Question

Unit Cell Calculations Name _____________________________
Unit Cells: The Simplest Repeating Unit in a Crystal

The structure of solids can be described as if they were three-dimensional analogs of a piece of wallpaper. Wallpaper has a regular repeating design that extends from one edge to the other. Crystals have a similar repeating design, but in this case the design extends in three dimensions from one edge of the solid to the other. We can unambiguously describe a piece of wallpaper by specifying the size, shape, and contents of the simplest repeating unit in the design. We can describe a three-dimensional crystal by specifying the size, shape, and contents of the simplest repeating unit and the way these repeating units stack to form the crystal. The simplest repeating unit in a crystal is called a unit cell. Each unit cell is defined in terms of lattice points, the points in space about which the particles are free to vibrate in a crystal. The unit cell can pack in a simple cube as represented in the image to the right with only the equivalent of 1 atom present in the unit cell, or in more complicated structures with multiple atoms. A body centered cubic unit cell has the equivalent of 2 atoms, while the face centered cubic or the cubic closest packed unit cell has the equivalent of 4 atoms.
These unit cells are important for two reasons. First, a number of metals, ionic solids, and intermetallic compounds crystallize in cubic unit cells. Second, it is relatively easy to do calculations with these unit cells because the cell-edge lengths are all the same dimension and the cell angles are all 90.
Molar mass is the mass in grams of a mole of the particles in a substance.
molar mass= mass of the element in grams1 mole of atoms× 1 mole6.02 ×1023 atoms= mass of one atom in grams 1 atom × # atoms in the unit cell 1 unit cell

If we use the atomic mass of the element, the mass of a mole of the atoms is equal to the mass represented on the periodic table. Using this mass and the relationship of the number of atoms in a mole, Avogadro’s number, Na, 6.02 x 1023 atom / mole, then we can determine the mass of a single atom. The Unit cell can have the equivalent of one or many atoms present in the cubic unit cell.
Since the unit cells are cubic, we can determine the volume of the unit cell if we know the atomic radius and the packing of the cell. Volume of a cube is the length of the unit cell cubed or Length3. The radius of the atom is related to the length of the unit cell as follows:

Packing	Number of atoms per unit cell	Length of the unit cell in terms of atomic radius
Simple cubic	1	Length = 2 radii
Face centered cubic	4	Length = 4 r2 or radius = 2 length4
Body centered cubic	2	Length = 4 r3 or radius = 3 length4

We can also determine the volume from the density of the material. Density equals mass divided by volume. This is a physical property of the substance and therefore is not dependent upon the amount of the substance present, so the unit cell that contains only 1 atom would have the same relative density as a ton of the substance that occupies a truck load.
density= mass of the element in grams 1 cm3 or 1 mL= mass of the atoms present in the unit cell in gramsvolume of the unit cell in cm3

This however involves some simple metric conversions because the radius of an atom can be reported in several units, the most common units are nanometers (1 nm = 10-9 m), picometers (1 pm = 10-12 m) or Angstroms (1 Å = 10-10 m).  

1. The atomic radii of most atoms range between 50 pm and 350 pm. The volume of the unit cell represents the volume of the atoms as they pack in the lattice. The atoms within the cube can pack to fill the space or there can be space left unfilled within the cube. The unit cell often holds two or more atoms on a side of the cube; therefore, the cell edge may be significantly larger than the size of the atom. If the unit cell of aluminum is a cube as described above with the radius of the aluminum atoms present in the cell is 125 pm, the actual unit cell edge would be at least 2 times the value of the radius of the atoms present.  

   1. The edge of the unit cube for Aluminum is 404.95 pm. Generally, volume is measured in L or dm3 for gases and mL and cm3 for liquids and solids. Convert the edge length of the unit cube from pm to cm.

2. What is the volume (V) of the unit cell?

3. Since the unit cell contains individual atoms, how can the mass of an individual atom be determined? Calculate the mass of an individual atom of aluminum in grams.   

4. Given the mass of an individual atom, assuming the atoms pack in a simple cube with only one atom present within a unit cell, determine the density of the unit cell.  

5. Aluminum packs in a Face centered cubic unit cell. How would this affect the above calculation? The correct density of Aluminum is 2.70 g/cm3.

2. If we reverse this calculation, given the density of the element and its unit cell packing, determine the atomic radius of the atom. Iron packs as a body –centered unit cell. The density of iron is 7.86 g/cm3, what is the atomic radius of an iron atom?  

   1. First calculate the mass of a single atom of iron.

2. In the packing of this unit cell, how many atoms are present? How would this affect the mass of the unit cell?

3. Third, calculate the volume of the unit cell.

4. Fourth, determine the relationship between the volume and the cell length, a. Calculate the cell length, a.

1. Fifth, Based upon the relationship of the packing, cell length and radius, determine the radius of a single atom in the unit cell.

Answer 1

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