# Mastering Chemistry Pre-Lecture Video Transcript

This is a transcript of the video on the Educator Features page and Student Features page.

### Unit Cells: Simple Cubic, Body-Centered Cubic, and Face-Centered Cubic

Hello. In this video, you’ll learn about three cubic unit cells.

We use unit cells to represent a crystalline lattice.

The unit cell is the smallest unit of volume that, when repeated over and over reproduces the crystalline lattice.

For example, consider the two-dimensional crystalline lattice shown here. The dark red square is the unit cell. When the unit cell is repeated over and over, it reproduces the entire lattice.

A number of different kinds of unit cells exist, but here we limit our discussion to three types of cubic unit cells: simple cubic, body-centered cubic, and face-centered cubic.

The simple cubic unit cell looks like this. It is a cube with one atom in each corner. But only the cube itself is the unit cell.

An important property of any unit cell is the length of the edge of the cell in terms of the radius of the atoms that compose it.

For the simple cubic unit cell, the atoms touch along the edge of the cube, so the edge length l is equal to twice the radius of the atom 2r.

Another important property of a unit cell is the number of atoms contained within it. Since only 1/8 of each corner atom is actually in the unit cell, and since a cube has eight corners, the simple cubic unit cell contains only one atom.

A third important property of a unit cell is the coordination number, the number of atoms in direct contact with any one atom.

Examine the central atom in this image which shows the simple cubic unit cell structure. That atom, like all others if we extended the structure, is in direct contact with six other atoms, so the coordination number is 6 for the simple cubic unit cell.

The body-centered cubic unit cell looks like this. It is a cube with one atom in each corner and one atom in the center. Again, the unit cell itself contains only the parts of the atom that are within the cube.

In these images, the different colors are here just to help you see the structure. All atoms are the same type.

For the body-centered cubic unit cell, the atoms touch along the cube diagonal, so the relationship between the edge length l and the radius is a bit more complicated.

You can determine this relationship by labeling the edge length l, the face diagonal b, and cube diagonal c.

Then by the Pythagorean theorem, c squared is equal to b squared plus l squared. But c is just 4 times the radius because the atoms touch along the line labeled c.

If you apply the Pythagorean theorem to the face diagonal, you calculate that b squared is equal to l squared plus l squared, so that b squared is equal to 2l squared.

Now you can make some substitutions and find that the quantity 4r squared is equal to 2l squared plus l squared.

Solve this for l and you find that l equals 4r divided by the square root of 3.

The body centered cubic unit cell contains two atoms within it: 1/8 of each corner atom and the one atom in the center which is entirely within the unit cell.

The coordination number for the body centered cubic unit cell is 8. You can see from this image that the central atom is in direct contact with 8 other atoms.

The face centered cubic unit cell looks like this. It is a cube with one atom in each corner and one atom in the center of each face.

Again, the unit cell itself contains only the parts of the atoms that are within the cube. And again, in these images, the different colors are just here to help you see the structure all atoms are of the same type.

For the face-centered cubic unit cell, the atoms touch along the face diagonal.

You can do a calculation like the previous one to find that the edge length is equal to 2 times the square root of 2 times r.

The coordination number for the face centered cubic unit cell is 12.

You can see from this image that the central atom is in direct contact with 12 other atoms.

Ok here is a practice question for you. How many atoms are in the face centered cubic unit cell? a 1 b 2 c 4 or d 14

The correct answer is c 4.

The face centered cubic unit cell contains four atoms within it: 1/8 of each corner atom and ½ of each atom on the six faces of the cube. So the total number of atoms per unit cell is four.