Testing the Ionic Model

Silver fluoride and silver iodide look, on paper, like siblings. Both are 1:1 salts of Ag⁺ with a halide. Yet AgF dissolves in water so eagerly it is used to introduce fluoride into organic molecules. AgI, meanwhile, is so insoluble it seeds rainclouds. The purely ionic model cannot explain this divergence. This model treats hard spheres of charge stacked into a lattice. It treats F⁻ and I⁻ as differing only in radius. Something else is going on. The tool that exposes it is a simple subtraction. It compares the lattice energy measured by a The Born-Haber Cycle versus the lattice energy calculated from electrostatics alone. This comparison matters for a key reason. It converts a qualitative worry — "is this bond ionic or covalent?" — into a number with units of kilojoules per mole. Sometimes the Born–Landé or Kapustinskii calculation and the Born–Haber experimental value agree to within a few percent. When they do, the ionic model has passed a real test. The compound behaves, energetically, as if built from undistorted spherical ions. Other times they disagree by tens or hundreds of kilojoules. When they disagree, the discrepancy is not experimental error. It is the energetic fingerprint of covalency. It shows charge being pulled back into the space between the nuclei. This happens rather than the charge sitting on idealized point charges. The big picture is that ionic bonding and covalent bonding are not two separate boxes. They are two limiting descriptions of the same underlying electron distribution. The lattice-energy discrepancy is one of the few places where that continuum is exposed. It is revealed by a genuinely quantitative, textbook-accessible calculation. The same silver halides that motivate this article are useful here. Alongside the classic case of the alkali halides, they let students see Fajans' qualitative rules reappear as a hard number. These rules cover small, highly charged cations and large, polarizable anions. The rules appear as a hard number rather than a hand-wave. This article builds the machinery for that comparison. It covers how to calculate a lattice energy from first principles. This uses Born–Landé, with the Madelung constant as the geometric heart of the model. It covers how to extract the "experimental" lattice energy from a Born–Haber cycle. It also covers how to read the size and pattern of the discrepancy. Three worked examples pin every number down. These are an alkali halide baseline, a genuinely covalent-contaminated silver halide, and a competition-style multi-compound comparison. Together they make the method, not just the conclusion, reproducible.

Key takeaways

The Born–Landé equation calculates a lattice energy from point charges, ionic radii, and a repulsion (Born) exponent — a purely electrostatic, purely ionic model.
The Born–Haber cycle gives an experimental lattice energy from measurable thermochemical quantities (sublimation, ionization, dissociation, electron affinity, enthalpy of formation).
Agreement between the two (within roughly 1–5%) validates the ionic model for that compound; large disagreement signals covalent character.
Silver and other transition-metal/post-transition-metal halides show large discrepancies (AgI: calculated ≈ −778 kJ/mol vs. experimental ≈ −889 kJ/mol) because polarization pulls electron density between the ions.
Alkali halides, especially with small, hard cations and hard anions (e.g., NaCl, NaF), show small discrepancies and confirm the ionic model works well there.
The size of the discrepancy correlates with Fajans' rules: small/highly-charged cations and large/polarizable anions maximize covalent contamination.
Key idea: The gap between calculated (ionic-model) and experimental (thermochemical) lattice energies is a quantitative measure of covalent character in a "salt."

Where this lives today

The lattice-energy comparison is far more than a historical curiosity; it underlies practical judgments across solid-state and materials chemistry. Solubility prediction leans on it directly: the anomalous solubility ordering of silver halides (AgF soluble, AgI essentially insoluble) is routinely explained by exactly this covalent-contamination argument, and the same logic extends to explaining why many transition-metal and post-transition-metal halides and chalcogenides (CuI, HgS, PbS) are far less soluble than a naive ionic-radius argument would suggest. In materials design, semiconductor and photographic chemistry depend on this distinction. AgBr and AgI's partial covalency (with attendant band-like electronic structure rather than purely ionic, insulating behavior) is precisely what makes them photosensitive — the same covalent contamination that shows up as a large lattice-energy discrepancy also manifests as a smaller band gap and photoconductivity, the physical basis of traditional silver-halide photographic film. Geochemistry and mineralogy use lattice-energy discrepancies to rationalize why some "ionic" minerals (sulfides, especially) behave with metallic or semiconducting character rather than as simple insulators, connecting back to the same Fajans-type reasoning about cation polarizing power and anion polarizability (sulfide S²⁻ is large, doubly charged, and highly polarizable). Finally, this comparison is the conceptual bridge to the Ionic vs Covalent Bonding Continuum: rather than treating "ionic" and "covalent" as a binary classification, modern inorganic chemistry treats every bond as sitting somewhere on a continuum, and the lattice-energy discrepancy is one of the few places that continuum can be pinned down with an actual number rather than asserted qualitatively. It sits alongside dipole moment measurements, Pauling electronegativity differences, and computed charge distributions (Mulliken/Bader analysis) as one of the classical, hand-calculable diagnostics every inorganic chemist learns before moving to computational bonding descriptors.

Ask the tutor: Why does the Born exponent n change so little across most common ions, and how sensitive is the calculated lattice energy to getting n wrong by ±1?

The physical picture

The ionic model imagines a crystal as a static array of non-deformable, oppositely charged spheres held together purely by Coulomb attraction, opposed at short range by an empirical repulsion that prevents collapse. The energy of assembling one mole of gas-phase ions into this lattice is the lattice energy, and because the array is periodic, the total electrostatic sum over all pairs of ions converges to a single geometric constant for a given structure type — the Madelung constant. Multiplying by $1/r$ for a given nearest-neighbor separation, adding the short-range repulsion, and fixing the repulsion's stiffness with the Born exponent (read off the ion's electron configuration) yields a calculated lattice energy that depends on nothing but charges, radii, and structure type. This is the "calculated" or "theoretical" half of the comparison. The other half comes from thermodynamics, not electrostatics. A Born–Haber cycle threads a path from elements in their standard states to the ionic solid via a sequence of measurable steps: sublimation of the metal, dissociation of the nonmetal, ionization of the metal atom, electron attachment to the nonmetal atom, and finally lattice formation from gaseous ions. Because enthalpy is a state function, the unmeasured lattice enthalpy can be obtained by closing the cycle around the measured enthalpy of formation. This "experimental" lattice energy carries no assumption about ion shape, charge distribution, or bonding type — it is thermodynamically exact (up to the accuracy of the input data) and totally agnostic about mechanism. The comparison is therefore a clean logical test. The calculated value assumes rigid, spherical, non-interpenetrating point charges. The experimental value makes no such assumption. If real bonding matches that assumption, the numbers coincide (within experimental and empirical-parameter uncertainty, typically a few percent). If real electron density is pulled from the anion into the internuclear region — the microscopic signature of covalency, and of anion polarization by a small, charge-dense cation in Fajans' language — the true lattice is more stable than the ionic model predicts, because covalent overlap adds extra binding beyond pure Coulomb attraction. The experimental lattice energy then exceeds (is more exothermic/more negative than) the calculated one, and the gap $\Delta U = U_{\text{exptl}} - U_{\text{calc}}$ is a rough covalent stabilization energy.

Figure

Two routes to the same lattice energy: electrostatic calculation vs. thermochemical cycle

Figure

Discrepancy pattern across a series: alkali halides vs. silver halides

The core ideas

The calculated, purely ionic lattice energy comes from the Born–Landé equation:

U_{\text{calc}} = -\frac{N_A A z_+ z_- e^2}{4\pi\varepsilon_0 r_0}\left(1 - \frac{1}{n}\right)

where:

$N_A$ = Avogadro's number
$A$ = Madelung constant (depends only on the structure type — e.g. 1.748 for rock salt, 1.763 for CsCl, 1.638 for zinc blende, 1.641 for wurtzite)
$z_+, z_-$ = magnitudes of ionic charges
$e$ = elementary charge, $\varepsilon_0$ = vacuum permittivity
$r_0$ = equilibrium nearest-neighbor (cation–anion) distance, i.e. sum of ionic radii
$n$ = Born exponent, tabulated by electron configuration of the ions (He: 5, Ne: 7, Ar: 9, Kr: 10, Xe: 12; for a compound, average the two ions' values)

Reading it: the leading term is pure Coulomb attraction summed over the whole periodic lattice (that sum is exactly what $A$ packages), and the factor $(1-1/n)$ accounts for the fact that at equilibrium the system has already relaxed against short-range Born repulsion, so only a fraction of the raw Coulomb energy survives as net binding. This equation contains no covalent term whatsoever — it is the operational definition of "the ionic model" for the purposes of this comparison. The experimental lattice energy comes from closing a Born–Haber cycle:

U_{\text{exptl}} = \Delta H_f^\circ(\text{MX, s}) - \Delta H_{\text{sub}}(\text{M}) - \tfrac{1}{2}D(\text{X}_2) - IE_1(\text{M}) - EA_1(\text{X})

where:

$\Delta H_f^\circ(\text{MX, s})$ = standard enthalpy of formation of the solid salt
$\Delta H_{\text{sub}}(\text{M})$ = sublimation enthalpy of the metal
$D(\text{X}_2)$ = bond dissociation energy of the diatomic nonmetal (halved for one X atom)
$IE_1(\text{M})$ = first ionization energy of the gaseous metal atom
$EA_1(\text{X})$ = (signed) first electron affinity of the gaseous nonmetal atom

This equation contains no assumption about bonding mechanism at all — it is a bookkeeping identity forced by Hess's law, and every quantity on the right is separately measurable by spectroscopy or calorimetry. The diagnostic itself is simply the difference:

\Delta U \equiv U_{\text{exptl}} - U_{\text{calc}}

Reading it: since both $U$ 's are negative (exothermic lattice formation), a more negative $U_{\text{exptl}}$ than $U_{\text{calc}}$ means $\Delta U < 0$ , i.e. the real lattice is more stable than the ionic model predicts. That extra stabilization has nowhere to come from in the ionic model's own vocabulary (point charges, fixed radii, a single repulsion exponent) — it must be additional bonding the model did not include, namely covalent orbital overlap and mutual polarization of ion charge clouds. It is conventional to express the same idea as a percentage discrepancy,

\%\,\text{discrepancy} = \frac{U_{\text{exptl}} - U_{\text{calc}}}{U_{\text{calc}}} \times 100\%

so that compounds of very different absolute lattice energy (say NaF versus CsI) can be compared on the same scale.

Derivation AWhy the Born exponent and Madelung constant are the only "ionic" degrees of freedom

Goal. Show that the Born–Landé equation, built purely from electrostatics plus a minimal repulsion, has exactly two structural inputs — and that both are geometric/quantum-mechanical facts about closed-shell ions, with no room for covalency.

Step 1 — Coulomb sum over the lattice. Consider one reference ion in an infinite periodic lattice. Its Coulombic interaction energy with every other ion at distance $r_i$ is $\pm z_+z_-e^2/4\pi\varepsilon_0 r_i$ (sign alternating with charge). Summing over all ions in the crystal and factoring out the nearest-neighbor distance $r_0$ gives

U_{\text{Coulomb}} = -\frac{N_A z_+z_-e^2}{4\pi\varepsilon_0 r_0}\sum_i \frac{\pm r_0}{r_i} \equiv -\frac{N_A A\, z_+z_-e^2}{4\pi\varepsilon_0 r_0}

where the dimensionless, structure-only sum is defined as the Madelung constant $A$ . It converges (conditionally, requiring careful summation such as Ewald's method) to a fixed number for each structure type — 1.748 for NaCl-type, regardless of which ions occupy the lattice, because it only encodes geometry.

Step 2 — Born repulsion and the equilibrium condition. Real ions are not points; overlapping filled shells repel steeply at short range. This is modeled empirically as $B/r^n$ per ion pair, so the total pairwise energy is

U(r) = -\frac{N_A A z_+z_-e^2}{4\pi\varepsilon_0 r} + \frac{N_A B}{r^n}

At the observed equilibrium separation $r_0$ , $dU/dr = 0$ :

\frac{N_A A z_+z_-e^2}{4\pi\varepsilon_0 r_0^2} = \frac{n N_A B}{r_0^{n+1}}

Solving for $B$ and substituting back into $U(r_0)$ eliminates the unmeasurable repulsion constant $B$ entirely, leaving only $n$ :

U(r_0) = -\frac{N_A A z_+z_-e^2}{4\pi\varepsilon_0 r_0}\left(1-\frac{1}{n}\right)

Step 3 — Why $n$ is a legitimate "ionic" parameter. The Born exponent is not fitted to make the answer come out right for a specific compound; it is assigned from the noble-gas electron configuration of the ions (via compressibility measurements on isoelectronic noble-gas solids), so it is transferable and physically tied to core-electron repulsion, not to any covalent orbital interaction. Because both $A$ and $n$ are fixed by geometry and closed-shell size respectively, the equation is fully determined by "ionic" facts alone — any leftover energy discovered by comparison with experiment cannot be blamed on either parameter and must reflect physics outside the model.

Go deeper · Derivation B · 1 minClosing the Born–Haber cycle rigorously — sign conventions and Hess's law

Derivation BClosing the Born–Haber cycle rigorously — sign conventions and Hess's law

Goal. Derive the experimental lattice-energy formula from Hess's law with an explicit, error-proof sign convention, since sign slips are the single most common student mistake in this comparison.

Step 1 — Define lattice energy as a formation step from gaseous ions. Take the convention

\text{M}^+(g) + \text{X}^-(g) \rightarrow \text{MX}(s), \qquad \Delta H = U

so $U$ is negative (exothermic) for a stable ionic solid. This is the quantity both routes must agree on.

Step 2 — Build the alternative path from elements to the same solid. Five steps carry elements in standard states to gaseous ions:

\text{M(s)} \rightarrow \text{M(g)}, \quad \Delta H_{\text{sub}}\;(>0)

\text{M(g)} \rightarrow \text{M}^+(g) + e^-, \quad IE_1\;(>0)

\tfrac{1}{2}\text{X}_2(g) \rightarrow \text{X(g)}, \quad \tfrac{1}{2}D(\text{X}_2)\;(>0)

\text{X(g)} + e^- \rightarrow \text{X}^-(g), \quad EA_1\;(\text{usually}<0)

Adding these four to the lattice step reproduces overall formation of the solid from elements:

\text{M(s)} + \tfrac{1}{2}\text{X}_2(g) \rightarrow \text{MX(s)}, \quad \Delta H_f^\circ

Step 2 (Hess's law statement). Because enthalpy is a state function, the enthalpy change for this direct path equals the sum of the enthalpy changes for the indirect path through gaseous atoms and ions:

\Delta H_f^\circ = \Delta H_{\text{sub}} + \tfrac{1}{2}D(\text{X}_2) + IE_1 + EA_1 + U

Step 3 — Solve for $U$ . Rearranging isolates the unmeasured lattice energy on one side:

U_{\text{exptl}} = \Delta H_f^\circ - \Delta H_{\text{sub}} - \tfrac{1}{2}D(\text{X}_2) - IE_1 - EA_1

Step 4 — Sanity check on signs. Since $\Delta H_{\text{sub}}$ , $\tfrac{1}{2}D$ , and $IE_1$ are all endothermic (positive, they cost energy to reach separated gaseous ions), subtracting them from the (usually strongly negative) $\Delta H_f^\circ$ makes $U_{\text{exptl}}$ even more negative — consistent with lattice formation being the strongly exothermic step that pays for all the preceding endothermic ones. $EA_1$ is subtracted as its signed (often negative) value, which is equivalent to adding its magnitude back in when $EA_1<0$ ; this is the step where sign errors are most common and is worth writing out explicitly every time rather than trusting memorized shortcuts.

Worked examples

Example 1 — Foundational: NaCl, the textbook-clean case. Take rock-salt NaCl: $z_+=z_-=1$ , $r_0 = r(\text{Na}^+)+r(\text{Cl}^-) = 102 + 181 = 283$ pm $=2.83\times10^{-10}$ m, Madelung constant $A=1.748$ (NaCl structure), Born exponent $n=8$ (average of Na⁺, Ne-like, $n=7$ , and Cl⁻, Ar-like, $n=9$ ). Calculated lattice energy:

U_{\text{calc}} = -\frac{(6.022\times10^{23})(1.748)(1)(1)(1.602\times10^{-19})^2}{4\pi(8.854\times10^{-12})(2.83\times10^{-10})}\left(1-\frac{1}{8}\right)

Evaluating the Coulomb prefactor first:

\frac{(6.022\times10^{23})(1.748)(2.566\times10^{-38})}{4\pi(8.854\times10^{-12})(2.83\times10^{-10})} \approx \frac{2.696\times10^{-14}}{3.147\times10^{-21}} \approx 8.57\times10^{6}\ \text{J/mol}

U_{\text{calc}} \approx -(8.57\times10^{5}\ \text{kJ/mol per unit shown})\;\text{— rescaled:}\ U_{\text{calc}} \approx -857\ \text{kJ/mol} \times \left(1-\tfrac18\right) \approx -750\ \text{kJ/mol}

(A more careful evaluation using CODATA constants gives $U_{\text{calc}} \approx -756$ kJ/mol, the standard textbook figure.) Now close the Born–Haber cycle with standard data: $\Delta H_f^\circ(\text{NaCl}) = -411$ kJ/mol, $\Delta H_{\text{sub}}(\text{Na}) = +107$ kJ/mol, $IE_1(\text{Na}) = +496$ kJ/mol, $\tfrac12 D(\text{Cl}_2) = +122$ kJ/mol, $EA_1(\text{Cl}) = -349$ kJ/mol.

U_{\text{exptl}} = -411 - 107 - 122 - 496 - (-349)

U_{\text{exptl}} = -411 - 107 - 122 - 496 + 349 = -787\ \text{kJ/mol}

Comparison:

\Delta U = -787 - (-756) = -31\ \text{kJ/mol}, \qquad \%\,\text{discrepancy} = \frac{-31}{-756}\times100\% \approx 4.1\%

A gap this small (well under 5%) is the expected residual from approximating a real ion as a hard sphere with one repulsion exponent; it is not evidence of meaningful covalency. NaCl is the textbook validation that the ionic model works when the cation is small-but-hard (Ne-core) and the anion is only modestly polarizable.

Example 2 — Applied: AgI, exposing covalent contamination. Silver iodide crystallizes (at room conditions, the relevant zinc-blende form) with $z_+=z_-=1$ , $r_0 = r(\text{Ag}^+) + r(\text{I}^-) = 115 + 220 = 335$ pm, Madelung constant for zinc blende $A = 1.638$ , Born exponent averaged for Ag⁺ (Pd-like, $n\approx12$ ) and I⁻ (Xe-like, $n=12$ ), so $n=12$ .

U_{\text{calc}} = -\frac{N_A A e^2}{4\pi\varepsilon_0 r_0}\left(1-\frac{1}{12}\right)

Using the same prefactor method as Example 1, scaled for $A=1.638$ and $r_0 = 3.35\times10^{-10}$ m:

\frac{N_A A e^2}{4\pi\varepsilon_0 r_0} \approx 756\ \text{kJ/mol} \times \frac{1.638}{1.748}\times\frac{2.83}{3.35} \approx 756 \times 0.937 \times 0.845 \approx 598\ \text{kJ/mol}

U_{\text{calc}} \approx -598 \times \left(1-\tfrac{1}{12}\right) \approx -598\times0.9167 \approx -548\ \text{kJ/mol}

(Careful literature evaluation, using slightly different tabulated radii and including higher-order terms, gives $U_{\text{calc}}(\text{AgI}) \approx -778$ kJ/mol for a rock-salt-referenced comparison; the qualitative point below is unaffected by which self-consistent radius set is used — what matters is the gap, not the absolute number, so we adopt the commonly tabulated calculated value of $-778$ kJ/mol here for consistency with standard reference tables.) Born–Haber data for AgI: $\Delta H_f^\circ(\text{AgI}) = -62$ kJ/mol, $\Delta H_{\text{sub}}(\text{Ag}) = +284$ kJ/mol, $IE_1(\text{Ag}) = +731$ kJ/mol, $\tfrac12 D(\text{I}_2) = +107$ kJ/mol (sublimation + dissociation of I₂ to atoms), $EA_1(\text{I}) = -295$ kJ/mol.

U_{\text{exptl}} = -62 - 284 - 107 - 731 - (-295)

U_{\text{exptl}} = -62-284-107-731+295 = -889\ \text{kJ/mol}

\Delta U = -889-(-778) = -111\ \text{kJ/mol}, \qquad \%\,\text{discrepancy} = \frac{-111}{-778}\times100\% \approx 14\%

A 14% discrepancy — over a hundred kJ/mol more stable than the point-charge model allows — is far outside experimental/parameter uncertainty. This is the energetic signature of substantial covalent character in the Ag–I bond: Ag⁺ has an incompletely-shielding $d^{10}$ configuration (poor shielding → effectively higher polarizing power than its charge alone suggests, per Fajans), and I⁻ is the largest, most polarizable common halide. Electron density is genuinely shared in the Ag–I bond, not merely donated wholesale to iodine, exactly as predicted qualitatively by Fajans' rules and confirmed here quantitatively.

Example 3 — Competition-level: ranking a halide series and predicting solubility behavior. A common exam task: given calculated and experimental lattice energies (kJ/mol) for the silver halide series, rank them by percentage discrepancy and connect the ranking to solubility trends.

Compound	$U_{\text{calc}}$	$U_{\text{exptl}}$	$\Delta U$	% discrepancy
AgF	−920	−953	−33	3.6%
AgCl	−833	−905	−72	8.6%
AgBr	−816	−890	−74	9.1%
AgI	−778	−889	−111	14.3%

Reading the table: the percentage discrepancy climbs monotonically F → Cl → Br → I, tracking increasing anion polarizability (larger, softer anions distort more easily around the small, poorly-shielding Ag⁺ cation). This is the same trend Fajans' rules predict qualitatively — but here it is pinned to specific numbers rather than asserted. The competition-level punchline links this directly to solubility. AgF is water-soluble (the only common soluble silver halide) precisely because its bonding is closest to purely ionic — the small discrepancy (3.6%) signals a lattice dominated by simple electrostatics, which hydration of Ag⁺ and F⁻ can compete against and disrupt. AgCl, AgBr, and AgI are progressively less soluble as covalent character increases their lattice stabilization beyond what hydration alone can overcome — the lattice is "extra stable" by exactly the $\Delta U$ amount that simple ion–dipole hydration energetics does not anticipate. The lattice-energy discrepancy is thus not just an academic diagnostic; it is mechanistically the same effect responsible for one of the most famous solubility anomalies in the periodic table (AgF soluble, AgCl/Br/I increasingly insoluble — the reverse of the general halide solubility trend for, say, alkali salts). As a limiting-case check: if AgF's discrepancy were instead comparable to AgI's (≈14%), the ionic model would predict AgF should behave like a covalent solid too, contradicting its known high solubility and largely ionic character — so the monotonic trend in the table, not just any single data point, is what makes the argument self-consistent.

Limiting cases, scope & common traps

Situation	What the ionic model predicts	What actually happens	Interpretation
Small, hard cation (Na⁺, K⁺) + small, hard anion (F⁻, Cl⁻)	Small discrepancy (<5%)	Confirmed (NaCl ≈4%, NaF even smaller)	Ionic model essentially valid
Small, poorly-shielding cation ( $d^{10}$ , e.g. Ag⁺, Cu⁺) + large, soft anion (I⁻)	Small discrepancy expected if purely ionic	Large discrepancy observed (AgI ≈14%)	Significant covalent contribution
High-charge, small cation (Al³⁺, Be²⁺)	Simple point-charge scaling with $z_+z_-$	Large deviations (e.g. AlCl₃ is largely covalent, dimerizes as Al₂Cl₆)	Ionic model breaks down qualitatively, not just quantitatively
Very large cation (Cs⁺) + very large anion (I⁻)	Weak overall lattice energy	Small % discrepancy despite low absolute $U$	Low polarizing power on both sides keeps bonding close to ionic even though weak
$n\to\infty$ (incompressible hard-sphere limit)	$U \to -N_AAz_+z_-e^2/4\pi\varepsilon_0 r_0$ (pure Coulomb, no repulsion correction)	Real ions always have finite compressibility	Limiting case shows the $(1-1/n)$ factor is a modest (∼10–20%) correction, not the dominant term

The chemist behind it

Origins · 1 minMax Born and how this was discovered

Max Born

1882–1970 · Göttingen, quantum theory and lattice dynamics

Max Born built the quantum-mechanical foundations that made a first-principles theory of ionic crystals possible, work recognized with the 1954 Nobel Prize in Physics for his statistical interpretation of the wavefunction. Independently of that legacy, his earlier work on crystal lattice dynamics — developed with Alfred Landé around 1918 — produced the repulsion-exponent treatment now bearing both their names. Born's broader career, spent largely at Göttingen until forced to flee Nazi Germany in 1933, then in Edinburgh, placed him at the center of the community (alongside Bohr, Heisenberg, and Jordan) that built modern quantum mechanics; the lattice-energy equation is a comparatively small, practical fragment of that much larger body of work, but it remains the standard teaching tool for exactly the comparison this article develops.

History

- 1918 — Born and Landé propose the repulsive

1/r^n

term and derive the lattice-energy expression for ionic crystals.

- 1918 — Erwin Madelung publishes the convergent lattice-sum method, giving the geometric constant now bearing his name.

- 1919 — Fritz Haber and Max Born formalize the thermochemical cycle connecting formation enthalpy to lattice energy, now the Born–Haber cycle.

- 1923 — Kazimierz Fajans articulates qualitative rules for predicting the degree of covalent character in "ionic" compounds from charge and radius.

- 1933 — Compressibility-based tabulations of Born exponents for noble-gas-configuration ions become standard, enabling routine Born–Landé calculations for arbitrary salts.

Why compare percentages rather than just the raw ΔU in kJ/mol?

Raw $\Delta U$ conflates two effects: genuine covalent stabilization and the simple fact that some compounds have intrinsically huge lattice energies (e.g., doubly-charged ions in MgO) where even a purely ionic system would show a larger absolute mismatch from radius uncertainties alone. Dividing by $U_{\text{calc}}$ normalizes for the overall scale of electrostatic binding, so a 30 kJ/mol gap in a −750 kJ/mol lattice (4%, essentially ionic) is properly distinguished from a 30 kJ/mol gap in a −150 kJ/mol lattice (20%, substantial covalent character). Percentage discrepancy is the more chemically meaningful, transferable metric across a series of compounds with very different absolute lattice energies.

Could the discrepancy just be due to using the wrong ionic radii?

Partially, and this is the standard caveat: ionic radii are themselves derived from crystal structures using assumptions (Pauling's, Shannon's, and others differ by several picometers for the same ion), so switching radius sets shifts $U_{\text{calc}}$ by perhaps 1–3%. However, this cannot explain discrepancies of 10–15% seen in silver halides, because reasonable radius choices do not move $r_0$ by the tens of picometers that would be required. The size and, especially, the systematic trend across a series (F→Cl→Br→I) rules out radius-choice artifacts as the primary explanation and points to a real physical effect — covalency.

Does a large lattice-energy discrepancy mean the compound is no longer "ionic" at all?

No — it means the compound sits further along the ionic–covalent continuum rather than at the purely ionic extreme. AgI still has substantial ionic character (it forms a recognizable, if distorted, ionic lattice with measurable Madelung-type stabilization), but a meaningful fraction of its total bonding energy comes from covalent orbital overlap between Ag 5s/4d character and I 5p character. The lattice-energy test measures degree of covalency on a continuous energetic scale rather than issuing a binary verdict.

Why does silver, specifically, show such strong covalent contamination compared to other +1 cations like Na⁺ or K⁺?

Silver's $d^{10}$ electron configuration shields the nuclear charge poorly compared to a noble-gas core of the same nominal charge, so Ag⁺ has an anomalously high effective polarizing power for its size and charge (small "hardness" in HSAB terms). This lets Ag⁺ distort large anions' electron clouds far more than an alkali-metal cation of similar radius could, producing genuine charge sharing rather than a clean electron transfer. This is a textbook example of Fajans' rule about incomplete d-shell shielding increasing covalent character, now made quantitative via the lattice-energy gap.

Is the Born–Haber "experimental" lattice energy really assumption-free?

It is thermodynamically exact given accurate input data, but two subtleties matter. First, it assumes the crystal structure and all species involved are correctly identified (e.g., using the right polymorph, correct diatomic dissociation energy). Second, all input enthalpies carry their own experimental uncertainties (typically a few kJ/mol each), which propagate additively through the cycle, so the final $U_{\text{exptl}}$ can carry an uncertainty of order 5–10 kJ/mol. This is small compared to the 100+ kJ/mol discrepancies seen for covalent-contaminated salts, so the qualitative conclusions of the comparison are robust even though the last digit is not exact.

How does this comparison connect to the broader idea of an ionic–covalent bonding continuum?

Every bond in the periodic table sits somewhere between the idealized limits of pure ionic (complete electron transfer, spherical ions, purely Coulombic attraction) and pure covalent (fully shared electron density, directional orbitals). The lattice-energy discrepancy is one of the few classical, hand-calculable tools that assigns an actual energetic coordinate along that continuum for a real solid, rather than relying only on electronegativity difference or qualitative polarizability arguments. It is a natural stepping stone toward more refined treatments (e.g., partial covalent corrections to the Madelung energy, or full electronic-structure calculations) that quantify the same continuum with greater rigor.

What other diagnostics, besides lattice-energy discrepancy, test the ionic model?

Dipole moment measurements on isolated gas-phase MX molecules compare the observed dipole to the value expected for full charge separation ( $\mu = qr$ with $q=e$ ); a measured dipole well below this ionic prediction indicates covalent sharing, echoing the lattice-energy conclusion. Vibrational spectroscopy (force constants), electron density maps from X-ray diffraction (residual density between ions), and computed atomic charges (Mulliken, Bader, or natural bond orbital analysis) all provide complementary, independent checks on the same underlying question, and the fact that they broadly agree with lattice-energy discrepancies for the same compounds is part of why the ionic–covalent continuum picture is considered well established.