Lattice Energy (1) Calculating Lattice Energy using Born-Haber cycles

Edexcel A level Chemistry (2017)

Topic 13A: Energetics (II): Lattice Energy

Here are the first learning objectives:

13A/1. To be able to define lattice energy as the energy change when one mole of an ionic solid is formed from its gaseous ions

13A/2. To be able to define the terms:

i) enthalpy change of atomisation, ΔatH 


ii) electron affinity 
E_aff

13A/3. To be able to construct Born-Haber cycles and carry out related calculations

Constructing Born Haber Cycles

The question that a Born Haber Cycle tries to answer is this:  Is it possible to measure the bond strength of an ionic bond?

Here is the ionic lattice for sodium chloride:

Is it possible to separate out the attraction of one sodium ion (in green) for one chloride ion (in orange) ?

When we look closely at the lattice we can see that the sodium ion is actually surrounded by six chloride ions and similarly the chloride ion is surrounded by six sodium ions!

This gives sodium chloride a coordination number of 6:6 but it also means that to measure the strength of a mole of sodium chloride ionic bonds is not possible.

The best we can do is measure the energy change when a mole of sodium ions and a mole of chloride ions form a mole of sodium chloride.

In this process energy is given out.

The bonds formed between ions are very strong and much energy is released.

Under standard conditions the energy released when one mole of a crystal lattice is formed from its constituent ions in the gaseous state is called the lattice energy ΔHlatt. 

We can represent this using this equation:

Na⁺ (g)    +     Cl^—  (g)     ⟶     Na⁺Cl^— (s)  

Confusion over the definition:

I ought to say at this point that there is some confusion over the definition of lattice energy.

You will find that some chemistry texts present the lattice energy as an endothermic value: the energy required to dissociate a mole of ions and separate them as far apart as possible so that they no longer interact with each other. 

The equation for this process is then:

Na⁺Cl^—(s)     ⟶       Na⁺(g)    +     Cl^—(g)

The numerical value of both lattice energy terms is the same only the sign of each value is different since one is exothermic and the other endothermic.

It is very important then at A level or high school/college level you clarify precisely how your course defines lattice energy.

I am posting on the English Edexcel A level course and that course adopts an exothermic definition of lattice energy.

Therefore, my approach to building an energy cycle to determine lattice energy assumes that we are discussing the calculation of an exothermic value.

It is not possible to measure this energy value by direct experiment but we can be obtain the value using an energy cycle based on Hess’s Law. 

This cycle is known as a Born-Haber cycle.

The Born-Haber cycle is formed from experimentally measurable enthalpy change values. 

The cycle breaks the process down into several steps.

We begin with a mole of sodium atoms and half a mole of chlorine molecules.

If we start with this material then

A) The first thing that happens is the sodium is atomised and ionised, processes that require the enthalpies of both atomisation and ionisation.

B) Then the chlorine is atomised and there is the addition of a mole of electrons to the mole of chlorine atoms forming a mole of chloride ions.  These processes require the electron affinity of chlorine and its atomisation energy.

C) If the gaseous sodium and chloride ions are then combined the result is a large exothermic energy change.  This energy change is the lattice energy as we have defined it above.

Here is the list of the values necessary to build a Born-Haber cycle.

1.   Atomisation energy  Δ_atH:

The standard enthalpy change of atomization of an element is the enthalpy change that takes place when one mole of gaseous atoms is made from the element in its standard physical state under standard conditions. 

Na(s)     ⟶   Na(g) 

½Cl₂(g)  ⟶   Cl(g)

2.   Successive Molar Ionization energy Emj

Thus is the energy needed to remove a mole of the jth electron from a mole of gaseous atoms or ions of an element specifically

Na(g)    ⟶    Na⁺(g)    +   e—

3.   Electron affinity ΔU is the molar internal energy change when a mole of atoms of an element in the gaseous state gains a mole of electrons, specifically

Cl(g)    +    e^—      ⟶     Cl^—(g)

4. Enthalpy of formation of the salt involved ΔH^⦵_formation.  This is the energy change when one mole of the compound is formed from its elements in their standard states under standard conditions.

Na(s)    +    ½Cl₂(g)     ⟶     NaCl(s)

Now there are many different ways in which texts and scholars have drawn up Born Haber cycles.

Each textbook and/or examination course seems to have its own brand for the process.

Basically you can take your pick or better check with your Chemistry examination specification or Chemistry exam provider for a take on their version of the Born-Haber cycle to follow.

You can probably find the version of the Born Haber cycle you need from a past paper question on the subject. 

Here are five different examples I found on the internet:

So let’s build up a Born Haber cycle for sodium chloride in stages.

We start with the elements in their standard states and atomise the sodium:  Na(s)  ⟶   Na(g)   Δ_atmΗ^⦵  =  +107.3 kJ.mol^—1

Next we ionise the sodium:  Na(g) ⟶  Na⁺(g)  +  e^—   E_m1 = +496 kJ.mol^—1

Then we add the standard enthalpy of atomisation of chlorine: 

½Cl₂(g) ⟶ Cl(g)      Δ_atmΗ^⦵  =  +121.7 kJ.mol^—1

All these values are endothermic.

Next we add in the electron affinity of chlorine:  Cl(g)   +  e^—  ⟶ Cl^— (g)                

E_aff   = — 348.8 kJ.mol^—1

Last but one we put in the lattice energy:  Na⁺ (g)  +  Cl^— (g) ⟶ NaCl (s)

This value we do not know directly.

And finally to complete the cycle we add in the enthalpy of formation of sodium chloride:   Na(s)    +   ½Cl₂(g)    ⟶    NaCl(s)

Δ_formationΗ^⦵  =  —411.2 kJ.mol^—1

These final three values are exothermic.

The final cycle can look like this if we add in the values of the energy changes mentioned above:

Now to calculate the lattice energy all we need to do is add up the numerical values (that is ignoring the signs) on the left hand side of the cycle and subtract the numerical value for the right hand electron affinity.

So the lattice energy = 121.7+ 496+107.3+411.2 — 348.4 = 787.8

Note the direction of the arrow which means that the value for the lattice energy of sodium chloride is

ΔH_latt = —787.8 kJ.mol^—1

All that remains is for me to challenge you to see if you can determine the lattice energy of compounds such as potassium iodide (KI) and magnesium oxide (MgO).