Born-Haber Cycle

Core idea

Overview

The Born-Haber cycle is a thermochemical application of Hess's Law used to calculate the lattice energy of ionic crystalline solids. It relates the standard enthalpy of formation of an ionic compound to the energy required to atomize and ionize the constituent elements.

When to use: Use this cycle when direct experimental measurement of lattice enthalpy is not feasible. It is applicable for calculating any missing energetic component of an ionic compound's formation when the other thermodynamic values are known.

Why it matters: This cycle allows scientists to evaluate the strength of ionic bonds and the stability of crystals. Discrepancies between theoretical lattice energy and values derived from the cycle often reveal the degree of covalent character in a bond.

Symbols

Variables

$Δ$ $H_{f}$ = Enthalpy of Formation, $Δ$ $H_{a t}$ (M) = Atomization (Metal), $Δ$ $H_{a t}$ (X) = Atomization (Non-metal), IE = Ionization Energy, EA = Electron Affinity

Δ H_{f}

Enthalpy of Formation

kJ/mol

Δ H_{a t} (M)

Atomization (Metal)

kJ/mol

Δ H_{a t} (X)

Atomization (Non-metal)

kJ/mol

IE

Ionization Energy

kJ/mol

EA

Electron Affinity

kJ/mol

Δ H_{L E}

Lattice Enthalpy

kJ/mol

Walkthrough

Derivation

Understanding the Born-Haber Cycle

Applies Hess’s Law to calculate lattice enthalpy by breaking ionic solid formation into gaseous steps.

Cycle steps are theoretical and use standard enthalpy values.

1

Use Hess’s Law Around the Cycle:

Formation enthalpy equals the sum of intermediate steps plus lattice enthalpy (with correct signs).

Δ_{f} H^{⊖} = \sum (atomisation, IE, EA) + Δ_{latt} H^{⊖}

Note: Exact steps depend on the ionic compound (number of ionisations/electron affinities).

Result

Δ_{f} H^{⊖} = \sum (atomisation, IE, EA) + Δ_{latt} H^{⊖}

Source: OCR A-Level Chemistry A — Energetics (Born–Haber cycles)

Free formulas

Rearrangements

Solve for $Δ H_{f}^{θ}$

Make Delta Hf^theta the subject

Δ H_{f}^{θ} = Δ H_{a t} (M) + Δ H_{a t} (X) + I E (M) + E A (X) + Δ H_{L E}

Start with the general Born-Haber Cycle equation and expand its terms to define the standard enthalpy of formation for a specific ionic compound.

Difficulty: 2/5

Solve for $Δ H_{L E}$

Born-Haber Cycle: Make Lattice Enthalpy the Subject

Δ H_{L E} = Δ H_{f}^{θ} - Δ H_{a t} (M) - Δ H_{a t} (X) - I E (M) - E A (X)

Rearrange the Born-Haber Cycle equation to isolate the Lattice Enthalpy ( $Δ$ $H_{L E}$ ), expanding general terms into specific components for the metal and non-metal.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph is a straight line with a negative slope of -1. As the ionization energy increases, the lattice enthalpy must decrease linearly to maintain the balance of the equation.

Graph type: linear

Why it behaves this way

Intuition

Imagine a closed energy cycle, like a multi-stage journey, where the total energy change for forming an ionic compound from its elements is the sum of the energy changes for each intermediate step of atomization

Δ H_{f}

Standard enthalpy of formation

The net energy change for the overall chemical reaction of forming the ionic compound from its basic elements in their standard states.

Δ H_{a t}

Enthalpy of atomization

The energy cost to break apart the elemental form (e.g., metallic bonds in Na, covalent bonds in Cl2) to get individual, isolated gaseous atoms ready to react. This step always requires energy input.

IE

Ionization energy

The energy cost to create a positive ion from a neutral gaseous atom by removing an electron. It reflects how tightly the outermost electron is held by the nucleus and always requires energy input.

EA

Electron affinity

The energy change associated with creating a negative ion from a neutral gaseous atom by adding an electron. A negative value means energy is released (favorable), while a positive value means energy is required

Δ H_{L E}

Lattice enthalpy (enthalpy of lattice formation)

The energy released when gaseous positive and negative ions come together to form the stable crystal lattice. A more negative value indicates stronger electrostatic attractions and a more stable crystal.

Free study cues

Insight

Canonical usage

All terms in the Born-Haber cycle equation represent enthalpy changes and must be expressed in consistent molar energy units, typically Joules per mole or kilojoules per mole.

Common confusion

Mixing units such as J/mol and kJ/mol within the same calculation without proper conversion, or using electron-volts per atom directly for ionization energy or electron affinity without converting to molar energy units.

Unit systems

$Δ H_{f}$ kJ/mol - Standard molar enthalpy of formation of the ionic compound.

$Δ H_{a t}$ kJ/mol - Total molar enthalpy of atomization for all constituent elements required to form gaseous atoms.

IEkJ/mol - Total molar ionization energy(ies) required to form the gaseous cation(s). If given in electron-volts per atom (eV/atom), convert to kJ/mol using Avogadro's constant and the electron-volt to Joule conversion factor (1 eV

EAkJ/mol - Total molar electron affinity(ies) required to form the gaseous anion(s). If given in electron-volts per atom (eV/atom), convert to kJ/mol using Avogadro's constant and the electron-volt to Joule conversion factor (1 eV

$Δ H_{L E}$ kJ/mol - Molar lattice enthalpy (or lattice energy) of the ionic compound.

Ballpark figures

Quantity:
Quantity:
Quantity:
Quantity:
Quantity:

One free problem

Practice Problem

Calculate the lattice enthalpy (LE) for Sodium Chloride (NaCl) using the following thermochemical data: enthalpy of formation (Hf) = -411 kJ/mol, enthalpy of atomization of Na (HatM) = 107 kJ/mol, enthalpy of atomization of Cl (HatX) = 121 kJ/mol, first ionization energy of Na (IE) = 496 kJ/mol, and electron affinity of Cl (EA) = -349 kJ/mol.

Enthalpy of Formation-411 kJ/mol

Atomization (Metal)107 kJ/mol

Atomization (Non-metal)121 kJ/mol

Ionization Energy496 kJ/mol

Electron Affinity-349 kJ/mol

Solve for: LE

Hint: Rearrange the equation to LE = Hf - (HatM + HatX + IE + EA).

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In explaining why NaCl is stable, Born-Haber Cycle is used to calculate Lattice Enthalpy from Enthalpy of Formation, Atomization (Metal), and Atomization (Non-metal). The result matters because it helps connect measured amounts to reaction yield, concentration, energy change, rate, or equilibrium.

Study smarter

Tips

Ensure stoichiometry is correct: if the formula is MX₂, ensure you double the EA and use appropriate atomization values.
Lattice enthalpy and enthalpy of formation are almost always negative (exothermic).
Ionization energy is always positive (endothermic), while electron affinity is usually negative for the first electron.
Check that all values use consistent units, typically kJ/mol.

Avoid these traps

Common Mistakes

Sign errors (endo vs exo).
Forgetting atomization of diatomic elements.
Wrong electron affinity values.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Applies Hess’s Law to calculate lattice enthalpy by breaking ionic solid formation into gaseous steps.

Use this cycle when direct experimental measurement of lattice enthalpy is not feasible. It is applicable for calculating any missing energetic component of an ionic compound's formation when the other thermodynamic values are known.

This cycle allows scientists to evaluate the strength of ionic bonds and the stability of crystals. Discrepancies between theoretical lattice energy and values derived from the cycle often reveal the degree of covalent character in a bond.

Sign errors (endo vs exo). Forgetting atomization of diatomic elements. Wrong electron affinity values.

In explaining why NaCl is stable, Born-Haber Cycle is used to calculate Lattice Enthalpy from Enthalpy of Formation, Atomization (Metal), and Atomization (Non-metal). The result matters because it helps connect measured amounts to reaction yield, concentration, energy change, rate, or equilibrium.

Ensure stoichiometry is correct: if the formula is MX₂, ensure you double the EA and use appropriate atomization values. Lattice enthalpy and enthalpy of formation are almost always negative (exothermic). Ionization energy is always positive (endothermic), while electron affinity is usually negative for the first electron. Check that all values use consistent units, typically kJ/mol.

References

Sources

Atkins' Physical Chemistry
IUPAC Gold Book
Wikipedia: Born-Haber cycle
P. W. Atkins, J. de Paula, J. Keeler, Atkins' Physical Chemistry, 11th ed., Oxford University Press, 2018
Atkins' Physical Chemistry, 11th Edition
IUPAC Gold Book (Compendium of Chemical Terminology)
OCR A-Level Chemistry A — Energetics (Born–Haber cycles)

Overview

Variables

Derivation

Use Hess’s Law Around the Cycle:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Enthalpy of Atomization

Frequently Asked Questions

Sources