Copper has an atomic radius of 0.128 nm, an FCC crystal structure, and an atomic weight of 63.5 g/mol. Compute its theoretical density and compare the answer with its measured density

d = 8.88 g/cm³

0.56 % difference

Explanation:

We need to compare the theoretical density of copper to its measured density which the literature says is 8.93 g/cm³.

The density is mass per volume so we need to determine the mass of the FCC unit cell and its volume.

In the face centered cubic unit cell we have 8 atoms at the 8 corners shared by 8 unit cells for a  total of one atom, and 6 atoms in the 6 faces of the cube shared by two face centered cubic unit cell, giving us a total of 4 atoms of copper per unit cell ( 8 x 1/8 + 6 x 1/2  = 4).

The mass of the unit cell will be the mass of 4 copper atoms:

mass = 4 atoms/unit cell x 63.5 g/mol x 1 mol /6.022 x 10²³ atoms

mass = 4.22 x 10⁻²² g

The volume of a cube is the length side, a , cubed.

There is the relation for a FCC unit cell which expresses this length as a function of the atomic radius as:

a =  √8 x r

converting r to cm for units consistency:

0.128 nm x 1 x 10⁻⁷ cm/nm = 1.28 x 10⁻⁸ cm

a = √8 x 1.28 x 10⁻⁸ cm = 3.62 x 10⁸ cm

V = a³ = ( 3.62 x 10⁸ cm)³ = 4.75 x 10⁻²³ cm³

Now we can finally compute the theoretical density as:

d = m/V = 4.22 x 10⁻²² g / 4.75 x 10⁻²³ cm³ = 8.88 g/cm³

finally comparing the theoretical with the measured density for copper, we get:

(8.93 - 8.88)/8.93 x 100 =0.56 %

There is a difference of just 0.56 % between the two values.

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