Determine the lattice energy of MX
Consider an ionic compound, MX2, composed of generic metal M and generic, gaseous halogen X.
The enthalpy of formation of MX2 is ?Hf = -975 kJ/mol.
The enthalpy of sublimation of M is ?Hsub = 133 kJ/mol.
The first and second ionization energies of M are IE1 = 751 and IE2 = 1412.
The electron affinity of X is EA = -323 kJ/mol.
The bond energy of X2 is BE = 203 kJ/mol.
Determine the lattice energy of MX2.
The Correct Answer and Explanation is:
To determine the lattice energy of the ionic compound MX₂, we use the Born-Haber cycle, which relates the enthalpy changes in the formation of an ionic compound from its elements.
Given:
- ΔHf (MX₂) = –975 kJ/mol (formation of MX₂ from M(s) and X₂(g))
- ΔHsub (M) = +133 kJ/mol (sublimation of M(s) to M(g))
- IE₁ (M) = +751 kJ/mol
- IE₂ (M) = +1412 kJ/mol
- EA (X) = –323 kJ/mol (per atom)
- Bond Energy (X₂) = +203 kJ/mol (per mole of X₂ molecules broken)
Step-by-step Breakdown:
Formation of MX₂:
- Sublimation of M(s) → M(g):
ΔH = +133 kJ/mol - Ionization of M(g) → M²⁺(g):
IE₁ + IE₂ = 751 + 1412 = 2163 kJ/mol - Dissociation of X₂(g) → 2 X(g):
X₂ → 2X:
ΔH = +½(203) = +101.5 kJ/mol - Electron gain by 2 X(g) → 2 X⁻(g):
2 × EA = 2 × (–323) = –646 kJ/mol - Formation of MX₂(s) from M²⁺(g) and 2X⁻(g):
This is the lattice energy (U), which we solve for.
Use the Born-Haber cycle equation:
ΔHf=ΔHsub+IE1+IE2+12BE+2EA+U\Delta H_f = \Delta H_{sub} + IE_1 + IE_2 + \dfrac{1}{2}BE + 2EA + U
Plug in values: −975=133+751+1412+101.5−646+U-975 = 133 + 751 + 1412 + 101.5 – 646 + U −975=1751.5+U-975 = 1751.5 + U U=−975−1751.5=−2726.5 kJ/molU = -975 – 1751.5 = \boxed{-2726.5\ \text{kJ/mol}}
Final Answer:
Lattice Energy (U) of MX₂ = –2726.5 kJ/mol
Explanation
Lattice energy is a measure of the strength of the forces between the ions in an ionic solid. It represents the energy released when one mole of an ionic compound forms from its gaseous ions. To determine it, we use the Born-Haber cycle, which breaks down the formation of the ionic compound into a series of energetic steps.
In this problem, we are given the enthalpy of formation of MX₂, along with the sublimation energy, ionization energies of the metal M, the bond energy of halogen X₂, and the electron affinity of X. Starting from the elements in their standard states, M (solid) and X₂ (gas), we proceed through the steps of sublimating M, ionizing it to M²⁺, breaking X₂ into two atoms, and adding electrons to form two X⁻ ions. The total of these steps gives the energy required to produce the gaseous ions.
The final step is the combination of M²⁺ and 2 X⁻ into solid MX₂, which releases a large amount of energy – the lattice energy. By using the enthalpy of formation, we can rearrange the Born-Haber equation to solve for the unknown lattice energy. The result, –2726.5 kJ/mol, indicates a highly exothermic process, which is typical for ionic compounds due to the strong electrostatic attraction between oppositely charged ions.
This lattice energy is crucial in understanding the stability, melting point, and solubility of the ionic compound. A more negative lattice energy generally means a more stable solid with stronger ionic bonds.
