Question

If the ratio of coordination no. of \(A\) to that of \(B\) is \(x: y\), then the ratio of no. of atoms of \(A\) to that no. of atoms of \(B\) in unit cell is: (a) \(x: y\) (b) \(y: x\) (c) \(x^{2}: y\) (d) \(y: x^{2}\)

Short answer

The ratio of the number of atoms of A to that of B in the unit cell is the reciprocal of the ratio of their coordination numbers, making the correct answer (b) y: x.

Step-by-step solution

Understand the Concept of Coordination Number and Number of Atoms

Coordination number refers to the number of nearest neighbors of a particle in a crystal lattice, not to the total number of particles. The number of atoms of an element in the unit cell is determined by its positions within the unit cell and not directly by the coordination number. Thus, the ratio of the coordination numbers does not necessarily correspond to the ratio of the number of atoms.

Relate Coordination Numbers to Number of Atoms

The ratio of coordination numbers reflects the ratio of the immediate neighbors in the structure and not the actual number of atoms present. To find the ratio of the number of atoms, one must consider the entire unit cell rather than just the local arrangements. Typically, the ratio of the number of atoms of A to B in the unit cell is reciprocal to the ratio of their coordination numbers, as each atom of A is surrounded by atoms of B, and vice versa.

Determine the Correct Ratio of Number of Atoms

Considering that the ratio of coordination numbers represents the surround relationships and each atom of A is in relation to multiple B atoms and vice versa, thereby each B atom enables multiple A atoms to exist in relation, the ratio of the number of atoms of A to B will be the reciprocal of the coordination number ratio. Hence, the correct ratio of the number of atoms of A to that of B in the unit cell is the inverse of the coordination number ratio, which is y: x.

Key concepts

Coordination Number

Imagine you're at a party and consider how many people you're directly interacting with; that's somewhat analogous to the coordination number in crystal structures. It represents the number of nearest neighbor atoms or ions that immediately surround a central atom.

When visualizing this in three dimensions, think of balls representing atoms, linked together with sticks, which represent chemical bonds. The balls directly connected to the central ball count toward its coordination number. For example, in a cubic close-packed structure, an atom typically has a coordination number of 12 because it has 12 immediate neighbors. In the context of our exercise, a different coordination number for elements A and B signifies that they are surrounded by a different number of neighboring atoms. Understanding that the coordination number is tied to the geometric arrangement in a crystal provides insight into the type of crystal structure formed but does not directly translate to the ratio of atoms in the unit cell.

Unit Cell

The intricate pattern of a snowflake reflects the repeating nature of crystal structures. In a similar way, a unit cell is the smallest repeating unit in a crystal lattice that, when stacked repeatedly, forms the entire crystal. Consider it like a 3D 'tile' that tessellates through space.

The geometry of the unit cell (whether it's cubic, tetragonal, orthorhombic, etc.) determines how atoms pack together. The proportion of the types of atoms within this unit cell will define the crystal's overall composition. In the exercise, understanding the number of atoms for A and B within the confines of the unit cell is crucial as it's mistakenly considered synonymous with the coordination number, which is incorrect. By examining the entire unit cell, and not just local atomic interactions, you will determine the true stoichiometry of the crystal.

Crystal Lattice

Behold a marvelous mosaic where patterns repeat over and over; this is much like a crystal lattice in the world of crystals. It is the framework, or spatial arrangement, of points depicting the positions of atoms, molecules, or ions in a crystal. Each point in the lattice represents the position of a particle, and the overall pattern can extend infinitely in three dimensions.

In each crystal lattice, the unit cells fit together without gaps or overlaps, creating a seamless pattern much like tiles on a floor. The organization of unit cells within the lattice determines the material's properties, such as density and porosity. In our textbook problem, this comprehensive structure is influenced by the ratio of the atoms of A and B but is visualized on a much grander scale than that of a single unit cell or the local coordination number.