Question

An alloy of copper, silver and gold is found to have copper constituting the ccp lattice. If silver atoms occupy the edge centres and gold is present at body centre, the alloy will have the 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) CuAgAu

Short answer

The formula for the alloy is \(\mathrm{Cu}_{4}\mathrm{Ag}_{3}\mathrm{Au}\), so the correct answer is (c) \(\mathrm{Cu}_{4}\mathrm{Ag}_{3}\mathrm{Au}\).

Step-by-step solution

Identify the Copper Atoms

Copper atoms constitute the ccp (cubic close-packed) lattice. In this type of packing, each unit cell contains 4 atoms of the element forming the lattice. So, there are 4 copper atoms in the unit cell.

Identify the Silver Atoms

Silver atoms occupy the edge centres of the unit cell. Since there are 12 edges in a cube, and each edge is shared by 4 adjacent cubes, each silver atom at an edge centre will contribute 1/4 of its presence to the unit cell. Therefore, there are 12 edges × (1/4 contribution per edge) = 3 silver atoms in the unit cell.

Identify the Gold Atom

Gold is present at the body centre of the unit cell. The body-centred position is completely inside the unit cell, contributing one full atom to it. Therefore, there is 1 gold atom in the unit cell.

Formulate the Alloy

Combining the findings from the previous steps, we determine that the unit cell consists of 4 copper atoms, 3 silver atoms, and 1 gold atom. So, the formula for the alloy is \(\mathrm{Cu}_{4}\mathrm{Ag}_{3}\mathrm{Au}\).

Key concepts

Cubic Close-Packed Structure

Understanding the cubic close-packed (ccp) structure in solids is crucial for comprehending the organization of atoms in some materials. In a ccp structure, atoms are arranged in a very specific manner that makes efficient use of space. This arrangement allows each atom to be in contact with 12 other atoms, providing a high packing efficiency of 74%. The ccp structure is one of the most compact and tightly packed arrangements possible, often found in metals such as copper and some types of alloys.

The ccp lattice is composed of layers of atoms that stack in the sequence ABCABC... This means that the third layer of atoms is directly above the first, creating a three-dimensional repeating pattern that constitutes the crystal structure. When we consider the ccp’s unit cell, which is the smallest repeated unit in the lattice, it consists of atoms located at each corner of a cube, along with atoms situated in each face center of the cube, summing to a total of four atoms per unit cell due to shared contributions with adjacent cells.

This close-packed structure has significant implications for the material properties, including its density, ductility, and electrical conductivity. Therefore, understanding the ccp structure is paramount for students looking to grasp the fundamental concepts of solid-state physics and material science.

Unit Cell Calculations

In the realm of crystallography, unit cell calculations involve determining the number of atoms present within the smallest repeatable unit of a crystal lattice. This calculation is not only crucial for deriving the formula of an alloy or compound but also for predicting properties like density and stoichiometry.

For a cubic close-packed (ccp) structure, as in the original exercise, the process involves identifying and counting atoms in different locations: corner atoms, face-centered atoms, edge-centered atoms, and body-centered atoms.

Roles of Atoms in a Unit Cell

Each corner atom is shared by eight adjacent unit cells, contributing 1/8th of an atom to the cell. The face-centered atoms are shared by two cells, contributing half an atom each. Edge-centered atoms, if present, would contribute 1/4th each since they are shared among four cells. Lastly, a body-centered atom lies entirely within the unit cell, contributing a full atom.

By calculating these contributions, we determine how many atoms exist in one unit cell, leading to the stoichiometry of the material. This step-by-step approach allows us to arrive at a precise and accurate formula for the compound in question, essential for both academic understanding and practical applications in engineering and science.

Alloy Composition

Alloy composition plays a pivotal role in materials science for it dictates the properties and suitability of an alloy for various applications. By adjusting the ratio of constituent metals, engineers can tailor an alloy's hardness, corrosion resistance, melting point, and other key characteristics.

In the context of the exercise provided, we explore an alloy made of copper, silver, and gold. Here, understanding both the crystal structure and precise counts of each type of atom in the unit cell is vital. The formula of the alloy reflects the proportions of these constituent metals within the crystal lattice. These ratios determine the physical and chemical properties of the resultant material. For instance, in the solved problem, the presence of copper in larger quantities could enhance ductility, while silver and gold could contribute to the alloy’s resistance to tarnishing and corrosion respectively.

The composition also affects how an alloy can be processed. Different melting points of the constituent metals mean that the alloy may be more suitable for certain forming techniques. As such, grasping the concepts behind alloy composition is valuable for students aiming to apply their knowledge in real-world material design and engineering contexts.