Born–Haber_cycle - Wikiparency

# Born–Haber cycle The **Born–Haber cycle** is an application of [[Hess's_law|Hess's law]] that breaks the formation of an ionic solid from its elements into steps whose enthalpies can be measured, so that the one step that cannot be measured, the [[Lattice_energy|lattice energy]], can be found by difference.[^b054-cycle][^openstax-lattice] For a metal M and a halogen X the steps are the atomization of the metal, its ionization energies IE₁ to IEₙ, the atomization of n halogen atoms and their electron affinities, and finally the assembly of the gaseous ions into the crystal, and Hess's law says that their sum is the standard [[Enthalpy|enthalpy]] of formation of the [[Chemical_compound|compound]]: ΔHf = ΔHa(M) + Σ IE₁…IEₙ + n × [ΔHa(X) + EA(X)] + LE.[^b054-cycle] The cycle is named for Max Born and [[Fritz_Haber|Fritz Haber]], whose papers on the cohesion of ionic crystals and on the heat of formation of salts appeared in the same December 1919 issue of the *Verhandlungen der Deutschen Physikalischen Gesellschaft*, alongside a third by Kasimir Fajans.[^morris-short] In the microsim below the reader sets the number of chlorine atoms per [[Magnesium|magnesium]], n = 1, 2 or 3, and watches the staircase of enthalpy bars climb and fall to a running total that lands at −94, −643 and +3,949 kJ/mol; the equation the sim evaluates is the sum above, and what it answers is why the [[Chemical_formula|formula]] of magnesium chloride is MgCl₂.[^b054-mgcl] On the Chemistry flagship this article serves Part II — Modern principles › Matter, at the section *Compound* (row K10), directly after the [[Periodic_table|periodic table]], whose [[Ionization_energy|ionization-energy]] landscape supplies the cycle's largest bars, and before the [[Molecule|molecule]], where bonds are [[Covalent_bond|covalent]] and the cycle no longer applies. The cycle is the flagship's first piece of energy accounting: it takes the atom-scale numbers of the preceding rows, ionization energies and electron affinities, and turns them into a macroscopic fact, the formula of a [[Salt_(chemistry)|salt]]. ## Examples Lattice enthalpies cannot be measured directly, because no experiment assembles a [[Crystal_structure|crystal]] from a gas of separated ions; every published value comes either from a cycle like this one or from a calculation based on [[Coulomb's_law|Coulomb's law]].[^b054-cannot] The cycle works because enthalpy is a [[State_function|state function]]: the heat of a change depends only on its start and end, so the direct route from elements to crystal and the roundabout route through gaseous atoms and [[Ion|ions]] must give the same total, and any one step can be solved for when the rest are known. It is the same bookkeeping that [[Thermochemistry|thermochemistry]] applies to any reaction path, with the difference that here one leg of the path is imaginary. The lattice formation enthalpy, which is negative, and the lattice dissociation enthalpy, which is positive, have the same magnitude, and texts differ in which they call "the lattice energy": the bonding textbook used here works with the formation value, so that LE enters the sum as a negative number, while OpenStax defines the lattice energy as the energy needed to separate a mole of solid into gaseous ions and reports positive values.[^b054-sign][^openstax-lattice] The staircase in the sim uses the formation convention throughout, since a bar that goes down must mean heat given out. Lattice enthalpy grows with the charge on the ions and falls as the ions get larger,[^b054-trend] which is why doubling the cation charge, as the sim does between n = 1 and n = 2, more than doubles the lattice bar. The same two variables, charge and radius, are the ones the periodic-table landscape displays, so the cycle is where the trends of that article are cashed. #### The magnesium chloride staircase in the microsim The sim's control is n, the number of [[Chlorine|chlorine]] atoms per magnesium, with the default at 2. Each setting draws a bar per term of the equation, and the running total is read from the top of the last bar. With the book's data, atomization of magnesium costs 148 kJ/mol and its first ionization 738 kJ/mol; each chlorine costs 122 kJ/mol to free from Cl₂ and repays 349 kJ/mol when it takes an electron.[^b054-mgcl] For n = 1 the ions Mg⁺ and Cl⁻ form a lattice worth −753 kJ/mol, and the sum is 148 + 738 + 122 − 349 − 753 = −94 kJ/mol: MgCl is exothermic to form, but only just.[^b054-mgcl] For n = 2 the second ionization energy, 1,451 kJ/mol, is added, two chlorines are atomized and reduced, and the lattice of the doubly charged cation with two anions is worth −2,526 kJ/mol, more than three times the MgCl value; the sum is 148 + 738 + 1,451 + 244 − 698 − 2,526 = −643 kJ/mol.[^b054-mgcl] For n = 3 the third ionization energy is 7,733 kJ/mol, because the [[Electron|electron]] now comes out of the closed [[Neon|neon]] core, and even a lattice of −5,440 kJ/mol cannot repay it: 148 + 738 + 1,451 + 7,733 + 366 − 1,047 − 5,440 = +3,949 kJ/mol.[^b054-mgcl] On the screen the IE₃ bar dwarfs everything else in the staircase. That is the aha the sim is built for: the second ionization is repaid by a lattice more than three times stronger, and the third reaches into the core, where nothing repays it. The most stable formula is the one with the most negative total, and it is MgCl₂; the independently tabulated enthalpy of formation of MgCl₂(s), −641.3 kJ/mol, agrees with the cycle to within rounding.[^openstax-appg] The sim's numbers carry one caveat that the book states and the HUD repeats. The lattice enthalpies for MgCl and MgCl₃ are not measured quantities, since neither compound exists; the book describes them as values "found on the web" and unconfirmed, and they are best read as Coulomb estimates of what such lattices would be worth.[^b054-web] The conclusion does not depend on their exact size: no plausible lattice repays 7,733 kJ/mol, and even a −753 kJ/mol lattice for MgCl leaves it 549 kJ/mol less stable than MgCl₂. #### Sodium chloride as the check The book's own worked example runs the cycle the other way, solving for the lattice enthalpy of [[Sodium_chloride|sodium chloride]] from the [[Calorimetry|calorimetric]] heat of formation. With atomization of [[Sodium|sodium]] at 107 kJ/mol, its ionization at 496 kJ/mol, atomization of chlorine at 122 kJ/mol, electron affinity at −349 kJ/mol and ΔHf(NaCl) = −411 kJ/mol, the unknown is 107 + 496 + 122 − 349 + LE = −411, so LE = −787 kJ/mol.[^b054-nacl] The same book quotes 769 kJ/mol for the NaCl lattice dissociation enthalpy in an earlier chapter, and OpenStax gives 769 kJ as well; the difference is a disagreement between data sets, not an arithmetic slip, and the article reports it rather than reconciling it.[^b054-769][^openstax-lattice] The OpenStax text carries its own worked cycle for [[Caesium|caesium]] fluoride, 76.5 + 375.7 + 79.4 − 328.2 with ΔHf = −553.5 kJ/mol, giving a lattice energy of 756.9 kJ/mol in its positive convention.[^openstax-lattice] A See-also variant of the sim runs the NaCl data through the same staircase and lands on −787 kJ/mol. ### Formation of LiF The pair's first example is [[Lithium|lithium]] fluoride, and it can be assembled from tabulated data. The standard enthalpy of formation of gaseous lithium atoms, 159.3 kJ/mol, is the atomization step; the first ionization energy of lithium is 5.392 eV, or 520 kJ/mol; the enthalpy of formation of gaseous [[Fluorine|fluorine]] atoms, 79.4 kJ/mol, is half the F–F bond energy of 160 kJ/mol; the electron affinity of fluorine is −328.2 kJ/mol; and the enthalpy of formation of solid LiF is −616.0 kJ/mol.[^openstax-appg][^nist-srd111][^openstax-lattice] The four measured steps sum to 159.3 + 520 + 79.4 − 328.2 = 430.5 kJ/mol, all of it uphill except the electron affinity, and Hess's law then requires LE = −616.0 − 430.5 = −1,047 kJ/mol (derived). The lattice is worth more than the NaCl lattice by a third, because lithium and fluoride are the smallest ions in their groups and sit closest together, which is the radius trend of the cycle at work.[^b054-trend] Lithium fluoride is for that reason the textbook case of a purely [[Ionic_bonding|ionic bond]]: small, hard ions, a large lattice term, and a cycle that closes without any covalent correction. OpenStax lists 1,023 kJ/mol for LiF in a table of lattice energies, a value from a different data set or method; the gap of about two percent between the cycle and the table is typical of the spread between Born–Haber and calculated lattice energies, which the bonding textbook notes grows large for solids with partly covalent bonding such as [[Silver|silver]] chloride.[^openstax-lattice][^b054-agcl] That gap is itself a diagnostic: when a Born–Haber lattice energy exceeds the Coulomb calculation by more than a few percent, the crystal is held by something beyond electrostatics. ### Formation of NaBr Sodium bromide adds a step that the chloride and fluoride cycles hide. [[Bromine|Bromine]] is a liquid in its standard state, so producing gaseous bromine atoms means first [[Enthalpy_of_vaporization|vaporizing]] Br₂(l), 30.91 kJ/mol per mole of Br₂, and then breaking the Br–Br bond, 190 kJ/mol; the enthalpy of formation of Br(g), 111.88 kJ/mol, is the sum of half of each, to within the rounding of the bond-energy table.[^openstax-appg][^openstax-lattice] With sodium's atomization at 107.3 kJ/mol and its ionization energy at 5.139 eV, or 496 kJ/mol, the electron affinity of bromine at −324 kJ/mol and ΔHf(NaBr, s) = −361.1 kJ/mol, the measured steps sum to 107.3 + 496 + 111.88 − 324 = 391 kJ/mol, and the lattice enthalpy follows as LE = −361.1 − 391 = −752 kJ/mol (derived).[^openstax-appg][^nist-srd111][^openstax-ea] The number sits below the −787 kJ/mol of sodium chloride, as the larger bromide ion requires, and above the lithium fluoride value by nearly 300 kJ/mol. An [[Iodine|iodide]] cycle would add a sublimation step in the same place, because iodine is a solid; the atomization term always means "whatever it takes to reach one mole of gaseous atoms from the standard state". Read together the three cycles show what the sim shows with one compound: the lattice term is the only large negative number in the sum, its size is set by charge and radius, and the formula of a salt is whichever ion combination leaves the total lowest. ## See also - [[Lattice_energy]] - [[Chemical_compound]] - [[Hess's_law]] - [[Chemical_formula]] - [[Ionic_bonding]] - [[Ionization_energy]] - [[Periodic_table]] ## Notes Explanatory notes on sign conventions and on the 769 versus 787 kJ/mol values for sodium chloride are given in the text and its footnotes under References. ## References [^b054-cycle]: Blackstock, Lindsay; Brewer, Sharon; Cinel, Bruno (2022). *Chemical Bonding and Organic Chemistry*. Chapter 4, "Chemical Bonding I: Basic Concepts", pp. 224–225 (Hess's law form of the cycle, ΔHf = ΔHa(M) + Σ IE + n ΔHa(X) + n EA(X) + LE). Open Textbook Library record: https://open.umn.edu/opentextbooks/textbooks/chemical-bonding-and-organic-chemistry [^b054-sign]: Blackstock, Brewer and Cinel (2022), *Chemical Bonding and Organic Chemistry*, Chapter 4, p. 222 (lattice formation and lattice dissociation enthalpies have the same magnitude and opposite sign). [^b054-trend]: Blackstock, Brewer and Cinel (2022), *Chemical Bonding and Organic Chemistry*, Chapter 4, p. 223 (lattice enthalpy grows with ionic charge and falls with ionic radius). [^b054-cannot]: Blackstock, Brewer and Cinel (2022), *Chemical Bonding and Organic Chemistry*, Chapter 4, p. 224 (lattice enthalpies cannot be measured directly). [^b054-nacl]: Blackstock, Brewer and Cinel (2022), *Chemical Bonding and Organic Chemistry*, Chapter 4, pp. 224–225 (NaCl cycle: 107 + 496 + 122 − 349 + LE = −411, LE = −787 kJ/mol). [^b054-769]: Blackstock, Brewer and Cinel (2022), *Chemical Bonding and Organic Chemistry*, Chapter 4, p. 219 (769 kJ/mol quoted for NaCl lattice dissociation). [^b054-mgcl]: Blackstock, Brewer and Cinel (2022), *Chemical Bonding and Organic Chemistry*, Chapter 4, pp. 226–228 (MgCl, MgCl₂ and MgCl₃ cycles: −94, −643 and +3949 kJ/mol; Mg ionization energies 738, 1451 and 7733 kJ/mol; lattice values −753, −2526 and −5440 kJ/mol). [^b054-web]: Blackstock, Brewer and Cinel (2022), *Chemical Bonding and Organic Chemistry*, Chapter 4, p. 226 (the MgCl and MgCl₃ lattice values described as found on the web and unconfirmed). [^b054-agcl]: Blackstock, Brewer and Cinel (2022), *Chemical Bonding and Organic Chemistry*, Chapter 4, p. 226 (theoretical and Born–Haber lattice energies differ for partly covalent solids such as AgCl). [^openstax-lattice]: Flowers, Paul; Neth, Edward; Robinson, William, et al. (2019). *Chemistry: Atoms First 2e*. OpenStax. Chapter 9, §9.4 "Strengths of Ionic and Covalent Bonds" (lattice energy defined as the energy to separate one mole of solid into gaseous ions, positive by convention; NaCl 769 kJ; the CsF cycle 76.5 + 375.7 + 79.4 − 328.2, ΔHf −553.5, lattice energy 756.9 kJ/mol; LiF 1023 kJ/mol in the lattice-energy table; Table 9.3 bond energies F–F 160, Br–Br 190 kJ/mol). https://openstax.org/books/chemistry-atoms-first-2e/pages/9-4-strengths-of-ionic-and-covalent-bonds (PDF pp. 421–474, page to pin) [^openstax-appg]: Flowers et al. (2019). *Chemistry: Atoms First 2e*. OpenStax. Appendix G, "Standard Thermodynamic Properties for Selected Substances" (ΔHf°: Li(g) 159.3; LiF(s) −616.0; F(g) 79.4; Na(g) 107.3; NaBr(s) −361.1; NaCl(s) −411.2; Br(g) 111.88; Br₂(g) 30.91; Cl(g) 121.3; Mg(g) 147.1; MgCl₂(s) −641.3 kJ/mol). https://openstax.org/books/chemistry-atoms-first-2e/pages/g-standard-thermodynamic-properties-for-selected-substances (PDF pp. 1093–1110, page to pin) [^openstax-ea]: Flowers et al. (2019). *Chemistry: Atoms First 2e*. OpenStax. Chapter 3, §3.5 "Periodic Variations in Element Properties", electron-affinity figure (Br −324 kJ/mol). https://openstax.org/books/chemistry-atoms-first-2e/pages/3-5-periodic-variations-in-element-properties [^nist-srd111]: National Institute of Standards and Technology. *Ground Levels and Ionization Energies for the Neutral Atoms*, NIST Standard Reference Database 111 (Li 5.392 eV, Na 5.139 eV; kJ/mol values derived at 96.485 kJ/mol per eV). https://www.nist.gov/pml/ground-levels-and-ionization-energies-neutral-atoms [^morris-short]: Morris, D. F. C.; Short, E. L. (1969). "The Born–Fajans–Haber Correlation." *Nature*, 224: 950–952 (Born's and Haber's 1919 papers in *Verhandlungen der Deutschen Physikalischen Gesellschaft* 21, pp. 679 and 750, published December 5, 1919, with Fajans's in the same issue). https://doi.org/10.1038/224950a0 ## External links - [Strengths of Ionic and Covalent Bonds](https://openstax.org/books/chemistry-atoms-first-2e/pages/9-4-strengths-of-ionic-and-covalent-bonds), *Chemistry: Atoms First 2e*, OpenStax - [Standard Thermodynamic Properties for Selected Substances](https://openstax.org/books/chemistry-atoms-first-2e/pages/g-standard-thermodynamic-properties-for-selected-substances), *Chemistry: Atoms First 2e*, Appendix G - *Chemical Bonding and Organic Chemistry* (Blackstock, Brewer and Cinel, 2022), Chapter 4, on the [Open Textbook Library](https://open.umn.edu/opentextbooks/textbooks/chemical-bonding-and-organic-chemistry) — Portal Book 054  **Microsim — three.js (Wikitube framework), pending deploy:** *Born–Haber cycle* will play here once `https://wikitube-3d-microsims.netlify.app/matter/Born–Haber_cycle.html` is live.   ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Born–Haber_cycle) : [Wikitube](https://en.wikitube.io/wiki/Born–Haber_cycle) · pinned revision [1345106754](https://en.wikipedia.org/w/index.php?oldid=1345106754) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Chemistry]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Chemistry row K10 · sim pending (matter/Born–Haber_cycle).*