Born-Haber Cycle Formula
The Born-Haber cycle uses Hess's law to calculate the lattice energy of ionic compounds from thermochemical data.
Explains ionic crystal stability.
The Formula
The Born-Haber cycle applies Hess's law to ionic compounds. Since enthalpy is a state function, the lattice energy (U) can be calculated indirectly from quantities that are easier to measure directly: formation enthalpy, sublimation enthalpy, bond energy, ionization energy, and electron affinity. This cycle was developed by Max Born and Fritz Haber in 1919.
Variables
| Symbol | Meaning | Typical sign |
|---|---|---|
| ΔH°_f | Enthalpy of formation of the ionic compound from elements | − (exothermic for stable compounds) |
| ΔH_sub | Enthalpy of sublimation of the metal (solid → gas) | + (endothermic) |
| ½BE | Half the bond dissociation energy of the diatomic nonmetal | + (endothermic) |
| IE | Ionization energy of the metal (gas atom → cation) | + (endothermic) |
| EA | Electron affinity of the nonmetal (anion formation) | − (exothermic) |
| U | Lattice energy (ionic solid → gaseous ions) | − (exothermic for stable lattice) |
Example — Sodium Chloride Lattice Energy
Calculate the lattice energy of NaCl using the following data (all in kJ/mol):
ΔH°_f (NaCl) = −411, ΔH_sub (Na) = +108, ½BE (Cl&sub2;) = +122
IE (Na) = +496, EA (Cl) = −349
Sum of steps (Na° → Na&sup+; → NaCl path): 108 + 122 + 496 + (−349) + U = −411
377 + U = −411
U = −788 kJ/mol (experimental value: −787 kJ/mol — excellent agreement!)
When to Use It
Use the Born-Haber cycle when:
- Calculating lattice energies that cannot be measured directly
- Predicting which ionic compounds are stable (large negative lattice energy = stable)
- Understanding why ionic compounds have high melting points — large lattice energies require much energy to overcome
- Determining electron affinities from known lattice energies and other thermochemical data
- Comparing the stability of different ionic compounds (NaF vs NaCl vs NaBr — why does stability decrease?)
The lattice energy depends on ionic charges and ionic radii: higher charges and smaller ions give larger lattice energies. MgO has a much larger lattice energy (−3795 kJ/mol) than NaCl (−787 kJ/mol) because Mg²&sup+ and O²− carry double charges.