Born-Haber Calculations

Step 1: Construct the Cycle: Draw the energy levels starting with elements in their standard states. Map out the endothermic steps (atomisation and ionisation) moving upwards and the exothermic steps (electron affinity and lattice formation) moving downwards.

Step 2: Identify the Routes: Define the 'Direct Route' as the enthalpy of formation ($\Delta H_f$). Define the 'Indirect Route' as the sequence of all other steps that lead from the same starting elements to the same final ionic solid.

Step 3: Apply the Equation: Use the general relationship: $\Delta H_f = \sum (\text{Indirect Steps}) + \Delta H_{latt}$

Step 4: Rearrange and Solve: To find lattice enthalpy, subtract the sum of the preparatory steps from the enthalpy of formation: $\Delta H_{latt} = \Delta H_f - [\Delta H_{at} + IE + EA]$

It is vital to distinguish between Lattice Formation Enthalpy (exothermic, ions to solid) and Lattice Dissociation Enthalpy (endothermic, solid to ions). Born-Haber cycles typically use the formation definition, resulting in negative values.

Stoichiometric Coefficients must be applied to enthalpy values when the chemical formula involves multiple moles of an ion. For example, in a compound like $MgCl_2$, the atomisation and electron affinity values for chlorine must be doubled because two moles of chloride ions are produced.

Use Brackets Religiously: When calculating the sum of the indirect route, place all terms inside brackets before subtracting them from the enthalpy of formation. This prevents common sign errors when dealing with negative electron affinities.

Check State Symbols: Ensure every species in the gaseous ion level has the $(g)$ symbol and the correct charge. Missing state symbols or incorrect charges are frequent sources of lost marks in cycle construction.

Verify Multipliers: Always look at the subscripts in the chemical formula (e.g., the '2' in $CaF_2$). If there are two moles of an atom or ion, you must multiply the corresponding $\Delta H_{at}$, $IE$, or $EA$ by two.

Sanity Check: Lattice enthalpies for ionic compounds should always be significantly negative (exothermic). If your calculation results in a positive value, you likely reversed a sign or a route.

The 'Second' Electron Affinity Trap: While the first electron affinity is usually exothermic (negative), the second electron affinity (e.g., $O^-$ to $O^{2-}$) is always endothermic (positive) due to the repulsion between the negative ion and the incoming electron. Students often incorrectly assign it a negative sign.

Atomisation vs. Bond Enthalpy: For diatomic gases like $Cl_2$, the enthalpy of atomisation is the energy to form one mole of gaseous atoms ($\frac{1}{2} Cl_2 \rightarrow Cl$). If a data table provides 'Bond Dissociation Enthalpy' ($Cl_2 \rightarrow 2Cl$), you must halve that value to get the atomisation energy for one mole of atoms.

Route Reversal: Students often confuse the starting and ending points of the cycle. Remember that the enthalpy of formation and the lattice enthalpy both 'end' at the same point: the solid ionic lattice.