LATTICE ENTHALPY (LATTICE ENERGY)

This page introduces lattice enthalpies (lattice energies) and Born-Haber cycles.

Lattice enthalpy and lattice energy are commonly used as if they mean exactly the same thing - you will often find both terms used within the same textbook article or web site, including on university sites.

In fact, there is a difference between them which relates to the conditions under which they are calculated. However, the difference is small, and negligible compared with the differing values for lattice enthalpy that you will find from different data sources.

Unless you go on to do chemistry at degree level, the difference between the two terms isn't likely to worry you.

In the Born-Haber cycles below, I have used numbers which give a consistent answer, but please don't assume that they are necessarily the most accurate ones. If you are doing a course for 16 - 18 year olds, none of this really matters - you just use the numbers you are given.

If you use my chemistry calculations book, you will find a slightly different set of numbers. These came from the Chemistry Data Book edited by Stark and Wallace, published by John Murray. Values from this now fairly old book often differ slightly from more recent sources.

What is lattice enthalpy?

Two different ways of defining lattice enthalpy

There are two different ways of defining lattice enthalpy which directly contradict each other, and you will find both in common use. In fact, there is a simple way of sorting this out, but many sources don't use it.

I will explain how you can do this in a moment, but first let's look at how the problem arises.

Lattice enthalpy is a measure of the strength of the forces between the ions in an ionic solid. The greater the lattice enthalpy, the stronger the forces.

Those forces are only completely broken when the ions are present as gaseous ions, scattered so far apart that there is negligible attraction between them. You can show this on a simple enthalpy diagram.

For sodium chloride, the solid is more stable than the gaseous ions by 787 kJ mol^-1, and that is a measure of the strength of the attractions between the ions in the solid. Remember that energy (in this case heat energy) is given out when bonds are made, and is needed to break bonds.

So lattice enthalpy could be described in either of two ways.

• You could describe it as the enthalpy change when 1 mole of sodium chloride (or whatever) was formed from its scattered gaseous ions. In other words, you are looking at a downward arrow on the diagram.

  In the sodium chloride case, that would be -787 kJ mol^-1.

• Or, you could describe it as the enthalpy change when 1 mole of sodium chloride (or whatever) is broken up to form its scattered gaseous ions. In other words, you are looking at an upward arrow on the diagram.

  In the sodium chloride case, that would be +787 kJ mol^-1.

Both refer to the same enthalpy diagram, but one looks at it from the point of view of making the lattice, and the other from the point of view of breaking it up.

Unfortunately, both of these are often described as "lattice enthalpy".

This is an absurdly confusing situation which is easily resolved. I suggest that you never use the term "lattice enthalpy" without qualifying it.

• You should talk about "lattice dissociation enthalpy" if you want to talk about the amount of energy needed to split up a lattice into its scattered gaseous ions.

  For NaCl, the lattice dissociation enthalpy is +787 kJ mol^-1.

• You should talk about "lattice formation enthalpy" if you want to talk about the amount of energy released when a lattice is formed from its scattered gaseous ions.

  For NaCl, the lattice formation enthalpy is -787 kJ mol^-1.

That immediately removes any possibility of confusion.

So . . .

Factors affecting lattice enthalpy

The two main factors affecting lattice enthalpy are the charges on the ions and the ionic radii (which affects the distance between the ions).

The charges on the ions

Sodium chloride and magnesium oxide have exactly the same arrangements of ions in the crystal lattice, but the lattice enthalpies are very different.

You can see that the lattice enthalpy of magnesium oxide is much greater than that of sodium chloride. That's because in magnesium oxide, 2+ ions are attracting 2- ions; in sodium chloride, the attraction is only between 1+ and 1- ions.

The radius of the ions

The lattice enthalpy of magnesium oxide is also increased relative to sodium chloride because magnesium ions are smaller than sodium ions, and oxide ions are smaller than chloride ions.

That means that the ions are closer together in the lattice, and that increases the strength of the attractions.

You can also see this effect of ion size on lattice enthalpy as you go down a Group in the Periodic Table.

For example, as you go down Group 7 of the Periodic Table from fluorine to iodine, you would expect the lattice enthalpies of their sodium salts to fall as the negative ions get bigger - and that is the case:

Attractions are governed by the distances between the centres of the oppositely charged ions, and that distance is obviously greater as the negative ion gets bigger.

And you can see exactly the same effect as you go down Group 1. The next bar chart shows the lattice enthalpies of the Group 1 chlorides.

Calculating lattice enthalpy

It is impossible to measure the enthalpy change starting from a solid crystal and converting it into its scattered gaseous ions. It is even more difficult to imagine how you could do the reverse - start with scattered gaseous ions and measure the enthalpy change when these convert to a solid crystal.

Instead, lattice enthalpies always have to be calculated, and there are two entirely different ways in which this can be done.

You can can use a Hess's Law cycle (in this case called a Born-Haber cycle) involving enthalpy changes which can be measured. Lattice enthalpies calculated in this way are described as experimental values.

Or you can do physics-style calculations working out how much energy would be released, for example, when ions considered as point charges come together to make a lattice. These are described as theoretical values. In fact, in this case, what you are actually calculating are properly described as lattice energies.

Experimental values - Born-Haber cycles

Standard atomisation enthalpies

Before we start talking about Born-Haber cycles, there is an extra term which we need to define. That is atomisation enthalpy, ΔH°_a.

You are always going to have to supply energy to break an element into its separate gaseous atoms.

All of the following equations represent changes involving atomisation enthalpy:

Notice particularly that the "mol^-1" is per mole of atoms formed - NOT per mole of element that you start with. You will quite commonly have to write fractions into the left-hand side of the equation. Getting this wrong is a common mistake.

Born-Haber cycles

I am going to start by drawing a Born-Haber cycle for sodium chloride, and then talk it through carefully afterwards. You will see that I have arbitrarily decided to draw this for lattice formation enthalpy. If you wanted to draw it for lattice dissociation enthalpy, the red arrow would be reversed - pointing upwards.

Focus to start with on the higher of the two thicker horizontal lines. We are starting here with the elements sodium and chlorine in their standard states. Notice that we only need half a mole of chlorine gas in order to end up with 1 mole of NaCl.

The arrow pointing down from this to the lower thick line represents the enthalpy change of formation of sodium chloride.

The Born-Haber cycle now imagines this formation of sodium chloride as happening in a whole set of small changes, most of which we know the enthalpy changes for - except, of course, for the lattice enthalpy that we want to calculate.

• The +107 is the atomisation enthalpy of sodium. We have to produce gaseous atoms so that we can use the next stage in the cycle.

• The +496 is the first ionisation energy of sodium. Remember that first ionisation energies go from gaseous atoms to gaseous singly charged positive ions.

• The +122 is the atomisation enthalpy of chlorine. Again, we have to produce gaseous atoms so that we can use the next stage in the cycle.

• The -349 is the first electron affinity of chlorine. Remember that first electron affinities go from gaseous atoms to gaseous singly charged negative ions.

• And finally, we have the positive and negative gaseous ions that we can convert into the solid sodium chloride using the lattice formation enthalpy.

Now we can use Hess's Law and find two different routes around the diagram which we can equate.

As I have drawn it, the two routes are obvious. The diagram is set up to provide two different routes between the thick lines.

So, here is the cycle again, with the calculation directly underneath it . . .

-411 = +107 + 496 + 122 - 349 + LE

LE = -411 - 107 - 496 - 122 + 349

How would this be different if you had drawn a lattice dissociation enthalpy in your diagram? (Perhaps because that is what your syllabus wants.)

Your diagram would now look like this:

The only difference in the diagram is the direction the lattice enthalpy arrow is pointing. It does, of course, mean that you have to find two new routes. You can't use the original one, because that would go against the flow of the lattice enthalpy arrow.

This time both routes would start from the elements in their standard states, and finish at the gaseous ions.

-411 + LE = +107 + 496 + 122 - 349

LE = +107 + 496 + 122 - 349 + 411

LE = +787 kJ mol^-1

Once again, the cycle sorts out the sign of the lattice enthalpy for you.

Theoretical values for lattice energy

Let's assume that a compound is fully ionic. Let's also assume that the ions are point charges - in other words that the charge is concentrated at the centre of the ion. By doing physics-style calculations, it is possible to calculate a theoretical value for what you would expect the lattice energy to be.

And no - I am not being careless about this! Calculations of this sort end up with values of lattice energy, and not lattice enthalpy. If you know how to do it, you can then fairly easily convert between the two.

There are several different equations, of various degrees of complication, for calculating lattice energy in this way. You won't be expected to be able to do these calculations at this level, but you might be expected to comment on the results of them.

There are two possibilities:

• There is reasonable agreement between the experimental value (calculated from a Born-Haber cycle) and the theoretical value.

  Sodium chloride is a case like this - the theoretical and experimental values agree to within a few percent. That means that for sodium chloride, the assumptions about the solid being ionic are fairly good.

• The experimental and theoretical values don't agree.

  A commonly quoted example of this is silver chloride, AgCl. Depending on where you get your data from, the theoretical value for lattice enthalpy for AgCl is anywhere from about 50 to 150 kJ mol^-1 less than the value that comes from a Born-Haber cycle.

  In other words, treating the AgCl as 100% ionic underestimates its lattice enthalpy by quite a lot.

  The explanation is that silver chloride actually has a significant amount of covalent bonding between the silver and the chlorine, because there isn't enough electronegativity difference between the two to allow for complete transfer of an electron from the silver to the chlorine.

Comparing experimental (Born-Haber cycle) and theoretical values for lattice enthalpy is a good way of judging how purely ionic a crystal is.

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© Jim Clark 2010 (modified July 2013)