Question

Gold is a face-centered cubic structure that has a unit cell edge length of 4.08 Å. There are four gold atoms per unit cell.

How many gold atoms are there in a sphere that is 13 nm in diameter? Recall that the volume of a sphere is 4/3πr^3.

Answer 1

Gold is a face-centered cubic structure. Hence, there are four gold atoms in a unit cell of gold. Volume of unit cell of gold can be calculated as follows:

Volume of unit cell = a³

Where, a = length of unit cell edge = 4.08 Å = 0.408 nm

Thus, volume of unit cell = (0.408 nm)³

Volume of unit cell = 0.06792 nm³

Volume of 13nm diameter sphere can be calculated as follows:

Volume of sphere = 4/3πr³

Where, r = radius = 13nm/2 = 6.5nm

Thus, volume of sphere = 4/3 (3.142) (6.5 nm)³

Volume of sphere = 1150.5 nm³

Thus, number of units cells in 1150.5 nm³ volume of sphere can be calculated as follows:

Number of unit cells = (1150.5 nm³) / (0.06792 nm³)

Number of unit cells = 16939

Since, each unit cell contains 4 atom, the number of atoms in 16939 unit cells can be calculated as follows:

Total number of atoms = 4 x 16939

Total number of atoms = 67756 atoms