6: Born Haber cycles

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In a Born-Haber cycle, a series of reactions are written which add up to the overall reaction—the formation of an ionic compound from the reaction of its constituent elements. Because enthalpy is a state function, the sum of the enthalpies of the individual steps is equal to the enthalpy change for the overall reaction.

Below is a description of the individual reactions for the production of NaCl.

• In the Table:
  • Write a chemical equation that corresponds to each step. Be sure to include the proper state of matter and charge on each species.

  • Then predict whether the enthalpy change for each step is positive or negative.

Step	Balanced equation	Sign of ΔH
Sublimation of sodium
First ionization energy for sodium
0.5 X Cl-Cl bond dissociation energy
Electron affinity for Cl
Lattice energy for NaCl
Net:	Na(s) + 0.5 Cl2(g) -> NaCl(s)

Follow-up questions:

• The net reaction is exothermic/endothermic (circle one). Which of the steps that add up to it is likely the “driving force”—the biggest contributor that makes it exothermic/endothermic?

In a Born-Haber cycle, one can’t directly measure the enthalpy change for the lattice energy.

• Provide a method for determining the lattice energy. Remember, the calculated value does not have to be for the “net” reaction.

Here are the actual values for the formation of NaCl:

Na_(s) + 0.5 Cl₂ -> NaCl_(s)

Step	Balanced equation	ΔH
Sublimation of sodium	Na(s) -> Na(g)	+108 kJ/mol
First ionization energy for sodium	Na(g) ->Na^+(g) + e^-	+502 kJ/mol
0.5 X Cl-Cl bond dissociation energy (only 1 Cl needed)	0.5Cl2(g) -> Cl(g)	+121 kJ/mol
Electron affinity for Cl	Cl(g) + e^- -> Cl^-(g)	-349 kJ/mol
Lattice energy for NaCl	Na^+(g) + Cl^-(g) -> NaCl(s)	kJ/mol
Net:	Na(s) + 0.5Cl2 -> NaCl(s) -411 kJ/mol

• Label the type of step occurring in process on the Born-Haber Cycle shown.

• Calculate the lattice energy for the formation of NaCl.

Your younger sibling is taking high school chemistry and tells you that “the reason sodium and chlorine react is that sodium wants to lose an electron and chlorine wants to gain one.”

• Is the ionization of sodium favorable? YES or NO

• What is the most exothermic step? Why?

• What two factors affect lattice energy?

• Complete the following table.

Step (kJ/mol)	MgF	MgF2	MgF3
Mg sublimation	+150
F-F bond energy		+160
Mg ionization (total)	+740	+2190	+9930
F electron affinity	-330
Lattice energy	-900 (est)	-2880	-5900 (est)
Net ΔH

(Table adapted from Rayner-Canham, G; Overton, T. “Descriptive Inorganic Chemistry”, 5th ed, 2010, W. H. Freeman.)

Compare the first and second ionization energies of Mg.

• Why is the third ionization energy so much larger than what you would predict based on these values?

The lattice energies for MgF and MgF₃ were estimated, since they do not exist.

• If you were going to provide an estimate by looking up the value for a known compound, which compound’s lattice energy would you use for each?

Mg reacts with fluorine to form MgF₂. MgF and MgF₃ are not formed. MgF would only involve one ionization to remove an electron. MgF₃ would have a greater lattice energy.

• Why is MgF₂ the favored product?

Summary

• Define the following terms:
  • State Function

  • Path Function

  • Hess’s Law

• Born-Haber Cycles often contain these individual steps. Predict whether each step would be exothermic or endothermic.
  • First Ionization Energy | Exothermic OR Endothermic

  • Second Ionization Energy | Exothermic OR Endothermic

  • Sublimation Energy | Exothermic OR Endothermic

  • Electron Affinity | Exothermic OR Endothermic

  • Bond Dissociation | Exothermic OR Endothermic

  • Boiling | Exothermic OR Endothermic

  • Lattice Energy | Exothermic OR Endothermic

• Born-Haber Cycles are often used to determine Lattice Energy. Show how this would be calculated.

Practice Problems

Ionization energies, electron affinities and bond dissociation energies are all reported for gas-phase atoms.

1. Draw a Born-Haber cycle for each of the following processes. Be sure to include all needed steps.
   1. Na_(a) + 0.5 Br_2(l) -> NaBr_(s)

   2. Na_(s) + 0.5 I_2(s) -> NaI_(s)

   3. Ca_(s) + Cl_2(g) -> CaCl_2(s)

   4. 2 Na_(s) + 0.5 O_2(g) -> Na₂O_(s)

2. Construct a Born-Haber cycle for the formation of magnesium sulfide. You do not need to perform a calculation. (Hint—elemental sulfur is a solid containing S₈ rings.)​​​​​​​

3. Calculate the lattice energy for CaH₂ using the following information. (E_EA for H = - 72.8 kJ/mol; E_i1 for Ca=590 kJ/mol; E_i2 for Ca =1145 kJ/mol; heat of subm for Ca =178 kJ/mol; bond dissociation energy for H₂ =436 kJ/mol; net energy for the formation of CaH₂ from its elements =-186 kJ mol.

4. In episodes of Star Trek, science fiction allows intergalactic travel via the power of dilithum crystals. Let’s assume that dilithium crystals are Li⁺Li^-(s). Construct a Born- Haber cycle and calculate the enthalpy of formation of dilithium crystals from lithium metal. Some data: E_EA for Li = -60 kJ/mol; E_i1 for Li=526 kJ/mol; heat of subm for Li =159 kJ/mol; lattice energy (est) -757 kJ/mol.