Determining the Density of a Face-Centered Aluminum Cell

Crystals come in unique shapes, but all usually have an organized, repeating structure known as a lattice. For more information about metal lattices check out: 

Given Problem: An aluminum crystal has a face-centered unit cell. An aluminum atom has a radius of 143 picometers. What is the density of the solid crystal aluminum in g/cm³? 

1.  It’s given that the radius of an aluminum atom is 143 picometers, the number of atoms per face center cell is 4. We’re also given that density is mass (grams)/volume (cm3).

 The mass of a unit cell is the number of atoms in the unit cell multiplied by the mass of each atom. The volume of a unit cell is the edge length of the cell cubed. The mass of Aluminum is 26.98 grams/mol.

2.We first convert the density of aluminum from grams per mole to grams per atom. We put moles of Aluminum on the top of the second part of our analysis because then it will cancel out and we can get our grams per atom unit.

From there, we can multiply 1.792*10^-22 grams per atom by 4 atoms to get the mass of four atoms of aluminum. This corresponds to the number of atoms per face centered cell of aluminum crystal.

3. The edge length for a face-centered cell is L=2√2r, or two times the square root of the radius multiplied by 2. Remember, the volume of a cell can be found by cubing the edge length. 

Plug in the radius of the aluminum crystal in meters to the edge length equation. Cube that to find the volume of the cell. 

4. Now we have all the components needed to find density, the mass and volume of the face centered aluminum crystal cell. D=m/v, so plug in your values. The final answer should be 2.71 grams/cm³. 

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