Concept:The Born-Haber cycle equates the standard enthalpy of formation of an ionic solid to the sum of the energies of sublimation, ionization, dissociation, electron affinity, and lattice energy.
Formula:$$ \Delta H_{\text{lattice}} = \Delta H_f - \Sigma \Delta H_{\text{indirect steps}} $$
Solution:- From the provided diagram's standard data:
\( \Delta H_f = -392 \text{ kJ/mol} \).
Indirect steps sum (\( \Delta H_x \)) = \( 112 \text{ (sub)} + 420 \text{ (IE)} + 90 \text{ (dissoc)} - 342 \text{ (EA)} = 280 \text{ kJ/mol} \).
- Apply the cycle equation:
\( \Delta H_{\text{lattice}} = -392 - (280) \)
- \( \Delta H_{\text{lattice}} = -672 \text{ kJ/mol} \).
Why other options are incorrect:Option B uses the wrong units (KCal instead of kJ). Option D uses the wrong units (Joules) and incorrect sign. Option C is the lattice energy for \( NaCl \), acting as a distractor.
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