Theory — Lattice Energy via the Born–Haber Cycle
Lattice energy in this calculator is defined as the enthalpy change for
the process “gaseous ions → ionic solid”. With this
formation-from-ions convention,
\(\Delta H_{\text{latt}}\) is typically negative (exothermic).
Many handbooks report a positive magnitude for the reverse process
(crystal → separated gaseous ions); always check the sign convention.
1) The Born–Haber (Hess) relation used here
For a binary formula \( \mathrm{M_{a}X_{b}} \) we consider a sequence of steps that leads
from the elements in their standard states to the crystal. By Hess’s law the sum of
step enthalpies equals the overall change. We collect all steps except the lattice
step on the right-hand side:
\[
\Delta H_{\text{overall}}
\;=\;
\Delta H_{\text{sub}}
\;+\;
\Delta H_{\text{atom}}
\;+\;
\sum \mathrm{IE}
\;+\;
\sum \mathrm{EA}
\;+\;
\Delta H_{\text{latt}}
\]
- \(\Delta H_{\text{sub}}\) — sublimation of the metal:
\(\mathrm{M(s)\to M(g)}\). Multiply by \(a\) if the step data are given per mole of M.
- \(\Delta H_{\text{atom}}\) — atomization of the non-metal:
for a diatomic halogen,
\(\tfrac{1}{2}\,\mathrm{X_{2}(g)\to X(g)}\). Multiply by \(b\) atoms required in the formula unit.
- \(\sum\mathrm{IE}\) — sum of ionization energies to reach the metal’s charge.
For \(\mathrm{M^{z_{M}+}}\): \(\mathrm{IE}_{1}+\mathrm{IE}_{2}+\cdots+\mathrm{IE}_{z_{M}}\)
(apply the factor \(a\) for \(a\) metal atoms).
- \(\sum\mathrm{EA}\) — sum of electron affinities to reach the anion’s charge.
For \(\mathrm{X^{z_{X}-}}\): \(\mathrm{EA}_{1}+\mathrm{EA}_{2}+\cdots+\mathrm{EA}_{z_{X}}\)
(apply the factor \(b\) for \(b\) non-metal atoms).
Note: \(\mathrm{EA}_{1}\) is usually negative (exothermic), whereas \(\mathrm{EA}_{2}\) is often positive (endothermic).
- \(\Delta H_{\text{latt}}\) — lattice energy for
\[
a\,\mathrm{M^{z_{M}+}(g)} \;+\; b\,\mathrm{X^{z_{X}-}(g)}
\longrightarrow \mathrm{M_{a}X_{b}}(s)
\]
(our sign convention makes this term negative).
In many textbook examples, \(\Delta H_{\text{overall}}\) is the standard enthalpy for
“elements in their standard states \(\to\) crystal at 298 K”.
In this tool you may either provide \(\Delta H_{\text{overall}}\) to find
\(\Delta H_{\text{latt}}\), or provide \(\Delta H_{\text{latt}}\) to recover
\(\Delta H_{\text{overall}}\).
2) What the calculator expects
- Enter a simple binary formula (e.g.,
NaCl, MgCl2, CaO).
Choose the electron counts \(z_{M}\) and \(z_{X}\) so that the charge balance holds:
\[
a\,z_{M} \;=\; b\,z_{X}.
\]
The tool checks this balance and shows the lattice step explicitly.
- Provide the four step energies
\(\Delta H_{\text{sub}}, \Delta H_{\text{atom}}, \sum\mathrm{IE}, \sum\mathrm{EA}\)
as kJ·mol\(^{-1}\) per mole of the crystal.
If a tabulated value is per atom/ion, scale by the stoichiometric coefficients \(a\) or \(b\).
- Optionally provide \(\Delta H_{\text{overall}}\) (if you want the calculator to solve
for \(\Delta H_{\text{latt}}\)); otherwise provide \(\Delta H_{\text{latt}}\) and the tool
will give \(\Delta H_{\text{overall}}\).
3) Solving for the lattice energy
Rearrange the Hess relation:
\[
\Delta H_{\text{latt}}
\;=\;
\Delta H_{\text{overall}}
\;-\;
\big(\Delta H_{\text{sub}} + \Delta H_{\text{atom}} + \sum \mathrm{IE} + \sum \mathrm{EA}\big)
\]
4) Worked example (like NaCl)
Example data (kJ·mol\(^{-1}\)): \(\Delta H_{\text{sub}}=107\),
\(\Delta H_{\text{atom}}=122\),
\(\sum\mathrm{IE}=496\),
\(\sum\mathrm{EA}=-349\),
\(\Delta H_{\text{overall}}=-411\).
Insert in the rearranged expression:
\[
\Delta H_{\text{latt}}
=
-411 - (107 + 122 + 496 - 349)
= -787\ \mathrm{kJ\,mol^{-1}}.
\]
5) Common pitfalls
- Sign convention: we use the formation from gaseous ions convention
(negative values). If you compare with a source that quotes a positive lattice energy,
it likely refers to the reverse process (crystal → ions).
- Scaling: multiply atomic steps by stoichiometric coefficients.
For \(\mathrm{MgCl_{2}}\), use \(2\times\) the halogen atomization and \(2\times\) the
electron affinity; for the cation use \(\mathrm{IE}_{1}+\mathrm{IE}_{2}\).
- Electron affinity sums: \(\mathrm{EA}_{2}\) for species like \(\mathrm{O^{2-}}\)
is positive (endothermic) and must be included.
- Units: keep everything in kJ·mol\(^{-1}\). The calculator reports per mole of the crystal \(\mathrm{M_{a}X_{b}}\).
6) Workflow with the tool
- Enter the formula and click Compute. Confirm the displayed lattice step and charge balance.
- Fill \(\Delta H_{\text{sub}}\), \(\Delta H_{\text{atom}}\), \(\sum\mathrm{IE}\), \(\sum\mathrm{EA}\)
(already scaled to the formula’s stoichiometry).
- Choose what to solve: \(\Delta H_{\text{latt}}\) or \(\Delta H_{\text{overall}}\).
The chosen unknown is disabled.
- Click Calculate to see the numerical result and a step-by-step LaTeX derivation.
- Use Fill example (NaCl) to load textbook-like data and verify the method.
