Answer:
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.
