Question

The CORRECT statement(s) for cubic close packed (ccp) three dimensional structure is(are) (A) The number of the nearest neighbours of an atom present in the topmost layer is 12 (B) The efficiency of atom packing is \(74 \%\) (C) The number of octahedral and tetrahedral voids per atom are 1 and 2 , respectively (D) The unit cell edge length is \(2 \sqrt{2}\) times the radius of the atom

Short answer

The correct statements for cubic close packed (ccp) three-dimensional structure are (A) The number of the nearest neighbours of an atom present in the topmost layer is 12, (B) The efficiency of atom packing is 74%, and (C) The number of octahedral and tetrahedral voids per atom are 1 and 2, respectively. Statement (D) is incorrect; the correct factor should be the square root of 2 (√2), not 2√2.

Step-by-step solution

Understanding coordination number

Coordination number is the number of nearest neighbours of an atom in a crystal lattice. For the cubic close packed (ccp) or face-centered cubic (fcc) structure, the coordination number is 12. This means each atom is in direct contact with 12 other atoms.

Calculating packing efficiency

Packing efficiency is the fraction of volume in a crystal structure that is occupied by atoms. For a ccp structure, the packing efficiency is indeed 74%. This is derived from the volume occupied by atoms in a unit cell divided by the total volume of the cell.

Determining the number of voids per atom

In a ccp structure, the number of octahedral voids per atom is equal to the number of atoms, which is 1. The number of tetrahedral voids per atom is twice the number of atoms, which is 2. This is characteristic of the ccp structure.

Relating unit cell edge length to atomic radius

The edge length of the unit cell in a ccp structure is related to the radius of the atom by the formula: unit cell edge length = 2 times the square root of 2 (2√2) times the radius of the atom. This relationship is based on the geometry of an fcc unit cell where atoms touch along the face diagonal.

Key concepts

Coordination Number

Imagine atoms within a crystalline structure as individuals within a social network: the coordination number reflects how many other 'friends' (or nearest neighbors) each atom has. In the context of a cubic close packed (ccp) structure, which is also known as a face-centered cubic (fcc) structure, each atom boasts a sociable circle of 12 immediate neighbors. This is significant because the coordination number affects the physical properties of the material, such as its density and how it deforms under pressure.

Understanding the coordination number is like recognizing the value in a busy networking event; it signifies how tightly the atoms are 'packed together', influencing the strength and stability of the material.

Packing Efficiency

If you've ever played Tetris, you know the satisfaction of efficiently occupying space with no gaps. The packing efficiency in crystals exhibits the same principle. It's the measure of how much space in the crystal is filled up by the atoms themselves. A cubic close packed structure has a commendable packing efficiency of 74%, which means that nearly three-quarters of the volume in the structure is taken up by atoms, leaving only a quarter as empty space. Understand this as a dense, well-packed suitcase – very few voids imply a good use of the available space.

The high packing efficiency explains why ccp structures have high density and are commonly found in metals like copper and aluminum, giving them their characteristic physical properties, such as durability and malleability.

Crystal Structure Voids

In a city full of buildings, there are spaces that lie unoccupied - parks, plazas, and playgrounds. Similarly, in the dense urbanization of atoms in a ccp structure, there lie crystal structure voids. These are the octahedral and tetrahedral gaps not filled by atoms. A ccp structure has a 1:2 ratio of octahedral to tetrahedral voids relative to atoms. For every atom, there's one octahedral void and two tetrahedral voids.

This careful accounting of voids is crucial not just for understanding the overall architecture of the structure, but also for predicting the behavior of materials when 'guest' atoms or defects are introduced. These voids can accommodate impurities or interstitial atoms, deeply affecting the material's electrical and mechanical properties.

Unit Cell Edge Length

Imagine building a 3D puzzle; the edge of each piece helps determine how they will fit together. Similarly, in a ccp structure, the unit cell edge length is a key measurement that tells us how atoms are packed. Derived geometrically, the edge length of a unit cell in a ccp structure is an elegant dance of mathematics and spatial relationships, amounting to precisely \(2\sqrt{2}\) times the radius of an atom.

This measurement is not arbitrary but emerges from the layout of atoms within the unit cell that define the structure's spatial configuration. Understanding this helps us visualize and calculate other critical aspects of the crystal, such as its volume, density, and how it interacts with light and other forms of energy.