The Born-Haber cycle is a hypothetical series of steps that represent the formation of an ionic compound
from its constituent elements in their standard states.
For example, we'll look at the formation of solid sodium chloride from solid sodium and chlorine gas.
The standard enthalpy of formation of solid sodium chloride is -411 kilojoules.
According to Hess's Law, this reaction can be viewed as a series of five steps.
In our first step, we'll change the solid sodium metal into gaseous sodium atoms.
The enthalpy change for this process is the enthalpy of sublimation of sodium
since sublimation is changing directly from a solid to a gas.
In our second step, we'll break the chlorine- chlorine single bond in Cl2 to produce two chlorine gas atoms.
The energy required to break the bond is called the bond energy.
Since we only need one half of the chlorine molecule from our enthalpy of formation equation,
we multiply this equation times ½, so the enthalpy change for this step is 1/2 of the bond energy for a chlorine-chlorine single bond.
In our next step, we can remove an electron from the gaseous sodium atom.
The energy required to remove an electron from a gaseous atom is the ionization energy.
So our enthalpy change for the third step is equal to the first ionization energy of sodium.
In our next step, we'll add an electron to the gaseous chlorine atom.
The energy change when we add an electron to a gaseous atom is the electron affinity.
So the enthalpy change for this step is the first electron affinity of chlorine.
Our last step is to combine the two gaseous ions to form a crystal lattice.
The energy released when gaseous ions combine in a crystal lattice to form an ionic solid is called the lattice energy.
So the enthalpy change for this step is the lattice energy of sodium chloride.
When we add all of these steps together we get the enthalpy of formation equation for sodium chloride,
so according the Hess's Law, if we add together the enthalpy changes for each of the steps,
we will get the overall enthalpy changed for the reaction.
The first three steps that were listed are endothermic;
the process of sublimation, breaking a bond, and removing an electron all require energy added to the system.
Adding electron to a gaseous atom is generally exothermic and the lattice energy is always highly exothermic.
Let's practice calculating the lattice energy for magnesium chloride.
We're given the sublimation energy for magnesium, first and second ionization energies of magnesium,
bond energy for the chlorine-chlorine single bond, the electron affinity of chlorine,
and the standard enthalpy of formation for magnesium chloride.
We'll start by writing the equation for the standard enthalpy of formation for magnesium chloride.
The standard state of magnesium metal is a solid and for chlorine is a diatomic gas, Cl2.
We have coefficients of one to balance the equation.
We want to use Hess's law to combine a series of steps to obtain the overall reaction for the formation of magnesium chloride
from its constituent elements in their standard states.
Our quation for the sublimation of magnesium is magnesium solid is converted to magnesium gas.
Since we have one mole of magnesium solid in the overall reaction, we multiply this equation and its enthalpy change by one.
The next information we're given is the first ionization energy of magnesium.
This is the energy needed to remove an electron from a gaseous magnesium atom.
In our previous equation, we produced one mole of magnesium gas which is not in the overall equation.
We need to cancel out one mole of magnesium gas product by adding one mole of magnesium gas reactant.
So we multiply this equation and its enthalpy change by one.
Our next equation is the second ionization energy of magnesium.
This is the energy needed to remove the second electron from a gaseous magnesium ion.
In our previous equation, we produced one mole of magnesium plus gaseous ions,
so we need to cancel it out by adding one mole of magnesium plus gaseous ions as a reactant.
Thus, we multiply this equation and its enthalpy change by one.
The next information we're given is the bond energy for a chlorine-chlorine single bond.
This is the energy required to break the bond in the chlorine gas molecule to produce two chlorine gas atoms.
Since we have one Cl2 gas in our standard enthalpy of formation equation,
we can add this equation as it is to get one mole of Cl2 gas as a reactant.
The next information we're given is the electron affinity for chlorine,
which is the energy released when we add an electron to a gaseous chlorine atom.
From our previous equation, we produced two moles of gaseous chlorine atoms
so we need to multiply this equation and its change in enthalpy by two
so we can cancel out the two moles of gaseous chlorine atoms as products.
The last step is to combine our gaseous magnesium ion with our two gaseous chloride ions to produce solid magnesium chloride.
The enthalpy change for this process is the lattice energy.
If we add all of these equations together, we should get our overall reaction
for the standard enthalpy of formation of magnesium chloride.
According to Hess's Law, the sum of the enthalpy changes for the steps equals the overall enthalpy change of the reaction.
Plugging in our numbers for each step, we can solve for the lattice energy.
Completing the calculation, we should get -2522 kilojoules.
We can also consider our trends in lattice energy.
Lattice energies become more exothermic with decreasing ionic radius
because smaller ions are brought closer together, thus releasing more energy
when these oppositely charged ions come together.
Lattice energies also become more exothermic with increasing magnitude of ionic charge.
So a two plus ion combining with a one minus ion gives off more energy than a one plus ion combining with a one minus ion.
Let's practice applying our trends for lattice energies.
We want to put these ionic compounds in order of increasing magnitude of lattice energies.
The most important factor in the magnitude of lattice energy is the charge on the ions.
So our first step is to determine the charges of the ions forming each ionic compound.
When we indicate the charge, we're looking at the charge of an individual cation or anion.
Our most exothermic lattice energy will come from the ionic compound with the largest charges.
Our least exothermic lattice energy will come from the ionic compound with the smallest charges.
Gallium oxide has a +3 charge on gallium and a -2 charge on oxygen, so gallium oxide is the most exothermic.
Next comes gallium chloride with a +3 charge and a -1 charge.
Next as calcium chloride with a +2 charge and a -1 charge.
And when comparing sodium chloride and potassium bromide, we have +1 and -1 for charges in both compounds.
The next thing we look at is the size of the ions.
Smaller ions release more energy when brought together and ionic size decreases going up a column of the periodic table.
So sodium chloride is more exothermic because sodium is smaller than potassium and chloride is smaller than bromide.