Question

In NaCl the centres of two nearest like-charged ions are present at a distance of: (a) \(\frac{1}{2} a \sqrt{2}\) (b) \(\frac{1}{2} a\) (c) \(\frac{\sqrt{3}}{2} a\) (d) \(\frac{1}{\sqrt{2}} 2 a\)

Short answer

\(\frac{1}{2} a \sqrt{2}\)

Step-by-step solution

Understanding the Structure of NaCl

Crystalline sodium chloride (NaCl), also known as table salt, adopts a cubic lattice structure. Each sodium ion (Na+) is surrounded by six chloride ions (Cl-), and each chloride ion is similarly surrounded by six sodium ions. This structure repeats in space to form the crystal.

Identifying the Nearest Like-Charged Ions

In a cubic crystal lattice, like-charged ions are found at the corners of the cube. To find the distance between two like charges (for instance, Na+ to Na+ or Cl- to Cl-), we consider not the nearest neighbors, which are oppositely charged, but the next nearest, which will be the same charge.

Calculating the Distance

In a face-centered cubic lattice, the like-charged ions along one face diagonal are separated by a distance of the cube's edge length, 'a'. Therefore, the distance between them is the length of the face diagonal which can be calculated using the Pythagorean theorem for a square with sides of length 'a': diagonal = \(a \sqrt{2}\). The nearest like-charged ions are located at half of this diagonal, so the distance between them is \(\frac{1}{2} a \sqrt{2}\).

Key concepts

Understanding the NaCl Lattice

The crystal lattice structure of sodium chloride, commonly known as table salt, is a fascinating example of nature's geometrical precision. In the NaCl lattice, each sodium ion (Na+) is masterfully organized around chloride ions (Cl-) to form a repeating pattern that extends throughout the crystal. Imagine a perfectly stacked set of blocks, where each block has an alternate kind of ion at each corner. This repeating structure is what gives NaCl its characteristic strength and shape.

The solidity and stability of the NaCl crystal arise from this meticulous arrangement, where the ionic bonds between the Na+ and Cl- ions create a strong and resilient network. As a result, NaCl crystals exhibit high melting and boiling points, making table salt resistant to changes in physical state under typical kitchen conditions.

Exploring Cubic Crystal Lattices

Cubic crystal lattices are a group of crystal structures that are defined by their cubical shape. There are several types of cubic lattices, but what they all have in common is that their constituent particles are arranged at the corners of a cube.

The beauty of cubic crystal lattices lies in their simplicity and symmetry. Due to their straightforward geometric shape, calculating distances and angles within these lattices is relatively easy, making them an excellent subject for study in crystallography and materials science. A cube's uniformity in all three dimensions imparts consistent properties in those directions, which is why cubic crystals like NaCl can form such perfect, natural cubes.

The Role of Face-Centered Cubic Structures in Crystals

Within the realm of cubic crystal lattices exists a sophisticated arrangement known as the face-centered cubic (FCC) structure. In this design, particles not only occupy each corner of the cube but are also placed at the center of all the cube's faces, hence the term 'face-centered'. This results in a dense packing of particles and contributes to the strong structural integrity of the crystal.

Materials with FCC structures, such as NaCl, often have unique attributes including high density and remarkable strength. Furthermore, the way atoms or ions are packed in this structure determines the material's physical properties, such as its ability to conduct electricity and heat, making the FCC lattice an important concept in the study of both metallic and ionic crystals.

Applying the Pythagorean Theorem in Crystals

The Pythagorean theorem is a powerful tool not just in geometry, but also in crystallography. This theorem applies to cubic crystals when we seek to find the distance between points in the lattice that are not linearly aligned. Specifically, in the context of a face-centered cubic (FCC) lattice like that of NaCl, the theorem helps us calculate distances across the face diagonal.

The classic equation, \(a^2 + b^2 = c^2\), dictates that the square of the length of the hypotenuse (c) equals the sum of the squares of the lengths of the other two sides (a and b). In the case of a FCC lattice, this allows us to determine that the face diagonal (hypotenuse) is \(a \sqrt{2}\), as each side of the 'square face' is length 'a'. Understanding this principle not only aids in visualizing the three-dimensional arrangements of atoms in a crystal but also illustrates the fundamental connections between mathematics and the physical sciences.