Question

Metallic gold crystallizes in FCC lattice with edge-length \(4.07\) ?. The closest distance between gold atoms is (a) \(3.525 \dot{\mathrm{A}}\) (b) \(5.714 \AA\) (c) \(2.857 \mathrm{~A}\) (d) \(1.428 \AA\)

Short answer

The closest distance between gold atoms is (c) \(2.857 \mathrm{~A}\).

Step-by-step solution

Identify the geometry of the lattice

Recognize that gold (Au) crystallizes in a face-centered cubic (FCC) lattice. In an FCC lattice, atoms are located at each corner of the cube and the centers of each face.

Compute the face diagonal

Since the closest distance between gold atoms will be along the face diagonal, we should calculate the length of the face diagonal using the Pythagorean theorem. In an FCC lattice, the face diagonal is equal to 4 times the radius (r) of the gold atom, and can be expressed as the square root of two times the edge length (a) of the unit cell: diagonal = \( \sqrt{2} \) * a.

Calculate the length of the closest distance

Divide the length of the face diagonal by 2 to get the closest distance (d) between the gold atoms, since the diameter is equal to 2 times the radius: d = diagonal / 2 = \( \sqrt{2} \) * a / 2.

Substitute the given edge-length and solve

Substitute a = 4.07 Å into the derived equation to calculate the closest distance. d = \( \sqrt{2} \) * 4.07 Å / 2.

Compute the final answer

Perform the calculation to find the closest distance between gold atoms: d = (4.07 Å * \( \sqrt{2} \)) / 2 = 2.87 Å (rounded to 3 decimal places).

Key concepts

Crystalline Structures

Understanding crystalline structures is integral to solid state chemistry, because the arrangement of atoms within a material defines many of its properties. A crystal lattice is a three-dimensional structure composed of a repeating pattern of a particular atom or molecule. The face-centered cubic (FCC) lattice is a common type of crystalline structure found in many metals, including gold as demonstrated in our example.

In an FCC lattice, atoms are not only at each corner of the cube, but also at the center of each face, leading to a highly dense packing. This contributes to the material's high melting point, ductility, and other characteristics that are important in materials science. The face diagonal plays a crucial role in determining the closest distance between atoms, which is a measure directly related to the atomic radius and packing efficiency in this type of lattice structure.

Solid State Physics

Solid state physics is the study of rigid matter, or solids. The bulk properties of solids are directly attributable to the microscopic properties of their crystalline structures. In the context of the FCC lattice with gold, solid state physics allows us to understand not just the geometry of the lattice, but also how the electrons are shared and how the lattice vibrations, or phonons, contribute to the material's properties.

The FCC structure's efficiency in packing, for example, affects the electron density and thus the metallic bonding strength. When we compute properties like the face diagonal and relate it to the closest atomic distances, we're engaging with core principles of solid state physics. Those principles explain how and why the atoms in a gold crystal interact and bind in the way that they do.

Physical Chemistry for Competitive Exams

When preparing for competitive exams in fields such as chemistry or materials science, understanding the principles behind crystalline structures and solid state physics is crucial. Questions on these topics test your ability to apply theoretical knowledge to practical problems, such as calculating the closest atomic distances in a metal lattice.

Through exercises like our gold FCC lattice example, candidates are assessed on their grasp of physical chemistry concepts and their mathematical prowess to solve real-world problems. For instance, using the Pythagorean theorem to determine face diagonal lengths and subsequently the closest interatomic distances are common problem-solving methods needed for these exams. Students should be proficient in these techniques and comfortable with lattice calculations to excel in their academic and professional pursuits.