Electronegativity — Pauling, Mulliken, Allred-Rochow scales: Question bank

Definition Symbol legend — read this first

Every trap below reuses these symbols. Fix their meaning once here so no reveal forces you to guess:

$χ$ — electronegativity: the relative, dimensionless tendency of a bonded atom to pull the shared electron pair toward itself.
IE — ionization energy: the energy needed to remove one electron from an isolated atom (how strongly it keeps electrons).
EA — electron affinity: the energy released when an isolated atom gains one electron (how much it wants electrons).
$E (A - B)$ — the measured bond energy of the bond between atoms A and B (kJ/mol).
$Δ$ — the ionic resonance energy: $Δ = E (A - B) - E (A - A) E (B - B)$ , the "extra glue" beyond a purely covalent bond.
$Z_{eff}$ — effective nuclear charge: the leftover nuclear charge a valence electron feels after inner-electron screening (Slater's Rules).
$r$ (a.k.a. $r_{cov}$ ) — the covalent radius: the effective distance from the nucleus to a bonding valence electron, in ångström (Å).

The figure below fixes the two visual anchors we lean on repeatedly — how the $χ$ gap maps to ionic character/dipole, and where the arithmetic vs geometric mean sit relative to the true bond energy.

Figure — Electronegativity — Pauling, Mulliken, Allred-Rochow scales

True or false — justify

True or false: Electronegativity is a directly measurable property of an isolated atom.

False. Electronegativity describes an atom while it is bonded and sharing electrons; there is no lab instrument that reads it out, which is exactly why Pauling, Mulliken and Allred–Rochow each estimate it from things we can measure.

True or false: Fluorine has both the highest electronegativity and the highest electron affinity of all elements.

False. F has the highest

χ

(3.98), but chlorine's Electron Affinity (349 kJ/mol) exceeds fluorine's (328) because F's tiny size crowds the incoming electron — so

χ

and EA rank atoms differently.

True or false: The Pauling equation gives absolute electronegativity values.

False. It yields only

∣ χ_{A} - χ_{B} ∣

, a difference. Absolute table values exist only because

χ (H)

is fixed at 2.20 as an anchor and everything is built outward from it.

True or false: Down a group, electronegativity rises because the nuclear charge

Z

rises.

False. Although

Z

grows, the covalent radius

r

grows and screening increases, so $Z_{\text{eff}}$ over

r^{2}

falls;

χ

therefore decreases down a group.

True or false: Mulliken electronegativity, being

(IE + EA) /2

, is already on the Pauling scale.

False. Raw

χ_{M}

is an energy in eV. To compare with Pauling numbers you must rescale with

χ_{P} \approx 0.187 (IE + EA) + 0.17

True or false: Because

Δ = E (A - B) - E (A - A) E (B - B)

can be positive, it can also legitimately come out negative for a real bond.

Mostly false, with a caveat. Pauling chose the geometric mean so that AM ≥ GM keeps the covalent estimate below the true value and

Δ \geq 0

for normal polar bonds. A small negative

Δ

(say within experimental error, a few kJ/mol) is just data noise; a large, reproducible negative

Δ

is chemically meaningful — it flags a bond destabilised by lone-pair repulsion or poor orbital overlap (e.g. F–F, N–N), not an electronegativity signal.

True or false: A larger

∣ χ_{A} - χ_{B} ∣

means a bond with more ionic character.

True. A bigger electronegativity gap means one atom hogs the shared pair harder, shifting charge and increasing ionic character (and the Dipole Moment) — see the rising curve in the figure.

True or false: Two atoms with equal electronegativity form a bond with zero ionic resonance energy

Δ

True. If

χ_{A} = χ_{B}

there is no extra pull, so

E (A - B)

equals the geometric mean and

Δ = 0

— a purely covalent bond.

Spot the error

"To find the expected covalent bond energy I average

E (A - A)

and

E (B - B)

arithmetically." — where's the slip?

Pauling uses the geometric mean

E_{AA} E_{B B}

, not the arithmetic one, because AM ≥ GM ensures the excess

Δ

is never forced negative (the figure shows GM sitting below AM).

"Mulliken divides by 2 because there are two atoms in the bond." — what's wrong?

The 2 is the arithmetic mean of one atom's two competing tendencies — its reluctance to lose electrons (IE) and its eagerness to gain them (EA) — not a count of atoms. The arithmetic mean is right here because IE and EA are two energies on the same footing (both describe the same single atom's grip on one electron), so their unweighted average measures its overall pull; there is no multiplicative structure that would call for a geometric mean.

"Allred–Rochow uses the full nuclear charge

Z

Z / r^{2}

." — correct it.

It uses

Z_{eff}

from Slater's Rules, because inner electrons screen the nucleus and the valence electron only feels the leftover charge.

"Since Coulomb's law has

r

, Allred–Rochow puts

r

in the denominator to the first power." — fix it.

The force law is inverse-square,

F \propto Z_{eff} / r^{2}

, so the covalent radius is squared in

χ_{A R} = 0.359 Z_{eff} / r_{cov}^{2} + 0.744

χ

has units of eV in every scale." — spot the confusion.

Only raw Mulliken

χ_{M}

carries eV. Pauling and rescaled values are dimensionless scale numbers; electronegativity as a concept is unitless.

"I'll compute

χ

(Cl) directly from the Pauling formula without any reference value." — what breaks?

The formula gives

∣ χ_{Cl} - χ_{H} ∣

only; you must add the anchor

χ (H) = 2.20

(and know Cl pulls harder, so you add rather than subtract).

"The

0.102

factor in Pauling is a physical constant of nature." — reframe.

It's a unit-conversion / fitting constant (

1/ 96.5

for kJ/mol) chosen so the scale reproduces H = 2.20 and F = 3.98 — a calibration, not a law.

Why questions

Why do we need three electronegativity scales instead of one measured value?

Because

χ

is a property of a bonded atom with no direct instrument; each scale reconstructs it from measurable proxies — bond energies, IE+EA, or Coulomb force — and they nearly agree.

Why does Pauling postulate

Δ \propto (χ_{A} - χ_{B})^{2}

rather than the first power?

The square makes the derived

χ

values self-consistent and additive across different atom pairs; a linear form would give inconsistent gaps depending on which pair you compared.

Why does Mulliken's formula correctly predict F as most electronegative?

F has both a huge Ionization Energy (hates losing electrons) and a large EA (loves gaining them); averaging two large numbers gives a large

χ_{M}

Why does electronegativity increase across a period?

Moving right,

Z_{eff}

rises while the atomic radius shrinks, so the pull on shared electrons (

\propto Z_{eff} / r^{2}

) grows.

Why is the geometric mean, not arithmetic, the "right" covalent baseline?

Because AM ≥ GM guarantees the estimate sits below the true bond energy, so the ionic excess

Δ

— the signal Pauling wants — stays non-negative and interpretable.

Why can Allred–Rochow be computed for many elements even without measured bonds?

It needs only

Z_{eff}

(from Slater's Rules) and a covalent radius, both tabulated per atom, so it doesn't require a specific compound's bond energy the way Pauling does.

Edge cases

For a homonuclear molecule like

Cl_{2}

, what does the Pauling difference give?

χ_{A} = χ_{B}

, so

Δ = 0

and

∣ χ_{A} - χ_{B} ∣ = 0

— the formula correctly reports no electronegativity gap and zero ionic character.

If an atom had EA = 0 (no energy released on gaining an electron), what happens to its Mulliken

χ

It reduces to

χ_{M} = IE /2

; the atom's pull comes only from its reluctance to lose electrons, giving a smaller but still defined electronegativity.

What does a negative electron affinity (energy absorbed to add an electron) do to Mulliken

χ

It lowers

χ_{M} = (IE + EA) /2

; if EA is negative and large enough in magnitude to outweigh IE,

χ_{M}

can in principle go negative. Physically that would describe an atom that both resists gaining and is happy to shed electrons — an idealised extreme (real atoms have IE

≫ ∣ EA ∣

, so

χ_{M}

stays positive), but it correctly signals a species with essentially no pull on shared electrons.

Which scale would you trust least for a heavy transition metal, and why?

Pauling struggles when reliable homonuclear/heteronuclear bond energies are scarce or the bond is far from covalent — but Mulliken is not automatically safe either, since accurate EA (and sometimes IE) data for heavy metals are often poorly defined or missing. Allred–Rochow is frequently the most robust because it needs only tabulated

Z_{eff}

and covalent radius.

At the limiting case of a bond between identical electronegativities, what is the ionic character and Dipole Moment?

Both are essentially zero — no charge is displaced, so the bond is purely covalent with no net dipole (barring geometric asymmetry).

Cl's EA exceeds F's, yet F's

χ

exceeds Cl's — how do all three scales stay consistent?

Mulliken adds F's much larger IE to its EA, Pauling reads F's larger bond-energy excesses, and Allred–Rochow reads F's higher

Z_{eff} / r^{2}

— each captures the bonded pull, not the isolated-atom EA anomaly.

When is a negative

Δ

"noise" versus a real chemical message?

∣Δ∣

is within the bond-energy measurement uncertainty (a few kJ/mol), treat it as noise and take

Δ \approx 0

. If it is large and reproducible, it is meaningful — the bond is genuinely weaker than the covalent baseline (lone-pair repulsion or weak overlap, as in F–F), and Pauling's electronegativity picture simply doesn't apply to that bond.

Recall One-line summary of the traps

Never confuse $χ$ (bonded, relative, unitless) with EA (isolated, measured energy); Pauling gives differences needing an anchor and uses the geometric mean; Mulliken's 2 is an arithmetic mean of two tendencies of one atom; Allred–Rochow uses $Z_{eff} / r^{2}$ (inverse-square, screened charge, covalent radius). Down a group $χ$ falls.

Connections

Effective Nuclear Charge, Slater's Rules — the $Z_{eff}$ behind Allred–Rochow.
Ionization Energy, Electron Affinity — Mulliken's inputs and the EA trap.
Bond Energy, Ionic Character of Bonds — Pauling's basis and its consequence.
Periodic Trends — Atomic Radius, Dipole Moment, Fajans Rules, Metallic vs Non-metallic Character — the trends $χ$ drives.