Question

The corundum (Al2O3) has a rhombohedral Bravais lattice, but it closely approximates a hexagonal lattice and is usually regarded as hexagonal. There are 30 ions per lattice site (and per unit cell). The Al2O3 formula requires that these 30 ions be divided as 12Al3+ and 18 O2-. The Al2O3 structure closely approximates close-packed O2- sheets with two-thirds of the small interstices between sheets filled with Al3+. In other words, close-packed O2- ion containing layers have repeated stacking and the Al3+ ions fill two-thirds of the octahedral interstices between adjacent layers. (22 points) The density of Al2O3 is Al2O3 = 3.8 gram/cm3, Atomic masses are : MAl = 26.98 gram/mol, MO = 16.00 gram/mol Ionic radii are : rAl3+ = 0.05 nm, rO2- = 0.14 nm a) Draw 3 layers/sheets of O2- ions and Al3+ ions in between them. b) Draw the corundum unit cell and show the O2- and Al3+ ions in the unit cell. c) How many Al2O3 molecules are there in 1 cm3 ? How many atoms are there in 1 cm3 of Al2O3? d) How many Al2O3 molecules are there in 1 gram? How many atoms are there in 1 gram of Al2O3? e)Calculate the atomic packing factor of Al2O3

Answer 1

In this hcp packing there is one octahedral void corresponding to each atom constituting the close packing. In corundum only 2/3rd of the octahedral voids are occupied. It means corresponding to each oxide are 2/3 aluminium ions. The whole number ratio of oxide and aluminium  ion in corundum is therefore 3:2 : formula of corundum is Al2O3.

c) How many Al2O3 molecules are there in 1 cm3 ?

The density of Al2O3 is , d = 3.8 gram/cm3, Atomic masses are : Al = 26.98 gram/mol, O = 16.00 gram/mol

Ionic radii are : rAl³⁺ = 0.05 nm, rO^2- = 0.14 nm

Al₂O₃ Formula weight : 101.96 amu

1 amu = 1.67*10^^-24 g

d = 3.8 gram/cm3

So, No. of Al2O3 molecules are there in 1 cm3 : 3.8 g / 101.96 *1.67*10^^-24 g/molecule = 2.23*10²² molecules

    How many atoms are there in 1 cm3 of Al2O3?

Each Al2O3 molecule : 5 atom ( 2 Al + 3 O)

thus, atoms in 1 cm3 of Al₂O₃ :1.12*10²³ atom

d) How many Al2O3 molecules are there in 1 gram? How many atoms are there in 1 gram of Al2O3?

No. of Al₂O₃ molecules are there in 1 gm :

1g / 101.96 *1.67*10^^-24 g/molecule = 5.87*10²¹ molecules

    How many atoms are there in 1 cm3 of Al2O3?

Each Al2O3 molecule : 5 atom ( 2 Al + 3 O)

thus, atoms in 1 cm3 of Al₂O₃ :2.94*10²² atom