Question

A solid has three types of atoms X, Y and Z. 'X' forms a FCC lattice with 'Y' atoms occupying all the tetrahedral voids and "Z' atoms occupying half the octahedral voids. The simplest formula of solid is (a) \(\mathrm{X}_{2} \mathrm{Y}_{4} \mathrm{Z}\) (b) \(\mathrm{XY}_{2} \mathrm{Z}_{4}\) (c) \(\mathrm{X}_{4} \mathrm{Y}_{2} \mathrm{Z}\) (d) \(\mathrm{X}_{4} \mathrm{YZ}_{2}\)

Short answer

The simplest formula of the solid is \( \mathrm{X}_{2}\mathrm{Y}_{4}\mathrm{Z} \).

Step-by-step solution

Understanding FCC Unit Cell

In a face-centered cubic (FCC) unit cell, there are 4 atoms of type 'X'. This is because each unit cell has atoms at each of the 8 corners, with each corner atom shared by 8 unit cells, and 6 face atoms, with each face atom shared by 2 unit cells. So the number of atoms in an FCC unit cell is calculated as: (1/8)*8 + (1/2)*6 = 4.

Tetrahedral Voids in FCC

In a FCC lattice, the number of tetrahedral voids is twice the number of atoms present in the lattice. Since the 'X' atoms form a FCC lattice and we already know they make up 4 atoms, there will be 4*2 = 8 tetrahedral voids available.

Filling the Tetrahedral Voids

'Y' atoms fully occupy the tetrahedral voids. Since there are 8 tetrahedral voids, we have 8 'Y' atoms within the FCC lattice.

Octahedral Voids in FCC

In a FCC lattice, the number of octahedral voids is equal to the number of atoms present in the lattice. Therefore, there will be 4 octahedral voids available.

Filling the Octahedral Voids

'Z' atoms occupy half the octahedral voids. Since there are 4 octahedral voids, half of them would be 4/2 = 2. Thus, there are 2 'Z' atoms within the FCC lattice.

Determining the Simplest Formula

For each unit cell, the ratio of 'X' atoms to 'Y' atoms to 'Z' atoms will be 4:8:2, which can be simplified to 1:2:0.5 by dividing all terms by 4. Multiplying these numbers by 2 to get whole numbers, the ratio becomes 2:4:1. Thus, the simplest formula for the solid is \( \mathrm{X}_{2}\mathrm{Y}_{4}\mathrm{Z} \).

Key concepts

FCC lattice

The face-centered cubic (FCC) lattice is one of the primary forms of atomic arrangement in crystalline solids. In this structure, atoms are placed at all the corners of a cube and at the centers of each face. The immediate visual effect is akin to a cube with an atom at each corner and one sitting squarely in the middle of every face.

Now, why is understanding the FCC lattice important in solid state chemistry? Well, it tells us a lot about the solid's properties, such as density, melting point, and how atoms interact with each other. Specifically, we can calculate the number of atoms per unit cell; for the FCC lattice, this number is always 4, derived from contributions of corner atoms and face atoms in specific proportions. These atoms are spatially distributed in such a fashion that they occupy the maximum volume, leading to tightly packed structures common in metals.

Tetrahedral voids

Next in the exploration of FCC lattices comes the concept of tetrahedral voids. Imagine a tetrahedron — a pyramid with a triangular base — nestled among the atoms. These voids, or small pockets of empty space, occur at points where atoms don't touch. In an FCC lattice, for every atom, there are two tetrahedral voids, leading to a fascinating 2:1 ratio of voids to atoms.

In the context of our problem, 'Y' atoms sit snugly inside these tetrahedral voids, fully packing them. Another compelling point about tetrahedral voids is that they contribute to the crystal's ability to incorporate different atoms, ultimately affecting the chemical and physical attributes of the material. Knowing the number of tetrahedral voids allows chemists and material scientists to manipulate and design new materials with desired properties.

Octahedral voids

Octahedral voids occupy another manner of empty space within the FCC lattice. If you visualize an octahedron, which is like two pyramids base to base, the space at the center of this shape is the octahedral void. In the face-centered cubic structure, the number of octahedral voids is equal to the number of atoms. Thus, in our case with the 'X' atoms forming an FCC lattice, there are 4 octahedral voids to consider.

However, not all voids need to be occupied. 'Z' atoms fill just half of these octahedral voids. The partial filling of octahedral voids is a key concept in determining the stoichiometry—the quantitative relationships between elements in compounds—of the solid. Understanding the occupancy of these voids can also explain the electric and magnetic properties of materials, as electrons can move through these voids depending on their occupancy.

Simplest formula of solid

The simplest formula of a solid represents the basic ratio of atoms present in the smallest repetitive unit of the solid, revealing the stoichiometry of the solid. For the solid described in our exercise, 'X' atoms form an FCC lattice, 'Y' fully occupies the tetrahedral voids, and 'Z' takes up half of the octahedral voids. The trick lies in understanding the distribution of atoms and voids to determine this formula.

To arrive at the simplest formula, we consider each type of atom and their corresponding positions within the unit cell. For every one 'X' atom, we have two tetrahedral voids (hence double the amount of 'Y' atoms), and for every two 'X' atoms, there's one 'Z' atom occupying the octahedral voids. Balancing this gives us the simplest ratio that governs the composition of our solid, crucial for predicting the material's elemental composition and inferring its various properties.