, \qquad Z_{\text{eff}} = 3-1.70 = 1.30
Beryllium, 1s^22s^2:
S = 1(0.35)+2(0.85) = 0.35+1.70 = 2.05, \qquad Z_{\text{eff}} = 4-2.05 = 1.95
Boron, 1s^22s^22p^1:
S = 2(0.35)+2(0.85) = 0.70+1.70 = 2.40, \qquad Z_{\text{eff}} = 5-2.40 = 2.60
Continuing across:
| Element | Z | S | Z_{\text{eff}} |
|---|---|---|---|
| Li | 3 | 1.70 | 1.30 |
| Be | 4 | 2.05 | 1.95 |
| B | 5 | 2.40 | 2.60 |
| C | 6 | 2.75 | 3.25 |
| N | 7 | 3.10 | 3.90 |
| O | 8 | 3.45 | 4.55 |
| F | 9 | 3.80 | 5.20 |
| Ne | 10 | 4.15 | 5.85 |
Z_{\text{eff}} rises by 0.65 per element — the proton adds 1.00 and the new same-shell electron screens only 0.35.
Now down a group. Compare lithium with sodium, 1s^22s^22p^63s^1:
S = 8(0.85)+2(1.00) = 6.80+2.00 = 8.80, \qquad Z_{\text{eff}} = 11-8.80 = 2.20
And potassium, from Chapter 9.3: Z_{\text{eff}} = 2.20.
Down a group, Z_{\text{eff}} barely changes (1.30, 2.20, 2.20) while n increases by one each time. So n dominates going down and Z_{\text{eff}} dominates going across. That single statement generates every trend.
Atomic radius
r \sim \frac{n^2a_0}{Z_{\text{eff}}}
Across a period: n fixed, Z_{\text{eff}} rising, so radius shrinks.
Down a group: n rising as n^2, Z_{\text{eff}} roughly fixed, so radius grows.
Measured values, in picometres:
| Li 152 | Be 112 | B 85 | C 77 | N 75 | O 73 | F 72 | Ne 71 |
| Na 186 | Mg 160 | Al 143 | Si 118 | P 110 | S 103 | Cl 100 | Ar 98 |
| K 227 | Ca 197 | Br 114 | Kr 112 |
Lithium to neon: 152 → 71 pm, a factor of more than two, while adding seven protons and seven electrons.
Lithium to potassium: 152 → 227 pm.
Check the model. From lithium to neon, Z_{\text{eff}} goes 1.30 → 5.85, a factor of 4.5, so the radius should fall by 4.5. It falls by 2.1. The model is qualitatively right and quantitatively rough, because Slater's rules are an empirical fit and because electron–electron repulsion pushes the shell outwards in ways a single screening constant cannot capture.
The lanthanide contraction
Something odd happens in period 6.
| Element | Radius |
|---|---|
| Zr (period 5) | 160 pm |
| Hf (period 6) | 159 pm |
| Nb | 146 pm |
| Ta | 146 pm |
Hafnium is the same size as zirconium, despite being a whole period lower.
The cause: between them the fourteen lanthanides filled the 4f subshell. f orbitals screen very poorly — they are diffuse and have most of their density away from the region an outer electron occupies — so fourteen protons were added while the screening rose by much less. Z_{\text{eff}} increased substantially and pulled the outer shells in, cancelling the expansion from the extra period.
Consequences that matter industrially:
Zirconium and hafnium are chemically almost indistinguishable and always occur together. Separating them is essential and difficult, because zirconium is transparent to neutrons and used for nuclear fuel cladding while hafnium absorbs them strongly and is used for control rods. Two elements with opposite nuclear uses and nearly identical chemistry.
The third-row transition metals are unusually dense. Osmium (22.6 g/cm³) and iridium (22.5) are the densest elements, and platinum and gold are far denser than their period-5 counterparts — same size, much more mass.
Ionic radius
Cations are much smaller than their parent atoms. Anions are much larger.
| Atom | Radius | Ion | Radius |
|---|---|---|---|
| Na | 186 pm | Na⁺ | 102 pm |
| Mg | 160 pm | Mg²⁺ | 72 pm |
| Al | 143 pm | Al³⁺ | 54 pm |
| Cl | 100 pm | Cl⁻ | 181 pm |
| O | 73 pm | O²⁻ | 140 pm |
Sodium shrinks by 45 % on losing one electron.
Two reasons, and the first is the big one. Sodium is [\text{Ne}]3s^1 and Na⁺ is [\text{Ne}] — the entire outer shell is gone, so the ion's size is set by n = 2 rather than n = 3. And with the same nuclear charge now pulling on fewer electrons, Z_{\text{eff}} for the remaining ones rises.
Anions grow because the added electron increases repulsion within an unchanged nuclear charge.
Isoelectronic series. N³⁻, O²⁻, F⁻, Ne, Na⁺, Mg²⁺, Al³⁺ all have ten electrons. Their radii fall monotonically — 146, 140, 133, 112, 102, 72, 54 pm — purely because Z rises from 7 to 13 while the electron count is fixed.
This is why crystal structures are what they are. In NaCl the chloride ion (181 pm) is far larger than sodium (102 pm), so the structure is essentially a close-packed array of chlorides with sodium ions in the holes.
Ionisation energy, including the irregularities
Chapter 9.3 showed the graph. The overall trend follows Z_{\text{eff}}^2/n^2, and the interesting part is where it does not.
Estimate lithium with Z_{\text{eff}} = 1.30, n = 2:
IE = \frac{13.6\times(1.30)^2}{4} = \frac{13.6\times1.69}{4} = 5.75\ \text{eV}
Measured 5.39. Within 7 %.
Estimate neon with Z_{\text{eff}} = 5.85:
IE = \frac{13.6\times34.2}{4} = 116\ \text{eV}
Measured 21.6. Off by a factor of five, and the failure is instructive: Slater's rules were fitted to give good sizes, not good energies, and for a nearly filled shell the mutual repulsion of eight electrons in the same shell is badly represented by a single average screening constant.
The two irregularities are the real content.
Beryllium to boron
\text{Be } (2s^2): 9.32\ \text{eV} \quad\to\quad \text{B } (2s^22p^1): 8.30\ \text{eV}
A drop, despite a larger nuclear charge.
Reason: the electron removed from boron is a 2p electron, and 2p is higher in energy than 2s because it penetrates less (Chapter 9.3). The subshell energy difference beats the extra proton.
This dip is direct experimental evidence that subshells exist, visible in a measurement anyone can make.
Nitrogen to oxygen
\text{N } (2p^3): 14.53\ \text{eV} \quad\to\quad \text{O } (2p^4): 13.62\ \text{eV}
Another drop.
Reason: nitrogen's three 2p electrons occupy three separate orbitals with parallel spins, by Hund's rule. Oxygen's fourth must pair with one of them, and two electrons in the same orbital repel strongly. The removed electron is the paired one, and its repulsion partner makes it easier to take.
This dip is direct evidence for Hund's rule.
The same two dips repeat at Mg→Al and P→S, and again in every subsequent period. The pattern is a fingerprint of orbital structure.
Electron affinity
The energy released when a neutral atom gains an electron:
\text{X}+e^- \to \text{X}^- + \text{energy}
Sign convention is a nuisance. This book uses positive = energy released = favourable.
| Element | EA (eV) |
|---|---|
| F | 3.40 |
| Cl | 3.62 |
| Br | 3.36 |
| I | 3.06 |
| O | 1.46 |
| N | \approx-0.07 |
| Ne | <0 |
Chlorine's affinity is larger than fluorine's, which breaks the expected trend and is worth understanding.
Reason: fluorine is very small (72 pm), so its 2p subshell is already crowded. Adding a ninth electron into that tight shell costs a lot of repulsion, which partly cancels the strong nuclear attraction. Chlorine's 3p shell is roomier.
The same anomaly appears at oxygen versus sulphur and at nitrogen versus phosphorus. The second-row elements are all unusually small, and it changes their chemistry — it is also why nitrogen and oxygen form strong double and triple bonds while phosphorus and sulphur prefer single bonds and larger structures.
Nitrogen's affinity is essentially zero or slightly negative. Adding an electron to 2p^3 forces pairing in a half-filled subshell, giving up the exchange stabilisation of Hund's rule. Half-filled shells resist accepting electrons.
Noble gases have negative affinity — you must supply energy to attach an electron, since it would have to start a new shell.
Electronegativity
The tendency of an atom in a bond to pull shared electrons towards itself. Unlike ionisation energy and electron affinity, it is not a directly measurable property of an isolated atom — it is a property of an atom in a molecule, which is why there are competing scales.
Pauling's scale
Linus Pauling's method, 1932, and still the most used.
The observation: a bond between different atoms is usually stronger than the average of the two like-bonds. Pauling attributed the excess to ionic character, and defined:
|\chi_A-\chi_B| = 0.102\sqrt{\Delta}
where \Delta = E_{AB} - \sqrt{E_{AA}E_{BB}} is the excess bond energy in kJ/mol.
Worked example: HF.
E_{\text{HF}} = 565, \quad E_{\text{HH}} = 436, \quad E_{\text{FF}} = 155\ \text{kJ/mol}
\Delta = 565-\sqrt{436\times155} = 565-\sqrt{67580} = 565-260 = 305
|\chi_F-\chi_H| = 0.102\sqrt{305} = 0.102\times17.5 = 1.78
With hydrogen fixed at 2.20, fluorine is 3.98 — and Pauling set fluorine as the maximum by construction.
Values:
| H 2.20 | ||||
| Li 0.98 | Be 1.57 | B 2.04 | C 2.55 | N 3.04 |
| O 3.44 | F 3.98 | |||
| Na 0.93 | Mg 1.31 | Al 1.61 | Si 1.90 | Cl 3.16 |
| K 0.82 | Br 2.96 | |||
| Cs 0.79 | I 2.66 |
Fluorine is the most electronegative element; caesium and francium the least.
Mulliken's scale
A more direct definition:
\chi_M = \frac{IE+EA}{2}
The average of how hard it is to take an electron and how much it wants one — which is close to what electronegativity means in words.
Worked example: fluorine. IE = 17.42 eV, EA = 3.40 eV:
\chi_M = \frac{17.42+3.40}{2} = 10.41\ \text{eV}
Rescaled to match Pauling's units it gives 3.9, agreeing closely.
Allred–Rochow
\chi_{AR} = \frac{0.359\,Z_{\text{eff}}}{r^2}+0.744
This is explicitly the electrostatic force the nucleus exerts on a bonding electron at the covalent radius. It is the most physically transparent of the three, and it ties electronegativity directly back to the Z_{\text{eff}} calculations at the start of this chapter.
All three scales agree to within about 0.2 units, which is reassuring given they measure different things.
What the difference tells you
\Delta\chi = |\chi_A-\chi_B|
| \Delta\chi | Bond type | Example |
|---|---|---|
| 0 | Pure covalent | H–H, Cl–Cl |
| 0–0.4 | Nearly covalent | C–H (0.35) |
| 0.4–1.7 | Polar covalent | H–O (1.24), H–Cl (0.96) |
| >1.7 | Largely ionic | Na–Cl (2.23), K–F (3.16) |
The 1.7 boundary is a convention, not a physical threshold. There is a continuum, and Chapter 10.1 develops it.
Percent ionic character, from Pauling:
\%\ \text{ionic} = 100\left[1-e^{-0.25(\Delta\chi)^2}\right]
For HF, \Delta\chi = 1.78:
\% = 100\left[1-e^{-0.25\times3.17}\right] = 100\left[1-e^{-0.792}\right] = 100(1-0.453) = 54.7\ \%
Even hydrogen fluoride, the most polar simple covalent bond, is only about half ionic.
Metallic character
Down and to the left: more metallic. Up and to the right: less.
The physical basis: metallic bonding requires atoms to give up electrons to a shared sea (Chapter 10.1). Low ionisation energy makes that easy. So low IE means metallic.
The staircase of metalloids — boron, silicon, germanium, arsenic, antimony, tellurium — runs diagonally through the p block, separating metals from non-metals. These are the semiconductors, and their position is not a coincidence: they have ionisation energies in the middle range, giving a small band gap rather than a large one or none at all (Volume III, Chapter 2).
Diagonal relationships. Lithium resembles magnesium, beryllium resembles aluminium, boron resembles silicon. Moving down increases size and decreases electronegativity; moving right does the reverse; so a diagonal step roughly cancels. Lithium and magnesium both form nitrides directly with N₂, which no other alkali metal does.
Where this shows up in your life
Why table salt is a solid and hydrogen chloride is a gas. \Delta\chi for Na–Cl is 2.23, giving an ionic lattice with a melting point of 801 °C. For H–Cl it is 0.96, giving discrete polar molecules that boil at −85 °C.
Why water is a liquid and hydrogen sulphide is a gas, despite sulphur being heavier. Oxygen's electronegativity of 3.44 makes O–H bonds polar enough for hydrogen bonding; sulphur's 2.58 does not (Chapter 10.3).
Galvanic corrosion. Bolt two metals of different electronegativity together in the presence of moisture and the less electronegative one corrodes preferentially. This is why you never use steel screws in aluminium boats.
Fluoride toothpaste works because fluorine's electronegativity of 3.98 makes fluorapatite more acid-resistant than the hydroxyapatite it replaces.
Semiconductor doping exploits the fact that adding a group 15 element to silicon gives one extra electron and a group 13 element gives one fewer — a direct application of periodic position.
And every drug molecule's behaviour — whether it dissolves in water or fat, whether it crosses a membrane, where it binds — is governed by the electronegativity differences that make some parts of it polar and others not.
What the next chapter fixes
Trends across a whole table are useful and abstract. Chapter 9.6 takes six elements that matter most — hydrogen, carbon, nitrogen, oxygen, silicon and iron — and works through each one specifically: what its configuration is, what that makes it do, why it is used where it is used, and in carbon's case why one element accounts for more compounds than all the others combined.