Lattice Enthalpy: Definitions and Conventions
Drop a crystal of sodium chloride into water and it dissolves. There is barely perceptible cooling of the solution. Yet tearing that same crystal apart into isolated gaseous Na⁺ and Cl⁻ ions takes about 787 kJ per mole of NaCl. That is roughly the energy released by burning 20 grams of propane. That enormous number, the lattice enthalpy, is the single most important energetic quantity in The Ionic Model of solids. It is also one of the most persistent sources of sign confusion in inorganic chemistry. Two chemically opposite processes share the same name in casual speech. Lattice enthalpy matters because it is the energetic price of assembling an ionic crystal. It is also the energetic reward of dismantling one. It feeds directly into every question a chemist asks about ionic solids. Why is MgO's melting point (2852 °C) so much higher than NaCl's (801 °C)? Why does LiF form at all, given that fluorine's electron affinity barely repays the energy needed to ionize lithium? Why are some hypothetical compounds, like "NeCl" or "MgCl", never observed? All of these questions are answered by comparing lattice enthalpies. Comparing them correctly requires getting the sign and the direction of the defining process exactly right, every time. The trouble is that textbooks and examiners use two conventions. These are numerically identical but algebraically opposite. The lattice formation enthalpy describes ions coming together; it is always exothermic, with ΔH negative. The lattice dissociation enthalpy describes a solid falling apart into gaseous ions; it is always endothermic, with ΔH positive. This is sometimes just called "the lattice energy" as a positive magnitude. Mixing these up is the single most common arithmetic error students make when running a Born-Haber Cycle. A sign flip in one step silently flips the final answer. This article builds the two conventions from the ground up. It uses Coulomb's law and the physical picture of an Ionic Solid. It derives the Born–Landé equation as the quantitative backbone of lattice enthalpy. Then it works through fully numerical examples. These include a careful side-by-side comparison of formation and dissociation values for the same compound. This way, the sign convention becomes a habit rather than a guess. It closes by setting up the machinery (Hess's law cycles) that the Born–Haber treatment will use. This machinery measures lattice enthalpies that cannot be measured directly.
- Lattice enthalpy describes the energy change when a solid ionic lattice forms from, or breaks into, its gaseous ions.
- Lattice formation enthalpy ΔH~lattice,f~: gaseous ions → solid lattice; always exothermic (negative), because oppositely charged ions attract.
- Lattice dissociation enthalpy ΔH~lattice,diss~: solid lattice → gaseous ions; always endothermic (positive), the exact reverse process.
- The two values have the same magnitude, opposite sign: ΔH~lattice,diss~ = −ΔH~lattice,f~.
- "Lattice energy" (U₀ or U) is usually quoted as a positive number equal in magnitude to the dissociation enthalpy, following the Born–Landé/Born–Mayer convention from lattice statics.
- Lattice enthalpy cannot be measured directly — it is obtained indirectly via a Born-Haber Cycle built from Hess's law.
- Key idea: lattice enthalpy is defined by which direction the arrow points between "gaseous ions" and "solid lattice" — get the direction right and the sign takes care of itself.
Where this lives today
Correctly signed lattice enthalpies are not a historical curiosity confined to worked exam problems; they underpin quantitative reasoning across modern inorganic and materials chemistry. In battery materials research, the lattice enthalpy of candidate cathode and solid-electrolyte compounds (lithium and sodium oxides, phosphates, sulfides) is used, via Born–Haber-type cycles, to predict ion mobility and thermodynamic stability — a smaller-magnitude lattice dissociation enthalpy generally signals a lattice that more easily releases mobile ions, a key design criterion for fast ion conductors. In pharmaceutical and agrochemical formulation, lattice enthalpy is compared against hydration enthalpy to rationalize and predict aqueous solubility of ionic active-pharmaceutical-ingredient salts, guiding which counter-ion (chloride, mesylate, citrate) is chosen for a drug's marketed salt form. In geochemistry and planetary science, lattice enthalpies of silicate and oxide minerals help explain phase stability under the extreme pressures found in planetary mantles, where the Born–Landé/Born–Mayer framework, extended with pressure–volume terms, predicts which mineral structures are thermodynamically favored at depth. In nuclear waste immobilization, the very large lattice enthalpies of actinide oxide and phosphate hosts are cited as part of the argument for their long-term chemical durability, since a deep formation-sense energy well corresponds to strong resistance against ion leaching. And in the undergraduate and graduate curriculum itself, correctly signed lattice enthalpies remain the entry point into the Born-Haber Cycle, which in turn is the standard tool used to back out otherwise unmeasurable quantities — electron affinities of unstable anions, or the thermodynamic feasibility of hypothetical compounds like "NeCl" or "MgCl" that are never observed because their predicted lattice enthalpies cannot compensate for the unfavorable ionization or electron-affinity steps.
Ask the tutor: In a Born–Haber cycle for MgCl₂, why does the lattice enthalpy have to be so large and negative for the compound to form at all, given how much energy the second ionization of magnesium costs?
The physical picture — thorough prose
Picture a mole of gaseous sodium cations and a mole of gaseous chloride anions, each ion infinitely far from every other ion, moving freely through space with no electrostatic interaction between them. This is the reference state: isolated gaseous ions, by convention at rest (0 K, though in practice referenced to 298 K with the appropriate heat capacities folded in). Now let these ions come together and condense into the sodium chloride rock-salt lattice, where each Na⁺ is surrounded by six Cl⁻ at 2.82 Å and vice versa, in the endlessly repeating three-dimensional array familiar from The Ionic Model. As the ions approach each other from infinite separation, opposite charges attract and the potential energy of the system falls — energy is released to the surroundings. This is exothermic by the ordinary sign convention (system loses energy, ΔH < 0). The energy released per mole when this exact process happens — gaseous ions to solid lattice — is called the lattice formation enthalpy, ΔH~lattice,f~(NaCl) = −787 kJ mol⁻¹. The negative sign is not a bookkeeping nuisance; it is the direct statement that bringing opposite charges together lowers energy, exactly as Coulomb's law says it must. Now run the film backward. Start with solid NaCl and ask how much energy must be supplied to rip every ion away from its neighbors, separate all of them to infinite distance, and leave them as free gaseous ions. This is the lattice dissociation enthalpy, ΔH~lattice,diss~(NaCl) = +787 kJ mol⁻¹, and it must be positive: you are fighting the same electrostatic attraction that made formation exothermic, so the process absorbs energy. Because "formation" and "dissociation" here are literally the same physical process run in opposite time directions, and enthalpy is a state function, the two numbers are forced to be equal in magnitude and opposite in sign — this is just Hess's law applied to a single reversible step. The confusion in practice comes from three surface-level issues that all trace back to this same underlying physics. First, spoken language: chemists often say "the lattice energy of NaCl is 787 kJ mol⁻¹" without specifying direction, silently using the dissociation magnitude as a positive number because that is the convention inherited from solid-state physics (where "lattice energy," symbol U₀, is defined as the energy required to separate the static lattice into infinitely distant ions, hence positive). Second, textbook tables differ: some tabulate ΔH~lattice,f~ (all negative), others tabulate "lattice energy" U (all positive, dissociation-sense). Third — and this is where marks are lost in a Born-Haber Cycle — the arrow direction used for the lattice step in the cycle must match whichever convention is stated, or the final answer comes out with the wrong sign entirely.
The physical origin of the magnitude — why it is 787 kJ mol⁻¹ and not, say, 78.7 or 7870 — lies in the electrostatic attraction between point charges packed into a specific geometry, moderated by short-range repulsion once electron clouds start to overlap. This is exactly the balance captured by the Born-Landé Equation, which sums the Coulombic attraction over the entire infinite lattice (via the Madelung constant) and subtracts a short-range repulsive term that prevents the ions from collapsing into each other.
The core ideas — the governing equations/relationships
The formal, IUPAC-consistent definitions are best stated as two explicit thermochemical equations, each with its enthalpy sign fixed by the direction written:
where:
- M^n+^(g), X^n−^(g) — isolated gaseous ions, infinitely separated, taken as the reference (zero) state of electrostatic energy
- MX(s) — one mole of the ionic solid in its standard crystalline form
- ΔH~lattice,f~ — lattice formation enthalpy; by definition always negative for a stable ionic solid
where:
- the same species appear, but the arrow — and hence the sign of ΔH — is reversed
- ΔH~lattice,diss~ — lattice dissociation enthalpy; always positive, since separating attracting charges costs energy
The two are tied together by the state-function property of enthalpy applied to a process and its reverse:
where:
- this equality holds exactly, with no approximation, because reversing a chemical equation exactly reverses the sign of ΔH (Hess's law, first law of thermodynamics)
- both quantities refer to the same physical lattice, at the same temperature and pressure
Reading: whichever convention a source uses, the number never changes — only the arrow and the sign. If a table lists "lattice energy of MgO = 3795 kJ mol⁻¹" with no sign, it is (by the near-universal convention in inorganic textbooks) reporting the dissociation magnitude; if it lists "ΔH~lattice,f~(MgO) = −3795 kJ mol⁻¹" it is the same number with the formation sign made explicit. A number without a stated direction is not wrong, but it is incomplete — always check which arrow is implied before plugging it into a cycle. The magnitude itself, for an idealized ionic solid, is captured by the Born–Landé equation, which will also make explicit why lattice enthalpy scales so steeply with ionic charge:
where:
- U₀ — lattice energy at 0 K, in the formation sign convention as written here (negative for a stable lattice); many sources define U₀ as the positive dissociation magnitude and drop the leading minus sign
- N~A~ — Avogadro's constant, so the equation gives a molar energy
- M — the Madelung constant, a pure geometry factor summing all electrostatic interactions (attractive and repulsive) over the infinite lattice (M = 1.7476 for rock salt)
- z~+~, z~−~ — the (positive) magnitudes of the ionic charges
- e — the elementary charge
- ε₀ — the vacuum permittivity
- r₀ — the equilibrium nearest-neighbor (cation–anion) distance
- n — the Born exponent, describing the steepness of short-range electron-cloud repulsion (typically 5–12, from compressibility data)
Goal: show, from first principles, that assembling an ionic lattice from gaseous ions is necessarily exothermic (ΔH~lattice,f~ < 0), and derive the magnitude in terms of measurable structural quantities.
Step 1 — Coulombic energy of one ion pair. Two point charges z~+~e and −z~−~e separated by distance r have electrostatic potential energy
This is negative for any r > 0 when the charges are opposite, meaning the pair is lower in energy than when infinitely separated — the physical origin of "opposite charges attract, and bringing them together releases energy."
Step 2 — Sum over the whole lattice: the Madelung constant. A single ion in the crystal does not interact with just one neighbor; it interacts with every other ion in the infinite lattice — nearest opposite-charge neighbors (attractive, negative contribution), next-nearest same-charge neighbors (repulsive, positive contribution), and so on, alternating and decaying with distance. Because the lattice geometry is fixed once the structure type is known, this infinite alternating sum can be written as a single number, the Madelung constant M, times the nearest-neighbor Coulomb term:
M depends only on geometry (6 for a simple cubic-like sum truncated at nearest neighbors, but the full alternating series converges to 1.7476 for rock salt, 1.7627 for CsCl, 1.6381 for zinc blende) — it is always the same for a given structure type regardless of which ions occupy the sites.
Step 3 — Short-range repulsion prevents collapse. If Coulomb attraction were the only force, r would shrink to zero and energy would diverge to −∞, which is unphysical: at short range, filled electron shells on adjacent ions begin to overlap, and the Pauli exclusion principle forces a steep repulsive energy. Born and Landé modeled this empirically as a power law,
with B a positive constant and n (5–12) large enough that this term is negligible at large r but rises very steeply as r shrinks — exactly the steepness needed to balance attraction and produce a stable minimum.
Step 4 — Minimize total energy to find equilibrium r₀. The total lattice energy per mole of ion pairs is
At equilibrium, dV/dr = 0 at r = r₀ (the experimentally known nearest-neighbor distance, from X-ray diffraction). Differentiating and setting to zero,
solves for B in terms of the known quantities M, z, r₀, and n:
Step 5 — Substitute back to get the equilibrium energy. Plugging this B back into V(r₀) and simplifying (the repulsive term becomes exactly 1/n of the attractive term at equilibrium) gives
The leading minus sign is not inserted by convention — it falls straight out of Step 1's minus sign, which itself comes from z~+~ and z~−~ having opposite algebraic sign in Coulomb's law. This is the rigorous statement that lattice formation must be exothermic: as long as the solid is held together by net electrostatic attraction between oppositely charged ions (true for any stable ionic solid), U₀ (formation-sense) is forced negative by the mathematics, not merely asserted by convention.
Go deeper · Derivation B · 2 minHess's-law proof that ΔH(lattice,diss) = −ΔH(lattice,f), with the enthalpy correction to 298 K
Goal: show rigorously (not just "physically obviously") that reversing the direction of the lattice process exactly reverses the sign of its enthalpy change, and identify the small correction needed to convert the 0 K lattice energy U₀ into a 298 K lattice enthalpy.
Step 1 — State the two processes as one Hess cycle. Consider the cycle: start at MX(s), go to Na⁺(g) + Cl⁻(g) via path (i) direct dissociation, or go around via path (ii) reforming the solid then... — but more simply, enthalpy H is a state function, so for any two states A and B,
This is a direct consequence of H being path-independent: if it were not true, you could go A → B → A and extract or generate energy from nothing, violating the first law of thermodynamics (a perpetual motion machine of the first kind).
Step 2 — Apply to A = MX(s), B = gaseous ions. Identify A → B as the dissociation process and B → A as the formation process:
No approximation has entered; this equality is thermodynamically exact at any fixed temperature and pressure, and holds independently of whatever numerical value the Born–Landé equation predicts.
Step 3 — Reconcile U₀ (Born–Landé, 0 K, no PV or heat-capacity terms) with ΔH~lattice~ (298 K, includes those terms). The Born–Landé/Born–Mayer treatment computes a static lattice energy U₀ that implicitly assumes stationary ions at 0 K. Real lattice enthalpies are quoted at 298 K, so a small correction is added for the translational/rotational degrees of freedom the gaseous ions gain on separating, and for the pΔV term converting internal energy to enthalpy:
where Δn~gas~ is the change in moles of gas (for MX(s) → M⁺(g) + X⁻(g), Δn~gas~ = +2, one mole of solid producing two moles of gas), and the factor 2RT per mole of gas approximates the combined translational contribution (3/2 RT) plus the pΔV = RT term, summed and rounded for a simple ionic solid. At 298 K, 2RT ≈ 4.96 kJ mol⁻¹ per mole of gas produced, i.e. roughly +10 kJ mol⁻¹ total for a 1:1 salt — a small correction (~1% for NaCl) compared to the ~800 kJ mol⁻¹ Coulombic term, which is why many treatments quote U₀ and ΔH~lattice~ interchangeably to three significant figures.
Step 4 — Conclusion. The sign relationship ΔH~lattice,diss~ = −ΔH~lattice,f~ is exact and unconditional; the magnitude relationship to the Born–Landé U₀ is approximate only because of the 0 K → 298 K correction, which is a genuinely separate physical effect (thermal population of gas-phase translational states) and not a sign ambiguity at all.
Worked examples
Example 1 (foundational): stating both conventions for NaCl and checking consistency. The accepted lattice dissociation enthalpy of sodium chloride is ΔH~lattice,diss~(NaCl) = +787 kJ mol⁻¹, meaning
By the Hess's-law relationship derived above, the formation enthalpy for the same lattice is obtained simply by reversing the arrow and flipping the sign:
Sanity check by magnitude: using the Born–Landé equation with M = 1.7476 (rock salt), z~+~ = z~−~ = 1, r₀ = 2.82 × 10⁻¹⁰ m, n ≈ 8 (typical for Na⁺/Cl⁻, both noble-gas-like ions of similar size), and e²/(4πε₀) = 2.307 × 10⁻²⁸ J·m,
Evaluating the prefactor first: (6.022 × 10²³)(1.7476)(2.307 × 10⁻²⁸ J·m) = 2.428 × 10⁻⁴ J·m mol⁻¹. Dividing by r₀:
Multiplying by (1 − 1/8) = 0.875 gives U₀ ≈ −754 kJ mol⁻¹ — within about 4% of the experimental −787 kJ mol⁻¹ (the small gap is exactly the kind of residual that motivates refined treatments such as Born–Mayer/Kapustinskii, and reflects the imperfect point-charge, purely-ionic approximation for NaCl, which does have a modest covalent contribution). The sign, −787 vs +787, must never be confused: writing "NaCl has a lattice enthalpy of −787 kJ mol⁻¹" is correct only if the reader understands "formation" is meant; writing it as "+787 kJ mol⁻¹" is correct only under "dissociation" or bare "lattice energy" convention.
Example 2 (applied): comparing MgO and NaCl to explain a melting-point gap using the correct sign throughout. Magnesium oxide has ΔH~lattice,diss~(MgO) = +3795 kJ mol⁻¹, i.e. ΔH~lattice,f~(MgO) = −3795 kJ mol⁻¹, roughly five times the magnitude for NaCl. Using the Born–Landé scaling,
For MgO, z~+~z~−~ = (2)(2) = 4, versus (1)(1) = 1 for NaCl — already a factor of 4 before accounting for r₀. The Mg–O distance (2.10 Å) is slightly shorter than Na–Cl (2.82 Å), adding a further factor of 2.82/2.10 = 1.343 to the ratio, giving a predicted ratio of
very close to the observed ratio 3795/787 = 4.82 (the small overshoot in the crude scaling reflects the fact that the Madelung constant and Born exponent also differ slightly between the two rock-salt-structured compounds, and MgO carries a somewhat larger covalent contribution that the purely ionic model does not capture). This directly explains the melting-point trend: melting requires disrupting the lattice enough for ions to move past one another, an endothermic process closely tracking the lattice dissociation enthalpy, so a lattice that is ~5× harder to pull apart (MgO) needs a correspondingly higher temperature to melt (2852 °C vs 801 °C) — both consistent with the dissociation-sign number being large and positive, i.e. a deep, hard-to-escape energy well in the formation-sign picture.
Example 3 (exam/competition level): setting up a Born–Haber cycle correctly by direction, without yet solving it fully. Suppose a problem gives: ΔH~sub~(K, s→g) = +89 kJ mol⁻¹; first ionization energy of K = +419 kJ mol⁻¹; ½ D(Br₂, l→g plus bond dissociation) effectively bundled as ΔH~atomization~(Br, ½Br₂→Br(g)) = +112 kJ mol⁻¹; electron affinity of Br = −325 kJ mol⁻¹ (energy released when Br(g) + e⁻ → Br⁻(g), so the value is negative in the "energy absorbed" convention); and ΔH~f~°(KBr, s) = −394 kJ mol⁻¹. Find ΔH~lattice~(KBr), and state explicitly which convention (formation or dissociation) the answer corresponds to. Set up the cycle so every arrow's direction is tracked explicitly, since a lattice step is exactly where sign errors creep in. By Hess's law, the direct formation route
must equal the sum of the indirect route through gaseous atoms and ions:
where ΔH~lattice,f~ is written as the formation step, K⁺(g) + Br⁻(g) → KBr(s), matching the direction the rest of the cycle is building toward (assembling the solid from atoms). Substituting the given numbers,
Sum the four known terms on the right first:
so
This is the formation-sign lattice enthalpy: K⁺(g) + Br⁻(g) → KBr(s), ΔH = −689 kJ mol⁻¹, consistent with the accepted literature value (lattice dissociation enthalpy of KBr is quoted around +670–690 kJ mol⁻¹ depending on the source's Born exponent/thermodynamic dataset). If instead the arrow in the cycle had been drawn as the dissociation direction, KBr(s) → K⁺(g) + Br⁻(g), the same algebra would demand ΔH~lattice,diss~ = +689 kJ mol⁻¹ — numerically identical magnitude, opposite sign, and both are correct provided the direction quoted matches the direction actually used in the cycle. The exam-level trap is writing down "−689 kJ mol⁻¹" as "the lattice energy" without specifying which arrow it belongs to; a grader working the other convention would mark that answer as having the wrong sign even though the number itself is right.
Limiting cases, scope & common traps
| Situation | Correct statement | Common error | ||
|---|---|---|---|---|
| Reporting "the lattice energy" with no direction stated | Acceptable only if the audience knows the implicit convention (usually dissociation, positive) | Copying a positive dissociation value straight into a Born–Haber cycle written in the formation direction, without flipping the sign | ||
| Comparing two salts' lattice enthalpies to rank stability/melting point | Compare magnitudes of either convention consistently; larger magnitude = stronger lattice, higher melting point, lower solubility (as a rough trend) | Comparing a formation value (negative) to a dissociation value (positive) and concluding the wrong salt is "more exothermic" | ||
| Very small, highly charged ions (e.g. Al³⁺, O²⁻ in Al₂O�3) | Purely ionic (Born–Landé) model increasingly underestimates the true | ΔH~lattice~ | because of growing covalent/polarization contributions — compare to a Born–Haber experimental value to detect this | Treating the Born–Landé prediction as exact for highly polarizing/polarizable ion pairs |
| Ions with noble-gas configuration vs transition-metal ions | Born exponent n and Madelung constant M depend on structure type, not on which convention is used — convention (sign) never changes with the chemistry, only the magnitude does | Assuming a different sign convention applies to "harder" or "softer" ions — it never does; sign depends only on direction of the arrow | ||
| Converting U₀ (0 K, static) to ΔH~lattice~ (298 K) | Small endothermic correction (~+10 kJ mol⁻¹ for a 1:1 salt) added when going to the dissociation-sense enthalpy at 298 K, reflecting extra translational degrees of freedom in the gas | Treating U₀ and ΔH~lattice~(298 K) as if the correction could ever flip the sign — it cannot; it is a few percent effect at most |
The chemist behind it
Lattice-energy theory as used today is a joint construction, but the two names permanently attached to the defining equation are Max Born and Alfred Landé, working in Germany in the years immediately following the emergence of X-ray crystallography as a tool for determining real interatomic distances in ionic solids.
Origins · 1 minMax Born and how this was discovered
Born was a central architect of quantum mechanics — he gave the probabilistic interpretation of the wavefunction, for which he won the 1954 Nobel Prize in Physics — but earlier in his career, working alongside crystallographers who were newly able to measure ionic separations directly from X-ray diffraction data, he turned to the classical problem of what holds an ionic crystal together. With Alfred Landé he combined the long-known Coulomb attraction between ions with an empirical short-range repulsion term modeled as a steep inverse power law, producing the equation that now bears their names. The work sits at the intersection of classical electrostatics and the then-nascent quantum picture of closed-shell repulsion, and it remains the standard first approximation taught for ionic lattice energies nearly a century later.
Why does lattice enthalpy have two conventions instead of just one universally agreed sign?
The dual convention exists because two different communities historically approached the same quantity from opposite directions. Thermochemists, following the ordinary sign rule for enthalpy of reaction, naturally write the formation direction (ions combining) and get a negative number, matching every other exothermic formation enthalpy in a Born–Haber cycle. Solid-state physicists and crystallographers, working from lattice statics and asking "how much energy is stored in this lattice, i.e. how much would it take to pull it apart," naturally define lattice energy as the dissociation magnitude, a positive number by construction. Both are self-consistent and both remain in active use, so the only real requirement is checking which one a given source intends before using the number in a calculation.
Is lattice enthalpy something you can measure directly in a calorimeter?
No — this is precisely why the Born–Haber cycle exists. You cannot simply place a crystal in a calorimeter and vaporize it cleanly into isolated gaseous ions without other processes (melting, atomization, ionization) getting entangled, and gaseous ionic species are not a state you can easily prepare and hold in isolation at standard conditions. Instead, lattice enthalpy is obtained indirectly, using Hess's law to close a thermochemical cycle built from experimentally accessible quantities: enthalpies of sublimation, atomization, ionization energies, electron affinities, and the standard enthalpy of formation of the solid. The lattice enthalpy is then the one unknown step needed to balance the cycle.
Does a larger lattice enthalpy magnitude always mean a higher melting point?
As a rough trend within a structurally similar family (same lattice type, similar ion sizes), yes — a deeper formation-sense energy well (larger |ΔH~lattice,f~|) corresponds to a lattice that is harder to disrupt thermally, tracking with higher melting and boiling points, as seen comparing NaCl (787 kJ mol⁻¹, mp 801 °C) to MgO (3795 kJ mol⁻¹, mp 2852 °C). However, melting point also depends on entropy changes on melting and on structural details (coordination number changes, partial covalency), so the correlation is a useful heuristic rather than an exact law, and exceptions exist when comparing chemically very different lattice types.
Why is the lattice enthalpy of MgO roughly five times, not four times, that of NaCl if both ions carry twice the charge?
The naive z~+~z~−~ scaling predicts exactly a factor of 4 (2×2 versus 1×1), but the Born–Landé equation also contains 1/r₀, and the Mg–O internuclear distance (2.10 Å) is shorter than the Na–Cl distance (2.82 Å) because O²⁻ and Mg²⁺ are both smaller than their Group 1/17 counterparts once account is taken of the higher effective nuclear charge experienced by their valence electrons. The shorter distance adds an extra multiplicative factor of about 1.34, bringing the naive ratio up close to the experimentally observed ratio of roughly 4.8. Small residual differences beyond that come from slightly different Madelung constants, Born exponents, and a modest covalent contribution in MgO that the purely ionic model does not capture.
If I am only given "lattice energy = 2526 kJ mol⁻¹" for CaO with no sign, how do I know what to do with it in a Born–Haber cycle?
Treat it as the dissociation-sense magnitude (the near-universal default when no sign is shown) and decide which direction your cycle needs. If your cycle is built in the "formation" direction (gaseous ions combining to make the solid, which is how most Born–Haber cycles are drawn when calculating an unknown formation enthalpy or electron affinity), insert it as −2526 kJ mol⁻¹ for that step. If instead your cycle happens to be drawn with the lattice step as dissociation (solid breaking into gaseous ions), insert +2526 kJ mol⁻¹. The physically important habit is to write the chemical equation for the lattice step explicitly, with its arrow, before assigning a sign — never assign a sign to a bare number without first writing what process it represents.
Is the negative sign in the Born–Landé equation just a convention chosen for convenience?
No — it is a direct mathematical consequence of Coulomb's law for two oppositely charged point charges, not an arbitrary choice. Writing V(r) = −z~+~z~−~e²/(4πε₀r) for a cation of charge +z~+~e and anion of charge −z~−~e, the product of the two signed charges is negative, forcing the overall potential energy to be negative for any positive separation r. This means the mathematics itself guarantees that assembling a stable ionic lattice from infinitely separated gaseous ions releases energy; the negative sign in ΔH~lattice,f~ is therefore a derived physical result, not a labeling convention layered on top afterward.
Why does the article say the two conventions have "the same magnitude" — is that always exactly true, or just approximately true?
It is exactly true, with zero approximation, as a consequence of enthalpy being a state function. If A → B has enthalpy change ΔH, then B → A must have enthalpy change exactly −ΔH; this is not an empirical regularity but a restatement of the first law of thermodynamics (energy conservation), since any inequality would allow a cyclic process A → B → A to produce or consume net energy from nothing. The only place approximation enters lattice-enthalpy work is in predicting the magnitude itself from a model like Born–Landé versus the true experimental value obtained via a Born–Haber cycle — that comparison can differ by a few percent, but the formation/dissociation sign symmetry itself never has any uncertainty.