Lattice structures — how ions pack together — Chemistry

Before this

This leads to

Properties of ionic compounds
Metallic bonding · coming soon

From one pair of ions to trillions

The last lecture followed one sodium atom and one chlorine atom. One electron moved, and a Na⁺ ion and a Cl⁻ ion attracted each other. It ended with a question: what happens when huge numbers of these ions pack together?

The numbers really are huge. A single grain of salt about half a millimetre across contains roughly $3 \times 1 0^{18}$ sodium ions and the same number of chloride ions. (That comes from salt's density, 2.165 g/cm³.) They do not sit in a random heap. They are arranged in a pattern so regular that you can see it with your own eyes: look closely at grains of salt, and many of them are tiny cubes.

This lecture explains that pattern, why the ions choose it, and why it makes salt so hard to melt.

A giant ionic lattice

Think about one Na⁺ ion in a crowd of ions. It attracts every Cl⁻ ion near it, not just the one that took its electron. So the best arrangement surrounds each positive ion with as many negative ions as will fit, and each negative ion with positive ions. Repeat this in all three directions, and you get a giant ionic lattice: a regular, repeating, three-dimensional arrangement of ions that continues to the edge of the crystal.

In sodium chloride, each Na⁺ ion has six Cl⁻ ions as its nearest neighbours: one above, one below, one to the left, one to the right, one in front and one behind. Each Cl⁻ ion, in the same way, has six Na⁺ neighbours.

A small block of the sodium chloride lattice. The ions alternate in every direction. The dark green Na⁺ ion in the centre is joined to its six nearest Cl⁻ ions. The ions are drawn smaller than real size so that the inside can be seen.

The number of nearest neighbours of oppositely charged ions is called the coordination number. In sodium chloride, it is 6 for both ions. We write this as 6 : 6.

Giant ionic lattice

An ionic compound is a giant lattice of positive and negative ions, held together by electrostatic attraction in every direction. There are no separate molecules. The formula gives only the ratio of the ions.

Stop and think

In the lattice, which Cl⁻ ion is a particular Na⁺ ion bonded to: the one that took its electron?

Show the answer

None in particular, and all six equally. The Na⁺ ion attracts each of its six Cl⁻ neighbours with the same force, because they are all the same distance away. Which chlorine atom happened to take the electron makes no difference: electrons are identical, and the ion does not remember where its charge came from. This is why "a molecule of NaCl" makes no sense. The formula NaCl means "one Na⁺ for every Cl⁻", nothing more.

Why this arrangement?

Opposite charges attract, but like charges repel. A good arrangement must bring opposite charges close while keeping like charges apart. Look at one layer of the lattice.

One layer of the sodium chloride lattice. Each Na⁺ ion is 0.282 nm from its nearest Cl⁻ ions, but about 0.399 nm (√2 times as far) from its nearest Na⁺ ions. The 0.282 nm is half the measured edge of the repeating cube, 0.5640 nm.

Every nearest neighbour has the opposite charge. The nearest ions with the same charge are diagonally across a square, which is $2 \approx 1.41$ times as far away. The attractions win, because they are both closer and more numerous. That is it. That is the whole idea. A lattice packs opposite charges close together and keeps like charges further apart, in every direction at once.

You can check the size of the grain from this distance too. Along an edge of half a millimetre, there are about $0.5 mm \div 0.282 nm \approx 1.77 \times 1 0^{6}$ ions. A cube of them holds about $(1.77 \times 1 0^{6})^{3} \approx 5.6 \times 1 0^{18}$ ions: half sodium, half chloride. That matches the density estimate.

Not every ionic compound has the sodium chloride arrangement. The number of neighbours that fit depends on the sizes of the ions. In caesium chloride, the caesium ion is much larger than the sodium ion, and eight chloride ions fit around it, so its coordination is 8 : 8. And when the formula is not 1 : 1, the two coordination numbers cannot be equal. In magnesium chloride, MgCl₂, each Mg²⁺ ion has 6 Cl⁻ neighbours, but each Cl⁻ ion has only 3 Mg²⁺ neighbours. There are twice as many chloride ions, so each one needs only half as many neighbours.

Lattice energy

In the last lecture, the energy diagram for sodium chloride had one large downward step: about 786 kJ/mol released when the separate gaseous ions came together as a solid. That step has a name.

Lattice energy

The lattice energy of an ionic compound is the energy released when 1 mol of the solid forms from its separate gaseous ions. The same amount of energy is needed to pull the solid apart into gaseous ions again.

Books disagree about its sign: some write it as a negative number (energy released), some as a positive number (energy needed to separate). In this course, we give its size, in kJ/mol. Signs come later, in the lectures on enthalpy. All the values below come from one table (LibreTexts, Chemistry: Structure and Properties, section 10.11), so they can be compared fairly.

The size of the lattice energy depends on how strongly the ions attract. That depends on two things:

The charges on the ions. Larger charges attract more strongly. Mg²⁺ has twice the charge of Na⁺.
The sizes of the ions. Smaller ions can get closer together, and closer charges attract more strongly. Down a group, ions get larger, as the lecture on atomic radius explained.

Before you explore the data, make a prediction.

Predict — what does doubling the charge do?

The lattice energy of sodium chloride, NaCl, is 786 kJ/mol. In magnesium chloride, MgCl₂, the metal ion has twice the charge. Roughly how large is the lattice energy of MgCl₂? Choose one, then read the explanation.

The lattice energies, from one table (LibreTexts, Tro, section 10.11):

NaCl: 786 kJ/mol.
MgCl₂: 2526 kJ/mol, about 3.2 times as large.

Doubling the charge on the metal ion more than triples the lattice energy. Each attraction involving Mg²⁺ is stronger, the Mg²⁺ ion is smaller than Na⁺, and each mole of MgCl₂ contains three ions instead of two.

Now explore the whole table: four metal ions and four halide ions, sixteen compounds in all.

Explore — lattice energies of sixteen compounds

Choose a metal ion and a halide ion. The bar for that compound is highlighted among all sixteen. First keep the metal and change the halide. Then keep the halide and change the metal.

Lattice energies in kJ/mol (LibreTexts, Tro, section 10.11):

Metal ion	F⁻	Cl⁻	Br⁻	I⁻
Li⁺	1036	853	807	757
Na⁺	923	786	747	704
K⁺	821	715	682	649
Mg²⁺	2957	2526	2440	2327

Along each row, the halide ion gets larger and the lattice energy falls. Down the first three rows, the metal ion gets larger and the lattice energy falls. The Mg²⁺ row, with a doubly charged ion, is about three times higher than the rest.

Did you see both trends? With the metal ion fixed, the lattice energy falls steadily from fluoride to iodide, as the halide ions get larger. With the halide fixed, it falls from lithium to potassium. The biggest jump of all comes from charge: every magnesium compound is around three times higher than its sodium partner.

Strong lattices are hard to melt

To melt an ionic solid, the ions must be shaken loose from their fixed places in the lattice. The stronger the attractions, the more energy that takes, and the higher the temperature needed. So, as a general pattern, a larger lattice energy goes with a higher melting point.

Compound	Ions	Melting point
NaCl	Na⁺, Cl⁻	801 °C
NaF	Na⁺, F⁻ (smaller anion)	996 °C
MgO	Mg²⁺, O²⁻ (both doubly charged)	about 2800 °C

(Melting points from LibreTexts, General Chemistry: An Atoms First Approach, section 4.2. It gives MgO as 2825 °C, and other sources differ slightly.) Magnesium oxide, with two doubly charged, small ions, has a lattice energy of about 3800 kJ/mol in the same source. It stays solid at temperatures that melt most metals.

Worked examples

Worked Example 1 — Ordering four compounds

Put NaCl, NaF, MgCl₂ and MgF₂ in order of increasing lattice energy. Then check with the table.

Charge first: the magnesium compounds have a doubly charged metal ion, so they will be far above the sodium compounds. Then size: F⁻ is smaller than Cl⁻, so each fluoride beats the matching chloride.

Prediction: NaCl < NaF < MgCl₂ < MgF₂.

Check: 786 < 923 < 2526 < 2957 kJ/mol. The prediction is right. Notice how much bigger the charge effect is than the size effect.

Worked Example 2 — Coordination numbers in MgCl₂

In magnesium chloride, each Mg²⁺ ion has 6 Cl⁻ neighbours. Show that each Cl⁻ ion must have 3 Mg²⁺ neighbours.

Count the Mg–Cl contacts in a large piece of crystal containing $N$ magnesium ions. Each Mg²⁺ has 6 contacts, so there are $6 N$ contacts in all.

The formula MgCl₂ says there are $2 N$ chloride ions. The same $6 N$ contacts are shared among them, so each Cl⁻ has $6 N \div 2 N = 3$ magnesium neighbours.

Check: the ratio of the coordination numbers (6 : 3 = 2 : 1) is the reverse of the ratio of the ions (1 : 2). The same counting gives 6 : 6 for NaCl.

Worked Example 3 — Explaining a melting point

Explain why sodium fluoride (996 °C) melts at a higher temperature than sodium chloride (801 °C).

Both have a 1+ ion and a 1− ion, so the charges are the same. The fluoride ion is smaller than the chloride ion, because fluorine is above chlorine in Group 17. So in NaF, the ions sit closer together, and the attraction between them is stronger.

A stronger lattice (923 against 786 kJ/mol) needs more energy to break up, so NaF has the higher melting point.

Three classic mistakes

Trap 1 — "Salt is made of NaCl molecules"

There are no molecules in an ionic lattice. Each ion is attracted equally to all its nearest neighbours of opposite charge. NaCl is a formula unit: the ratio 1 : 1.

Trap 2 — "Both ions always have the same coordination number"

Only when the formula is 1 : 1. In MgCl₂, the numbers are 6 and 3, because there are twice as many chloride ions to share the contacts.

Trap 3 — "A bigger ion has a stronger pull"

For ions with the same charge, the reverse is true. A larger ion keeps its neighbours further away, so the attraction is weaker and the lattice energy is smaller: KI (649) is weaker than LiF (1036).

Try it yourself

Work each one on paper first. Open a hint only when you are stuck.

Predict the order of lattice energy for KF, KCl and KBr, largest first, and explain. Then check with the table in the explorer.

Hint 1
The metal ion and the charges are the same in all three.

Hint 2
Which halide ion is smallest?

Full answer
KF > KCl > KBr. The charges are the same, and the halide ion gets larger down Group 17, so the ions sit further apart and attract less. Check: 821 > 715 > 682 kJ/mol.
Without data, predict which has the larger lattice energy in each pair, and give the main reason: (a) CaO or NaCl (b) MgO or CaO.

Hint 1
Compare the charges first. If they are the same, compare the sizes.

Hint 2
Ca²⁺ and O²⁻ are both doubly charged. Mg is above Ca in Group 2.

Full answer
(a) CaO, because both of its ions are doubly charged, while NaCl's are singly charged. (b) MgO: the charges are the same, but Mg²⁺ is smaller than Ca²⁺, so the ions sit closer together.
In calcium fluoride, CaF₂, each Ca²⁺ ion has 8 F⁻ neighbours. How many Ca²⁺ neighbours does each F⁻ ion have? Explain.

Hint 1
Use the counting from Worked Example 2.

Hint 2
$N$ calcium ions make $8 N$ contacts, shared among $2 N$ fluoride ions.

Full answer
Each F⁻ has $8 N \div 2 N = 4$ calcium neighbours. The coordination is 8 : 4, the reverse of the 1 : 2 ratio of ions.
The repeating cube of the sodium chloride lattice has an edge of 0.5640 nm, so neighbouring Na⁺ and Cl⁻ ions are 0.282 nm apart. Estimate the number of ions in a salt crystal that is a cube 1.0 mm on each side, and give your answer in standard form. (1 mm = $1 0^{6}$ nm.)

Hint 1
First find how many ions lie along one edge: $1 0^{6} \div 0.282$ .

Hint 2
A cube has that number of ions along each of its three directions, so cube the answer.

Full answer
Along one edge: $1 0^{6} \div 0.282 \approx 3.55 \times 1 0^{6}$ ions. In the cube: $(3.55 \times 1 0^{6})^{3} \approx 4.5 \times 1 0^{19}$ ions, about $2.2 \times 1 0^{19}$ of each kind. Check: this is 8 times the half-millimetre grain ( $5.6 \times 1 0^{18}$ ), because doubling the edge multiplies the volume by $2^{3} = 8$ .

The bigger idea

The regular lattice explains what you can see. Salt grains are cubes because the lattice is built from cubes. And the lattice explains what salt does. Because every ion is held by strong attractions in every direction, ionic compounds have high melting points. Because the ions are fixed in place, solid salt cannot conduct electricity. And because the layers line up so precisely, a sharp blow that shifts one layer brings like charges face to face, and the crystal splits.

Each of those properties follows from the same picture, and the next lecture works through them one by one. The idea of a giant lattice also returns later in a different form: in metals, which are lattices of positive ions in a sea of electrons, and in giant covalent structures such as diamond.

Key concepts in this lesson

An ionic compound is a giant lattice of positive and negative ions held together by electrostatic attraction in every direction, with no separate molecules.
In sodium chloride each ion has six nearest neighbours of opposite charge, so its coordination is 6 : 6, while in a 1 : 2 compound such as MgCl₂ the coordination numbers are in the ratio 2 : 1.
The lattice places opposite charges close together and like charges further apart, so the attractions outweigh the repulsions.
Lattice energy is the energy released when 1 mol of an ionic solid forms from its gaseous ions, and it is larger when the ions have higher charges or are smaller.
A larger lattice energy generally means a higher melting point, which is why magnesium oxide melts at about 2800 °C and sodium chloride at 801 °C.