Concept:The Born-Haber cycle utilizes Hess's law to equate the direct enthalpy of formation (\( \Delta H_f \)) to the sum of all individual step enthalpies: atomization, ionization, electron affinity, and lattice energy.
Formula:$$ \Delta H_f = \Sigma \Delta H_{\text{steps}} + \Delta H_{\text{lattice}} $$
$$ \Delta H_{\text{lattice}} = \Delta H_f - (\Delta H_{at}[Na] + \Delta H_{at}[Cl] + IE[Na] + EA[Cl]) $$
Solution:- Sum the indirect steps (excluding lattice):
\( 107 \text{ (atom Na)} + 122 \text{ (atom Cl)} + 496 \text{ (IE Na)} + (-349) \text{ (EA Cl)} = 376 \text{ kJ/mol} \).
- Apply Hess's Law equation:
\( \Delta H_{\text{lattice}} = \Delta H_f - \Sigma(\text{other steps}) \)
- Substitute values:
\( \Delta H_{\text{lattice}} = -411 - (376) \)
- \( \Delta H_{\text{lattice}} = -787 \text{ kJ/mol} \).
Why other options are incorrect:\( +787 \) is the magnitude but reversed sign, representing lattice dissociation instead of formation. \( 376 \) and \( -376 \) are merely the sum of the non-lattice steps in the cycle, omitting the crucial \( \Delta H_f \) term.
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