Lattice Energy (3) Measuring Theoretical Lattice Energies

Edexcel A level Chemistry (2017)

Topic 13A: Energetics (II): Lattice Energy

Here are two learning objectives:

13A/4. To know that lattice energy provides a measure of ionic bond strength

13A/5. To understand that a comparison of the experimental lattice energy value (from a Born-Haber cycle) with the theoretical value (obtained from electrostatic theory) in a particular compound indicates the degree of covalent bonding

Measuring theoretical lattice energies

The Born Haber experimental cycle may give us a measure of the lattice energy but the value needs to have a theoretical underpinning. 

In other words, what is the model of bonding used to provide a lattice energy?

If we get good agreement between the theoretical model for lattice energy and the experimental value of the lattice energy of an actual compound then it is likely that the model fits the real compound. 

It means that the assumptions made to create the theoretical value from the model are reasonable assumptions about . 

Here goes then with the theoretical model used to calculate lattice energy

First let’s assume that there are two point charges separated by a distance r.

.← r →.

+          ^—

there will be a force f  between these point charges which is given by Coulomb’s law:

where e is the size of the charge on the electron.

Now the potential energy V needed to separate these charges to an infinite distance is given by:

The sign is negative because when the ions are infinitely separated the potential energy between then is zero.  

However in a crystal of salt there is not just one pair of oppositely charged ions, there are billions of pairs of ions.

Just consider one ion X in a line of ions each separated by distance r:

.← r →.← r →.← r →.← r →.← r →.

+           ^—            +            ^—           +          ^—

              X

Then the energy of an ion such as X, because of the presence of the other ions that also exert forces of attraction on it, is:

But ions exist not in 2D but 3D environments. 

Crystal structures are large networks of ions held in place by electrostatic forces of attraction. 

Take the sodium chloride structure as a prime example.

Each ion has six near neighbours, twelve next nearest and so on…. as the illustration shows:

So for the potential energy of the ion in grey Vg 

is given by:

where r is the distance between the centres of nearest neighbour ions. 

The quantity is brackets needs to be summed to infinity.

The value is determined by the arrangement of the ions so each crystal form (NaCl, CsCl, fluorite, zinc blende etc.) will have its own constant M.

This is called the Madelung constant. 

The following table shows crystal structures and their Madelung constant.

Structure	Illustration	Madelung constant
NaCl		1.748
CsCl		1.763
CaF2		2.519
Zinc blende		1.638
Wurtzite		1.641

In summary, the expression for the coulombic forces within a crystal group is given by:

where the charges on the ions are represented by z and the interionic distance is r

If only the coulombic forces were to be considered then the crystal structure would collapse under the se strong attractive forces. 

We must also realise that there are repulsive forces at work and the two forces in tension allow the crystal structure to stabilise. 

It might seem unusual to think of two oppositely charged ions repelling each other but we need to see that if the two ions were to come very close together their electron shells would eventually repel each other. 

We can trace the two forces at work in this illustration:

At very short distances, this repulsive force rises rapidly with distance.

The coulombic force of attraction changes as a function of 1/r whereas you can see that the interionic repulsive force alters as a function of 1/rⁿ  where n is usually taken to be about 12.

Allowing then for this additional term covering the repulsive forces in the crystal form, the expression for the lattice energy per mole becomes:

But note the assumptions in this model of ionic bonding.

The ions are point charges, the electron shells of the ions do not interact with each other and the charge on each ion is an integer not a fractional charge. 

Now as we can see these assumptions are not the case in many supposedly ionic compounds and that is because the experimentally determined lattice energy turns out to be higher (that is more exothermic) than the theoretical value calculated from this model and its equation. 

You can find a discussion of the experimental lattice energies and theoretical lattice energies here.

Lattice Energy (2) Polarisation of ions

 Edexcel A level Chemistry (2017)

Topic 13A: Energetics (2): Lattice Energy

Here are the learning objectives relating to bond polarization:

13A/4. To know that lattice energy provides a measure of ionic bond strength.

13A/5. To be able to understand that a comparison of the experimental lattice energy value (from a Born-Haber cycle) with the theoretical value (obtained from electrostatic theory) in a particular compound indicates the degree of covalent bonding

13A/6. To be able to understand the meaning of polarization as applied to ions

13A/7. To know that the polarizing power of a cation depends on its radius and charge

13A/8. To know that the polarizability of an anion depends on its radius and charge

Consequences from the calculation of lattice energies:

Effects of charge and size of the ions involved:

If the experimentally determined lattice energy from a Born-Haber cycle provides a measure of the strength of an ionic bond then we would expect two things to be true:

First, it should be true that ions with double the charge would have higher lattice energies:

Comparing the lattice energy of sodium chloride and magnesium oxide we find this result:

ΔHº_L [MgO] = –3850 kJ.mol^–1        ΔHº_L [NaCl] = –786 kJ.mol^–1

Second it should also be true that compounds containing smaller ions have a more exothermic lattice energy than compounds containing larger ions.

And as we can see lithium fluoride with the smallest group1 and group 7 ions has a larger lattice energy than potassium fluoride where the positive ion is larger.

ΔHº_L [LiF] = –1037 kJ.mol^–1         ΔHº_L [KF] = –821 kJ.mol^–1

Comparing experimental with theoretical lattice energies:

The critical thing to do then is to compare the experimental lattice energy with that value determined from a theoretical model of an ionic compound.

The theoretical model from electrostatic theory makes certain assumptions about the ionic compound.

The ions are assumed to be perfect spheres and the ions are assumed to in no way interact with each other their electron shells being quite separate and distinct from each other.

In other words, it is assumed that there has a been a complete and irreversible exchange of outer shell electrons between the two elements in the compound so that one ion is negatively charged and the other ion positively charged and the charges are whole values. 

So when comparisons are made between experimental and theoretical lattice energies some interesting outcomes are observed.

Consider

Compound	Born Haber lattice energy (kJ.mol–1)	Theoretical lattice energy (kJ.mol–1)	% difference in Lattice energies
Lithium iodide	759	738	2.76
Sodium chloride	780	770	1.28
Potassium bromide	679	674	0.74
Silver iodide	889	778	12.5
Copper iodide	963	833	13.5

Now if there is a large difference between the theoretical and experimental values of the lattice energy, this suggests that the ionic model does not fit the way the ions bond to each other.  

A higher experimental lattice energy means that the bonding is stronger than would be expected from the ionic model alone.

Therefore, if the ionic model does not fit well then it could suggest that the bonding is stronger because there is some interaction between the electron shells of the ions—a degree of covalent bonding.

Electron density maps suggest this to be the case as can be seen below in this example. 

Even in sodium chloride, a “completely ionic” compound, we can that there is some interaction of one ion with another as the map reveals.

Clearly, the situation of ions interacting with one another is more significant in the case of those compounds where the difference between experimental and theoretical lattice energy is great.

In such compounds as silver iodide or copper iodide the difference is over 10%.

At this point, we can begin to model what might be happening in these and related compounds.

The breakdown of the ionic model occurs because one ion is much smaller than the other.

More specifically, the breakdown in the ionic model for some compounds occurs when the positive ion is much smaller than the negative ion.

Take the examples above of silver iodide and copper(II) iodide.

Here are the ionic radii: 

silver:  Ag+ 0.115nm   iodide:  I^—  0.215nm

copper: Cu²⁺  0.073nm  iodide:I^—  0.215nm

Both have much smaller positive ions than negative ions.

The suggestion is that the small positive ion polarises the larger negative ion.

Also the suggestion is that this process of polarisation is more effect at distortion of the ionic model if the positive ion is small and highly charged and the negative ion is large with a low charge.

The polarising power of a positive ion (cation) depends on its size and charge and is larger the larger the charge and the smaller the radius.

The polarisability of the negative ion (anion) also depends on its size and charge.  The larger the anion radius and the lower its charge the more polarisable it is.

We see in the graphic below a stylised picture of how the polarisation of an anion by a cation can happen.

But it is merely a very stylised, approximate attempt at revealing this effect.