The Born-Haber cycle for magnesium oxide (MgO) is a thermodynamic cycle that meticulously breaks down the overall formation of this ionic compound into a series of individual energetic steps. This powerful tool is primarily used to calculate the lattice energy of an ionic solid, which represents the energy released when gaseous ions combine to form a stable crystalline lattice. For MgO, this cycle illuminates the significant energy required to form the highly charged Mg²⁺ and O²⁻ ions in the gas phase, offset by the immensely exothermic lattice energy that drives its formation.
Understanding the Born-Haber Cycle
The Born-Haber cycle applies Hess's Law, stating that the total enthalpy change for a reaction is the same regardless of the path taken. For MgO, the cycle relates the standard enthalpy of formation (ΔHf°) to several other enthalpy changes:
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Enthalpy of Atomisation/Sublimation (ΔHsub): The energy required to convert one mole of a solid metal (Mg) into gaseous atoms.
Mg(s) → Mg(g)
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First and Second Ionisation Energies (IE₁ & IE₂): The energy required to remove one or more electrons from one mole of gaseous atoms to form gaseous positive ions. For Mg, two electrons are removed to form Mg²⁺.
Mg(g) → Mg⁺(g) + e⁻(IE₁)Mg⁺(g) → Mg²⁺(g) + e⁻(IE₂)
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Enthalpy of Atomisation/Dissociation (½ΔHdiss): The energy required to break the bonds in one mole of a gaseous diatomic element (O₂) into individual gaseous atoms. Since MgO requires one oxygen atom, we use half of the dissociation energy.
½O₂(g) → O(g)
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First and Second Electron Affinities (EA₁ & EA₂): The energy change when one or more electrons are added to one mole of gaseous atoms or ions to form gaseous negative ions. For oxygen, two electrons are added to form O²⁻.
O(g) + e⁻ → O⁻(g)(EA₁) - This step is usually exothermic.O⁻(g) + e⁻ → O²⁻(g)(EA₂) - Crucially, this step is endothermic for oxygen due to repulsion between the incoming electron and the already negatively charged O⁻ ion.
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Lattice Energy (ΔHlattice): The energy released when one mole of an ionic compound is formed from its constituent gaseous ions. This is a highly exothermic process, particularly for compounds like MgO with high ionic charges (Mg²⁺ and O²⁻).
Mg²⁺(g) + O²⁻(g) → MgO(s)
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Standard Enthalpy of Formation (ΔHf°): The overall enthalpy change when one mole of a compound is formed from its elements in their standard states.
Mg(s) + ½O₂(g) → MgO(s)
Applying Hess's Law for MgO
According to Hess's Law, the sum of the enthalpy changes for the individual steps in the cycle equals the overall standard enthalpy of formation:
ΔH<sub>f</sub>°(MgO) = ΔH<sub>sub</sub>(Mg) + IE₁(Mg) + IE₂(Mg) + ½ΔH<sub>diss</sub>(O₂) + EA₁(O) + EA₂(O) + ΔH<sub>lattice</sub>(MgO)
This equation allows us to calculate any one of the enthalpy terms if all the others are known. Most commonly, it is used to determine the lattice energy, as it is difficult to measure directly.
Typical Energy Values for MgO
Here's a table illustrating the approximate energy changes involved in the Born-Haber cycle for magnesium oxide:
| Energy Term | Process | Typical Value (kJ/mol) | Nature |
|---|---|---|---|
| Overall Formation | Mg(s) + ½O₂(g) → MgO(s) |
-601 | Exothermic |
| Sublimation of Mg | Mg(s) → Mg(g) |
+148 | Endothermic |
| 1st Ionisation Energy of Mg | Mg(g) → Mg⁺(g) + e⁻ |
+738 | Endothermic |
| 2nd Ionisation Energy of Mg | Mg⁺(g) → Mg²⁺(g) + e⁻ |
+1451 | Endothermic |
| Dissociation of O₂ (½ mole) | ½O₂(g) → O(g) |
+249 | Endothermic |
| 1st Electron Affinity of O | O(g) + e⁻ → O⁻(g) |
-141 | Exothermic |
| 2nd Electron Affinity of O | O⁻(g) + e⁻ → O²⁻(g) |
+744 | Endothermic |
| Lattice Energy (calculated) | Mg²⁺(g) + O²⁻(g) → MgO(s) |
-3790 | Exothermic |
(Note: Values are approximate and can vary slightly depending on the source.)
The sum of the endothermic steps (sublimation, two ionization energies, dissociation, and the second electron affinity) is very large and positive. However, the extremely large and negative lattice energy for MgO, due to the strong electrostatic attraction between the doubly charged Mg²⁺ and O²⁻ ions in the crystal lattice, more than compensates for these energy inputs, making the overall formation of MgO exothermic and stable.
Importance and Practical Insights
- Lattice Energy Calculation: The primary purpose is to indirectly determine lattice energies, which are crucial for understanding the stability of ionic compounds. For MgO, the lattice energy is exceptionally high compared to 1:1 ionic compounds like NaCl due to the higher charges (+2 and -2), leading to much stronger electrostatic attractions.
- Predicting Stability: By analyzing the balance of energy terms, the Born-Haber cycle helps predict the feasibility and stability of hypothetical ionic compounds. For instance, it shows why MgO is stable despite the high energy cost of forming O²⁻ ions.
- Thermodynamic Analysis: It provides a comprehensive thermodynamic picture of ionic bond formation, detailing the contributions of various energy changes.
In essence, the Born-Haber cycle for MgO beautifully illustrates how a seemingly unfavorable series of individual steps (forming highly charged gaseous ions) can lead to a highly stable ionic compound due to the overwhelming energy released during the formation of its crystal lattice.
