# Question #bbf6e

Aug 11, 2017

WARNING! Long answer! There are four ion pairs in a unit cell of $\text{NaCl}$.

#### Explanation:

The unit cell

A unit cell is the smallest group of particles that constitutes the repeating pattern in a crystal.

Each unit cell is surrounded by other unit cells in all directions to make up a crystal lattice. The $\text{NaCl}$ unit cell

Here is a diagram of the NaCl unit cell. It is called a face centred cubic cell because the large chloride ions (green) are arranged at the faces of a cube.

The smaller sodium ions (grey) fill in the holes around the edges of the cube.

Here is an expanded view of the unit cell. You can see that there is also a sodium ion in the centre of the cube, and it is not shared with any other unit cell.

Number of $\text{Na"^"+}$ ions in unit cell

Each edge of the cube is shared by four other cubes, so there is $\frac{1}{4}$ of a sodium ion at each edge.

There are four edges at the top, four at the bottom, and four at the bottom.

$\text{Number of Na"^"+"color(white)(l) "ions" = 12 color(red)(cancel(color(black)("edges"))) × (1/4 color(white)(l)"ion")/(1 color(red)(cancel(color(black)("edge")))) + 1 color(red)(cancel(color(black)("centre")))× "1 ion"/(1 color(red)(cancel(color(black)("centre")))) = "3 ions + 1 ion" = "4 ions}$

Thus, there are four sodium ions in a unit cell of sodium chloride.

Number of $\text{Cl"^"-}$ ions in unit cell

Each corner of the cube is shared with 7 other cubes, so there is $\frac{1}{8}$ of a chloride ion at each edge.

Each face of the cube is shared by one other cube, so there is $\frac{1}{2}$ of a chloride atom at each face

$\text{Number of Cl"^"-" color(white)(l)"ions" = 8 color(red)(cancel(color(black)("corners"))) × (1/8 color(white)(l)"ion")/(1 color(red)(cancel(color(black)("corner")))) + 6 color(red)(cancel(color(black)("faces")))× (1/2color(white)(l) "ion")/(1 color(red)(cancel(color(black)("face")))) = "1 ion + 3 ions" = "4 ions}$

Thus, there are four chloride ions in a unit cell of sodium chloride.

Number of NaCl ion pairs

There are four sodium ions and four chloride ions in a unit cell, so there are four $\text{NaCl}$ ion pairs.