Question

A compound alloy of gold and copper crystallises in a cubic lattice in which gold atoms occupy corners of cubic unit cells and copper atoms occupy the centre of faces of cube. What is the formula of the alloy compound?

Answer 1

Hint: As per given, gold atoms occupy corners of cubic unit cell and copper atoms occupy the centre of faces of cube, it shows the Face-centered cubic lattice, An \[FCC\] unit cell contains atoms at all the corners of the crystal lattice and at the centre of all the faces of the cube. With the help of Face-centered cubic lattice structure we can determine the formula of alloy compound.


Complete step by step answer: In \[FCC\] unit cell atoms are present in all the corners of the crystal lattice and also, there is an atom present at the centre of every face of the cube, This face-centre atom is shared between two adjacent unit cells
In face-centered cubic unit cell, we have:
\[8\] corners \[ \times {\text{ }}\dfrac{1}{8}\]per corner atom \[ = 8{\text{ }} \times {\text{ }}\dfrac{1}{8} = 1{\text{ }}atom\]
\[6\]Face-centered atoms \[ \times {\text{ }}\dfrac{1}{2}\]atom per unit cell = \[3{\text{ }}atoms\]
Therefore, the total number of atoms in a unit cell = \[4{\text{ }}atoms\]
Given,
Gold atoms occupy the corners of the lattice point and copper atoms are in the center of each cubic face. So the gold occupies the corner of the unit cell.
There are\[\;8\]corners and contribution of each corner = \[1/8\]​.
Therefore total gold atom is \[8 \times \dfrac{1}{8} = {\text{ }}1.\]
Copper occupies faces of the cube. There are 6 faces and contribution of each face is \[ = 1/2.\]
Total atoms of copper are \[6 \times \dfrac{1}{2} = {\text{ }}3.\]
Putting these values in compound of gold and copper we get,

The final Formula as \[\left( {AuC{u_{3}}} \right)\] .

Note: Atoms in an \[FCC\] arrangement are packed as closely together as possible, with atoms occupying \[74\% \] of the volume. This structure is also called cubic closest packing, in\[FCC\] metals are deform easier than \[BCC\] metals and thus they are more ductile. So\[BCC\] Metals are stronger than \[FCC\] metals.
Metals that possess face-centered cubic structure include copper, aluminum, silver, and gold.