Calculate lattice energy from a Born–Haber cycle
Use the formation enthalpy and the other cycle steps to solve for the lattice enthalpy, carrying every sign through the sum.
Section 23 · Chemical energetics
A Level; Cambridge International Chemistry 9701, 2028–2030 syllabus, Topic 23. Use the syllabus sign convention for lattice enthalpy (gaseous ions forming the ionic solid); state phases, signs, units and temperature. Written practice is self-checked
Before you begin
- Use AS enthalpy changes and Hess's law, formulae for ions, balanced equations, and temperature in kelvins.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What does Hess's law require when a formation route is replaced by a cycle?
What you will learn
- Use Hess's law to solve for a missing lattice enthalpy
- Add atomisation, ionisation and electron-affinity steps with signs
- Check the result against the exothermic lattice-formation direction
Write the Hess equation before substituting
For NaCl, ΔHf°(NaCl) = ΔHₐₜ(Na) + ½D(Cl₂) + IE₁(Na) + EA₁(Cl) + ΔHₗₐₜ. Keep the electron affinity negative when it releases energy, and use lattice formation from gaseous ions to solid.
If the lattice term is unknown, rearrange only after writing the full cycle. The resulting lattice-formation value should be negative. A positive answer may mean the lattice arrow was reversed or another term's sign was copied incorrectly.
Check stoichiometry and units
Each enthalpy value must refer to the amount in the balanced cycle. Coefficients multiply the step enthalpies; for example, two atoms require two atomisation steps and two electron-affinity steps.
Use kJ mol⁻¹ for all terms and report the result per mole of formula units formed. Check that the direct and indirect routes have the same net equation.
Pause and explain
If all known Born–Haber steps sum to +376 and ΔHf° is −411 kJ mol⁻¹, what is the lattice-formation term?
Worked example
For NaCl, ΔHf° = −411, ΔHₐₜ(Na) = +108, ½D(Cl₂) = +121, IE₁(Na) = +496 and EA₁(Cl) = −349 kJ mol⁻¹. Calculate ΔHₗₐₜ.
Show the worked solution
- Write −411 = 108 + 121 + 496 − 349 + ΔHₗₐₜ.
- The known terms sum to +376 kJ mol⁻¹.
- Rearrange and check that formation is exothermic.
Answer ΔHₗₐₜ = −411 − 376 = −787 kJ mol⁻¹ for Na⁺(g) + Cl⁻(g) → NaCl(s).
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A hypothetical MX cycle has ΔHf° = −390 kJ mol⁻¹ and the atomisation, ionisation and electron-affinity steps sum to +340 kJ mol⁻¹. Find the lattice-formation enthalpy.
Give me a hint
Set the known cycle total plus ΔHₗₐₜ equal to ΔHf°.
Compare my reasoning
- Write −390 = +340 + ΔHₗₐₜ.
- Subtract 340 from both sides.
- Check that the formation value is exothermic.
ΔHₗₐₜ = −730 kJ mol⁻¹.
Look for these in your work
- Hess equation set up
- Rearrangement correct
- Sign matches lattice formation
For a salt MX₂, two electron-affinity terms are each −300 kJ mol⁻¹. What is their combined contribution to the cycle?
Give me a hint
Multiply the per-mole step by the number of X atoms in one formula unit.
Compare my reasoning
- There are two X atoms per MX₂ formula unit.
- Include two first electron-affinity steps.
- Add the signed values.
The combined contribution is 2×(−300) = −600 kJ mol⁻¹.
Look for these in your work
- Stoichiometric count correct
- Both steps included
- Signed sum correct
A learner obtains +795 kJ mol⁻¹ for Na⁺(g) + Cl⁻(g) → NaCl(s). Give two checks before accepting the value.
Give me a hint
Compare the sign with the physical direction and substitute back into the Hess sum.
Compare my reasoning
- Lattice formation should release energy, so expect a negative sign.
- Check whether the lattice term was reversed.
- Substitute the result into all cycle terms and compare with ΔHf°.
Do not accept the positive value without checking it. Gas ions forming the solid should give an exothermic, negative lattice-formation enthalpy; verify the arrow and all signed terms by reproducing ΔHf°.
Look for these in your work
- Physical sign checked
- Reversed definition considered
- Cycle sum used for verification
Common mistakes
- Changing the sign of EA₁ without reversing its reaction
- Using positive lattice dissociation enthalpy for the formation arrow
- Forgetting stoichiometric coefficients
- Accepting a positive lattice-formation result without checking the cycle
What can you explain now?
What does Hess's law require when a formation route is replaced by a cycle?
Compare with the explanation
The signed enthalpy sum must equal the same overall formation change
Both routes connect identical initial and final states.
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Make it stick
How do you use the direct formation enthalpy and signed alternative steps to check a lattice-energy calculation?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.