Question

An alloy of copper, silver and gold is found to have copper constituting the face-centred cubic (FCC) lattice. If silver atoms occupy the edge centres and gold is present at body centre, the alloy has a formula (a) \(\mathrm{Cu}_{4} \mathrm{Ag}_{2} \mathrm{Au}\) (b) \(\mathrm{Cu}_{4} \mathrm{Ag}_{4} \mathrm{Au}\) (c) \(\mathrm{Cu}_{4} \mathrm{Ag}_{3} \mathrm{Au}\) (d) \(\mathrm{CuAgAu}\)

Short answer

\(\mathrm{Cu}_{4} \mathrm{Ag}_{3} \mathrm{Au}\) is the formula of the alloy with copper in the FCC lattice, silver at edge centres, and gold at the body centre.

Step-by-step solution

Calculate the number of Copper atoms

In the face-centred cubic (FCC) lattice, there are 4 copper atoms per unit cell. This includes 1/8th of each of the 8 corner atoms and 1/2 of each of the 6 face-centred atoms. Total copper atoms in one unit cell = 8 * (1/8) + 6 * (1/2) = 1 + 3 = 4.

Calculate the number of Silver atoms

Silver atoms occupy the edge centres. There are 12 edges in an FCC lattice and each edge is shared by 4 unit cells, so each edge contributes 1/4 of an atom to the cell. Total silver atoms in one unit cell = 12 * (1/4) = 3.

Calculate the number of Gold atoms

Gold is present at the body centre of the FCC lattice. There is one body centre per unit cell and it is not shared with any other unit cells, so it contributes one full atom. Total gold atoms in one unit cell = 1.

Determine the formula of the alloy

Combining the number of atoms for copper, silver, and gold in the FCC lattice, we get 4 copper atoms, 3 silver atoms, and 1 gold atom. Therefore, the formula of the alloy is \(\mathrm{Cu}_{4} \mathrm{Ag}_{3} \mathrm{Au}\).

Key concepts

Face-Centred Cubic Lattice

The face-centred cubic (FCC) lattice is one of the most prevalent crystal structures found in metals and alloy systems. In an FCC lattice, atoms are arranged in a way that each of the six faces of the cube has an atom at its center, in addition to the eight corner atoms. However, a unique aspect of this arrangement is that individual atoms at the corners and face centers are shared among adjacent unit cells.

This sharing means corner atoms are counted as 1/8th of an atom and face atoms as 1/2 when attributed to a single unit cell. This results in a total of 4 complete atoms per FCC unit cell (8 corners at 1/8 each plus 6 face-centers at 1/2 each). This information is fundamental when determining formula units in alloys, calculating density, or understanding various material properties such as slip systems during deformation.

Unit Cell Calculations

In solid state chemistry, understanding unit cell calculations is essential for grasping the composition and properties of materials. A unit cell represents the smallest repeating unit that exhibits the full symmetry of a crystal structure and its contents are key to determining the chemical formula of a substance.

For alloys and compounds, the occupancy of different sites within the unit cell—corners, edge centers, face centers or body centers—gives insight into the stoichiometry of the material. As seen with the FCC lattice, careful consideration of the contributions from shared lattice positions (1/8th for corners, 1/4th for edges, 1/2 for faces, and fully for the body center) is necessary. Once the quantity of each type of atom in a unit cell is computed, we derive at the empirical formula, which is vital for both theoretical and practical applications such as material design and industrial processes.

Solid State Chemistry

Solid state chemistry is the study of the synthesis, structure, and properties of solid materials. It focuses on the arrangement of atoms in solids and how this affects their behavior. A crucial aspect of this field is understanding how the structure of solids is related to their properties, such as electrical conductivity, strength, and optical properties.

Within solid state chemistry, analyzing crystal lattices is essential—specifically the arrangement of atoms within these lattices. In the context of alloys, the distribution of different atoms within the lattice has a direct impact on the alloy's characteristics. For example, the placement of different metal atoms in the FCC lattice determines the properties and uses of the alloy, like the inherent strength of steel or the electrical conductivity of copper. Analyzing and manipulating these atomic arrangements allows chemists and materials scientists to innovate new materials with desirable properties for technological advancements.