Consider an ionic compound, MX2, composed of generic metal M and generic, gaseous halogen X.
The enthalpy of formation of MX2 is ΔHf° = –799 kJ/mol.
The enthalpy of sublimation of M is ΔHsub = 157 kJ/mol.
The first and second ionization energies of M are IE1 = 785 kJ/mol and IE2 = 1451 kJ/mol. The electron affinity of X is ΔHEA = –349 kJ/mol.
The bond energy of X2 is BE = 157 kJ/mol.
Determine the lattice energy of MX2.
To determine the lattice energy of MX2, we need to use the Born-Haber cycle, which involves a series of steps to determine the overall energy change involved in the formation of an ionic compound from its constituent elements.
The Born-Haber cycle includes the following steps:
1. Formation of gaseous atoms from their respective elements.
2. Ionization of the metal atoms to form cations.
3. Electron affinity of the halogen atom to form an anion.
4. Formation of the solid ionic compound from the gaseous ions.
Let's break down each step and calculate the energy changes involved:
Step 1: Formation of gaseous atoms from their elements.
This step involves the sublimation of the metal and halogen atoms. Given:
ΔHsub = 157 kJ/mol for M (sublimation of metal)
ΔHEA = -349 kJ/mol for X (electron affinity of halogen)
The energy change for this step is the sum of the sublimation energy and the electron affinity:
ΔH1 = ΔHsub + ΔHEA
ΔH1 = 157 kJ/mol + (-349 kJ/mol)
ΔH1 = -192 kJ/mol
Step 2: Ionization of the metal atom.
This step involves the removal of one electron from the metal atom. Given:
IE1 = 785 kJ/mol (first ionization energy of metal)
Step 3: Electron affinity of the halogen atom.
This step involves the addition of one electron to the halogen atom. Given:
ΔHEA = -349 kJ/mol (electron affinity of halogen)
Step 4: Formation of the solid ionic compound.
This step involves the formation of the MX2 compound. Given:
ΔHf° = -799 kJ/mol (enthalpy of formation of MX2)
BE = 157 kJ/mol (bond energy of X2)
The energy change for this step is the sum of the lattice energy, the enthalpy of formation, and the bond energy:
ΔH4 = Lattice Energy + ΔHf° + 2BE
Now, using the Born-Haber cycle, we can write the overall energy change equation:
0 = ΔH1 + IE1 + ΔH4
To find the lattice energy, rearrange the equation:
Lattice Energy = -(ΔH1 + IE1 + ΔH4)
Substituting the given values, we get:
Lattice Energy = -(ΔH1 + IE1 + ΔHf° + 2BE)
Lattice Energy = -(-192 kJ/mol + 785 kJ/mol + (-799 kJ/mol) + 2(157 kJ/mol))
Calculate the value to find the lattice energy.