In simple terms
A friendly intro before the formal notes — no formulas yet.
Building Ionic Crystals, Step by Step
A Born-Haber cycle is a clever energy diagram that breaks down the formation of an ionic solid into a series of simpler, measurable steps. By applying Hess's Law, we can calculate the lattice energy, which is too difficult to measure directly.
Imagine building a complex LEGO model. The total energy you use is the same whether you build it in one go (direct formation) or follow a multi-step instruction manual: opening the bags, sorting the pieces, and assembling them bit by bit (the Born-Haber cycle). The cycle simply shows that the energy change for the direct route must equal the sum of the energy changes for the indirect route.
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Lattice energy (ΔH°_latt) is the exothermic enthalpy change when one mole of a solid ionic lattice is formed from its gaseous ions. It's more exothermic for smaller ions with higher charges.
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A Born-Haber cycle applies Hess's law, stating the total enthalpy change is independent of the route taken. It equates the standard enthalpy of formation (ΔH°_f) with the sum of all steps in an alternative pathway.
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The magnitude of lattice energy is determined by the strength of electrostatic attraction. It becomes more exothermic (stronger) as ionic charges increase and ionic radii decrease.
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Comparing the experimental ΔH°_latt (from a Born-Haber cycle) with a theoretical value (from a pure ionic model) reveals covalent character. A large discrepancy suggests significant orbital overlap, often due to a highly polarising cation and a polarisable anion.
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Defining the Key Enthalpy Changes
To construct a Born-Haber cycle, we need to be precise about several key enthalpy definitions. Each step in the cycle corresponds to a specific, defined process. Remember that 'standard conditions' refers to 100 kPa pressure and a specified temperature, usually 298 K.
Standard Lattice Energy (ΔH°_latt): Formation of 1 mole of solid ionic lattice from its gaseous ions. E.g., Na⁺(g) + Cl⁻(g) → NaCl(s). This is always exothermic.
Standard Enthalpy of Formation (ΔH°_f): Formation of 1 mole of a compound from its elements in their standard states. E.g., Na(s) + ½Cl₂(g) → NaCl(s). Can be exothermic or endothermic.
Standard Enthalpy of Atomisation (ΔH°_at): Formation of 1 mole of gaseous atoms from an element in its standard state. E.g., Na(s) → Na(g). This is always endothermic.
First Ionisation Energy (IE₁): Removal of 1 mole of electrons from 1 mole of gaseous atoms. E.g., Na(g) → Na⁺(g) + e⁻. This is always endothermic.
First Electron Affinity (EA₁): Addition of 1 mole of electrons to 1 mole of gaseous atoms. E.g., Cl(g) + e⁻ → Cl⁻(g). This is usually exothermic for non-metals.
The Born-Haber Cycle: An Application of Hess's Law
A Born-Haber cycle is an energy level diagram that illustrates the formation of an ionic compound. It connects the standard enthalpy of formation (the 'direct route') with a series of steps that form the gaseous ions from the elements and then combine them to form the lattice (the 'indirect route'). According to Hess's Law, the total enthalpy change for the direct route must equal the sum of the enthalpy changes for the indirect route. This allows us to calculate one unknown value, which is typically the lattice energy, as it cannot be measured directly.
For a simple MX compound like NaCl:
Pay close attention to the direction of arrows in a Born-Haber cycle diagram. Upward arrows represent endothermic processes (energy input), while downward arrows represent exothermic processes (energy release). The sum of clockwise energy changes must equal the sum of anticlockwise energy changes.
Factors Affecting Lattice Energy
The magnitude of lattice energy is a direct measure of the strength of the electrostatic forces within the ionic lattice. These forces are described by Coulomb's Law, which states that the force is proportional to the product of the charges and inversely proportional to the square of the distance between them. In chemistry, this translates to two key factors: ionic charge and ionic radius.
Ionic Charge: A greater charge on the ions leads to a much stronger electrostatic attraction. Therefore, lattice energy becomes significantly more exothermic. For example, the lattice energy of MgO (Mg²⁺O²⁻) is about -3795 kJ mol⁻¹, far more exothermic than NaCl's (Na⁺Cl⁻) -787 kJ mol⁻¹.
Ionic Radius: Smaller ions can get closer together, decreasing the distance between their centres of charge. This results in stronger attraction and a more exothermic lattice energy. For example, LiF has a more exothermic lattice energy than KCl because Li⁺ is smaller than K⁺ and F⁻ is smaller than Cl⁻.
Covalent Character and Polarisation
The Born-Haber cycle provides an experimental value for lattice energy. We can also calculate a theoretical value based on a perfect ionic model, assuming ions are perfect, hard spheres with evenly distributed charge. Often, there is a good agreement between the two. However, if there is a significant discrepancy, with the experimental value being more exothermic than the theoretical one, it suggests the bonding is not purely ionic. This difference is attributed to covalent character, which arises from the polarisation of the anion's electron cloud by the cation. A small, highly charged cation is highly polarising, and a large anion is easily polarisable. This distortion and sharing of electrons adds a covalent nature to the bond, making it stronger than the purely ionic model predicts.
When asked to compare lattice energies, always discuss both ionic charge and ionic radius. Charge is almost always the more significant factor. A perfect answer will state which factor is dominant.
Worked examples
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Calculate the standard lattice energy of sodium chloride, NaCl, using the data provided below.
| Enthalpy Change | Value / kJ mol⁻¹ |
|---|---|
| Standard enthalpy of formation of NaCl(s) | -411 |
| --- | --- |
| Standard enthalpy of atomisation of Na(s) | +107 |
| First ionisation energy of Na(g) | +496 |
| Standard enthalpy of atomisation of Cl₂(g) | +122 |
| First electron affinity of Cl(g) | -349 |
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State the Born-Haber cycle equation based on Hess's Law:
Explain why the lattice energy of magnesium oxide, MgO, is significantly more exothermic than that of sodium fluoride, NaF, given the following data.
| Ion | Ionic Radius / nm |
|---|---|
| :-- | :-- |
| Na⁺ | 0.102 |
| Mg²⁺ | 0.072 |
| F⁻ | 0.133 |
| O²⁻ | 0.140 |
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Ionic Charge: The primary reason is the magnitude of the ionic charges. MgO is formed from Mg²⁺ and O²⁻ ions. NaF is formed from Na⁺ and F⁻ ions. The product of the charges in MgO is (2+) × (2-) = 4, whereas in NaF it is (1+) × (1-) = 1. The much larger product of charges in MgO leads to far stronger electrostatic forces of attraction between the ions.
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What is Standard Lattice Energy (ΔH°_latt)?
The standard enthalpy change when one mole of an ionic solid is formed from its constituent gaseous ions under standard conditions (298 K, 100 kPa). It is always an exothermic process.
Key takeaways
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Standard Lattice Energy (ΔH°_latt): Formation of 1 mole of solid ionic lattice from its gaseous ions. E.g., Na⁺(g) + Cl⁻(g) → NaCl(s). This is always exothermic.
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Standard Enthalpy of Formation (ΔH°_f): Formation of 1 mole of a compound from its elements in their standard states. E.g., Na(s) + ½Cl₂(g) → NaCl(s). Can be exothermic or endothermic.
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Standard Enthalpy of Atomisation (ΔH°_at): Formation of 1 mole of gaseous atoms from an element in its standard state. E.g., Na(s) → Na(g). This is always endothermic.
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First Ionisation Energy (IE₁): Removal of 1 mole of electrons from 1 mole of gaseous atoms. E.g., Na(g) → Na⁺(g) + e⁻. This is always endothermic.
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First Electron Affinity (EA₁): Addition of 1 mole of electrons to 1 mole of gaseous atoms. E.g., Cl(g) + e⁻ → Cl⁻(g). This is usually exothermic for non-metals.
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