SCIENCE · INVESTIGATE AND EXPLAIN

Infer a signed lattice term from a formation cycle

Use a reaction-defined energy cycle rather than a memorized sign for lattice energy.

Learning outcome: Capture ionization450,550,650 at formation−350; then hold ionization550 and vary formation. Retain the full cumulative cycle, not only the inferred last number.

Starting recap

Review the atomic electron ledger, signed energy changes and dimensional conversions. Distinguish a mathematical parameter from a unique material identification.

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Try

At ionization550 and formation−350 kJ/mol, gas-ion preparation is150+100+550−300=500; lattice formation is−850 kJ/mol.

Open the experiment

Predict

If ionization becomes more costly while overall formation stays fixed, must lattice formation become more or less exothermic?

Your prediction and reason

Experiment

Capture ionization450,550,650 at formation−350; then hold ionization550 and vary formation. Retain the full cumulative cycle, not only the inferred last number.

Close a fictional formation cycle

Worked teaching model · synthetic values · Fictional MX(s) cycle per mole formula units: M(s)→M(g) +150, half X2(g)→X(g) +100, M(g)→M+(g) selected positive ionization, X(g)+e→X−(g) −300 kJ/mol. Infer signed lattice formation enthalpy from gaseous ions; same reference conditions throughout. No actual synthesis.

Ionization input: 550 kJ/mol Formation enthalpy: -350 kJ/mol

Reset keeps your prediction, captures and practice history.

Axes rescale with settings. Compare numerical values, not screen slopes. Discrete stems show steps, contributors or partial sums; connected samples show the declared parameter relationship, not a measured time trajectory.

Numerical diagram coordinates

completed cycle step	cumulative enthalpy (kJ/mol)
1	150
2	250
3	800
4	500
5	-350

Calculated values under declared assumptions
Quantity	Value	Unit
Metal atomization	150	kJ/mol
Nonmetal atomization	100	kJ/mol
Ionization	550	kJ/mol
Electron attachment	-300	kJ/mol
Gaseous-ion preparation subtotal	500	kJ/mol
Lattice formation enthalpy	-850	kJ/mol
Lattice dissociation enthalpy	850	kJ/mol
Formation enthalpy	-350	kJ/mol

Specify the lattice reaction before assigning its sign.

The signed lattice formation term brings separated gaseous ions into the solid; its reverse is positive here. Electron attachment is supplied as a signed process enthalpy, avoiding conflicting electron-affinity sign conventions. All terms refer to one mole MX and the same thermodynamic reference states.

Capture contrasting evidence and retain the interpretation limits. These supplied models do not identify an actual material or certify practical laboratory skill.

Specify the lattice reaction before assigning its sign.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

State the endpoint reaction

For fictional MX(s), the formation reaction begins with M(s) and half a mole of X2(g) per mole of formula units. We construct an alternate route through gaseous atoms and gaseous ions. Because enthalpy is a state function, the sum along that route must match the supplied formation enthalpy when all terms use consistent reference conditions.

Make the signed ledger explicit

Atomizing M costs150 kJ/mol, atomizing half X2 costs100, ionizing M costs a selected positive amount I and attaching an electron to X has supplied process enthalpy−300. The subtotal for separated gaseous ions is I−50. The final lattice formation process is M+(g)+X−(g)→MX(s). Its enthalpy L must satisfy ΔHf=(I−50)+L.

Solve a new cycle

At I=550 and ΔHf=−350 kJ/mol, the subtotal is500 and L=−850 kJ/mol. The reverse lattice dissociation reaction has +850 kJ/mol. Both signs are correct for their explicitly different reactions. Calling either value simply ‘lattice energy’ without the process can hide a sign error.

Predict a constrained change

Increase I by100 while holding the overall formation enthalpy fixed. The gaseous-ion subtotal rises by100, so L must become100 kJ/mol more negative. This is a consistency requirement on the fictional packet, not a claim that changing an actual atom’s ionization energy leaves all its other properties independently adjustable.

Audit conventions and missing evidence

Some electron-affinity tables report a positive energy released where a process-enthalpy convention reports a negative value. Here the actual attachment process and its sign are supplied. Do not reverse it twice. Real cycle analysis also needs compatible states, stoichiometric factors and uncertainties. If atomization were supplied per mole X2 rather than per mole X atoms, the half-mole factor would matter. Reconstruct the endpoint equation before accepting an impressive-looking final magnitude.

Evidence and safety boundary

Conceptual sources checked 30 October 2026: Born–Haber cycles. All prose, numerical packets and diagrams are original; no textbook exercises or images are reproduced. The activity is supplied mathematical or synthetic evidence, not a performed materials experiment. No unknown substances, heating, high voltage or chemical synthesis are requested. Retain prior predictions and a changed-case explanation; numerical agreement alone does not certify independent understanding or practical competence.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Explain why electron-affinity sign conventions cannot replace the explicit signed attachment reaction.

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Build an ionic array ledger without double-counting pairs Next: Separate resonance, formal charge and measured electron density

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