Gallium has an orthorhombic structure, with a0 = 0.45258 nm, b0 = 0.45186 nm, and c0 = 0.76570 nm. The atomic radius is 0.1218 nm. The density is 5.904 g/cm3, and the atomic weight is 69.72 g/mol. Determine (a) the number of atoms in each unit cell; and (b) the packing factor in the unit cell.

a) atoms/cell =  8 atoms

b) FE = 0.3876

Explanation:

You have the following data:

ao=0.45258 nm

bo=0.45186 nm

co=0.76570 nm

density=p=5.904 g/cm^3

atomic weight=69.72 g/mol

Avogadro number = 6.02x10^23 atom/mol

a) The number of atoms per cell is calculated by clearing it from the density formula:

atoms/cell = (px(cell volume)x(Avogadro number))/atomic weight

cell volume = aoxboxco = (0.45258x10^-7cm)x(0.45186x10^-7cm)x(0.76570x10^-7cm) = 1.5659x10^-22 cm^3

replacing values:

atoms/cell = ((5.904 g/cm^3)x(1.5659x10^-22 cm^3)(6.02x10^23 atoms/cell))/69.72 g/mol = 7.9 atoms = 8 atoms

b)the packing factor is calculated by:

FE = ((atoms/cell)x(atom volume))/cell volume

atom volume = 4pixr^3/3 = 4pix(0.1219x10^-7 cm)^3/3 = 7.5875x10^-24 cm^3

FE = (8x7.5875x10^-24)/(1.5659^-22 cm^3) = 0.3876

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