The Born-Haber Cycle: Unlocking the Energy of Ionic Bonds
The Building Blocks: Key Energy Terms
Before we can build the cycle, we need to understand the individual energy changes involved. Think of these as the ingredients for our recipe.
| Term | Symbol | Definition | Endo/Exothermic |
|---|---|---|---|
| Enthalpy of Formation | $\Delta H_f$ | The enthalpy change when 1 mole of a compound is formed from its elements in their standard states. | Usually Exothermic |
| Atomization/Enthalpy of Formation of Atoms | $\Delta H_{at}$ | The enthalpy required to form 1 mole of gaseous atoms from an element in its standard state. | Endothermic |
| Ionization Energy | $IE$ | The energy required to remove one mole of electrons from one mole of gaseous atoms. | Endothermic |
| Electron Affinity | $EA$ | The energy change when one mole of electrons is added to one mole of gaseous atoms. | Usually Exothermic |
| Lattice Energy | $U$ or $\Delta H_{lat}$ | The energy released when one mole of an ionic solid is formed from its gaseous ions. | Exothermic |
Hess's Law: The Foundation of the Cycle
Imagine you are on the first floor of a building and you want to get to the third floor. You can take the elevator directly, or you can walk to the second floor and then take the stairs to the third. The total energy you expend (or gain) will be the same regardless of the path you take. This is the core idea behind Hess's Law.
Hess's Law states that the total enthalpy change for a reaction is independent of the pathway taken. It depends only on the initial and final states. The Born-Haber cycle applies this law to the formation of an ionic compound. It creates an alternative, multi-step pathway from the elements to the ionic solid, allowing us to calculate the lattice energy, which is the "direct elevator" step that we cannot measure easily.
Constructing the Cycle: A Step-by-Step Journey
Let's build a Born-Haber cycle for sodium chloride (NaCl), common table salt. The overall reaction we care about is:
$Na(s) + \frac{1}{2}Cl_2(g) \rightarrow NaCl(s)$ $\Delta H_f = -411$ kJ/mol
This is the direct, one-step formation. The Born-Haber cycle creates a different path:
- Atomize the solid metal: Convert solid sodium into gaseous sodium atoms. This is the enthalpy of atomization.
$Na(s) \rightarrow Na(g)$ $\Delta H = +108$ kJ/mol - Atomize the non-metal: Break the $Cl_2$ molecule into gaseous chlorine atoms. This is half the bond dissociation energy since we only need half a mole of $Cl_2$.
$\frac{1}{2}Cl_2(g) \rightarrow Cl(g)$ $\Delta H = +122$ kJ/mol - Ionize the metal atom: Remove an electron from the gaseous sodium atom to form a sodium ion ($Na^+$). This is the first ionization energy.
$Na(g) \rightarrow Na^+(g) + e^-$ $\Delta H = +496$ kJ/mol - Add an electron to the non-metal atom: Add an electron to the gaseous chlorine atom to form a chloride ion ($Cl^-$). This is the electron affinity.
$Cl(g) + e^- \rightarrow Cl^-(g)$ $\Delta H = -349$ kJ/mol - Form the ionic lattice: The gaseous $Na^+$ and $Cl^-$ ions come together to form the solid ionic lattice, NaCl(s). This is the lattice energy ($U$), the value we want to find.
$Na^+(g) + Cl^-(g) \rightarrow NaCl(s)$ $\Delta H = U$
According to Hess's Law, the sum of the enthalpy changes for this indirect path must equal the direct enthalpy of formation ($\Delta H_f$).
$\Delta H_f = \Delta H_{at}(Na) + \Delta H_{at}(Cl) + IE(Na) + EA(Cl) + U$
We can rearrange this to solve for the lattice energy, $U$:
$U = \Delta H_f - [\Delta H_{at}(Na) + \Delta H_{at}(Cl) + IE(Na) + EA(Cl)]$
A Practical Application: Calculating the Lattice Energy of NaCl
Now, let's plug the known values from our steps into the equation to find the lattice energy for sodium chloride.
Given Data:
- $\Delta H_f(NaCl) = -411$ kJ/mol
- $\Delta H_{at}(Na) = +108$ kJ/mol
- $\Delta H_{at}(Cl) = +122$ kJ/mol
- $IE(Na) = +496$ kJ/mol
- $EA(Cl) = -349$ kJ/mol
Calculation:
$U = -411 - [(+108) + (+122) + (+496) + (-349)]$ kJ/mol
$U = -411 - (+377)$ kJ/mol
$U = -411 - 377$ kJ/mol
$U = -788$ kJ/mol
The large negative value for lattice energy (-788 kJ/mol) confirms that a tremendous amount of energy is released when the ionic lattice forms. This immense energy is the main reason why ionic compounds like NaCl are so stable and have high melting points.
Important Questions
It is impossible to directly combine gaseous ions in a lab to form a perfect crystal and measure the heat released. The ions would repel each other due to their like charges before they could form an ordered lattice. The Born-Haber cycle provides a clever, indirect method to calculate this crucial value using other measurable quantities.
How does the size and charge of ions affect the lattice energy?
Lattice energy depends heavily on the charges of the ions and the distance between them, as described by Coulomb's Law.
- Charge: Higher charges on the ions lead to a much stronger attraction. For example, the lattice energy of $MgO$ (with $Mg^{2+}$ and $O^{2-}$) is much larger than that of $NaCl$ (with $Na^+$ and $Cl^-$).
- Size: Smaller ions can get closer together, which also increases the attractive force and the lattice energy. For example, $LiF$ has a larger lattice energy than $LiI$ because fluoride ions are much smaller than iodide ions.
The Born-Haber cycle is specifically designed for ionic compounds. It can be applied to any ionic solid, such as potassium bromide ($KBr$), calcium fluoride ($CaF_2$), or magnesium oxide ($MgO$). For covalent compounds like water ($H_2O$) or methane ($CH_4$), the concept of lattice energy doesn't apply in the same way, so the cycle is not used.
Footnote
1 Hess's Law: A law in thermochemistry stating that the total enthalpy change for a reaction is the same regardless of the number of steps the reaction is carried out in.
2 Lattice Energy (U or $\Delta H_{lat}$): The energy released when one mole of an ionic crystal is formed from its constituent gaseous ions.
3 Enthalpy ($\Delta H$): A measurement of the total heat content in a system at constant pressure; the change in enthalpy ($\Delta H$) tells us if a process releases heat (exothermic, negative $\Delta H$) or absorbs heat (endothermic, positive $\Delta H$).
4 Ionization Energy (IE): The minimum energy required to remove the most loosely bound electron from an isolated gaseous atom to form a cation.
5 Electron Affinity (EA): The energy change that occurs when an electron is added to an isolated gaseous atom to form an anion.