Density problem

Question

Sodium metal (atomic weight 22.99 g/mol) adopts a body-centered cubic structure with a density of 0.97 g/cm3.

 

If it didn't react so vigorously, sodium could float on water. Use the answer from part A to estimate the density of Na if its structure were that of a cubic close-packed metal.

 

Express your answer to two significant figures and include the appropriate units.

Answer 1

Pia,

 

To calculate an expression for the density of atoms on a lattice, count the number of atoms (and partial atoms) that appear in the recurrent volume of lattice (unit cell).

 

Look in your book at the pictures for CCP and BCC lattices.

 

Notice that for the CCP lattice that the recurrent volume is a cube with an atom at each of the eight corners, and notice that each atom is shared by the eight adjacent volumes (cubes). So, at the eight atoms at the corners of the cube, there are one eighth of the masses of the atoms inside of the cube. The density of atoms on a CCP lattice is one atom for each recurrent volume.

 

Notice that for the BCC lattice that there is one whole atom in the center of the recurrent volume in addition to the atoms at the eight corners. So, the density of atoms on a BCC lattice is two atoms for each recurrent volume, and it is twice the density of the atoms on a CCP lattice.

 

Density of the atoms on a CCP lattice = Density of atoms on a BCC lattice / 2

 

For sodium in a CCP lattice, the density of the atoms on a CCP lattice = 0.97 g/cm^3 /2 = 0.49 g/cm^3.

 

(While you are at it, take a look work out the number of atoms within the recurrent volume of an FCC lattice.)

 

Michael.