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 . . .
The lattice dissociation enthalpy is the enthalpy change needed to convert 1 mole of solid crystal into its scattered gaseous ions. Lattice dissociation enthalpies are always positive.
The lattice formation enthalpy is the enthalpy change when 1 mole of solid crystal is formed from its scattered gaseous ions. Lattice formation enthalpies are always negative.
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