Question

Polonium crystallizes in a simple cubic structure. The edge of the unit cell is \(0.236 \mathrm{~nm}\). What is the radius of the polonium atom: (a) \(0.144 \mathrm{~nm}\) (b) \(0.156^{\circ} \mathrm{nm}\) (c) \(0.118 \mathrm{~nm}\) (d) \(0.102 \mathrm{~nm}\)

Short answer

The radius of the polonium atom is 0.118 nm, which corresponds to answer (c).

Step-by-step solution

Understanding the simple cubic structure

In a simple cubic structure, each corner of the cube is occupied by an atom. These atoms touch along the edge of the cube. The diameter of the atom is equal to the edge length of the cube.

Calculating the radius of the atom

Since the diameter of the atom is equal to the edge length of the cube, we can find the radius by dividing the edge length by 2. The edge length given is 0.236 nm.

Performing the calculation

Divide the edge length by 2 to find the radius of the polonium atom. Radius = Edge length / 2 = 0.236 nm / 2 = 0.118 nm.

Key concepts

Crystal Lattice

The concept of a crystal lattice is fundamental when studying the structure of solid materials. Essentially, a crystal lattice is an ordered arrangement of atoms, ions, or molecules in three-dimensional space. This pattern repeats itself at regular intervals, creating the solid structure of a crystal.

A great way to picture a crystal lattice is by imagining it as a three-dimensional grid where each intersection point, called a lattice point, represents the position of an atom or a group of atoms. In the context of our exercise, polonium atoms are arranged in such a lattice, specifically in a pattern known as a simple cubic structure.

The simple cubic structure is one of the simplest forms of crystal lattices, where each atom rests at the corners of a cube-shaped cell, allowing for a straightforward understanding of the material's spatial properties. It's a perfect starting point for students beginning their journey into materials science.

Atomic Radius Calculation

Understanding the calculation of the atomic radius is crucial in determining the size of atoms within a crystal structure. The atomic radius is half of the distance between the centers of two atoms that are just touching each other. In a simple cubic structure, this is typically the distance from the center of an atom to the center of the atom at the adjacent corner of the cube.

In solving for the atomic radius from the given edge length of a unit cell, as shown in our exercise, one simply divides the edge length by 2. This is based on the fact that in a simple cubic lattice, the diameter of an atom is equivalent to the edge length of the cube since the atoms are assumed to touch each other along the cube's edge. Therefore, the atomic radius is just half of that diameter. The ability to perform this calculation is foundational for students analyzing crystal structures.

Unit Cell

The unit cell is the smallest repeating unit that shows the full symmetry of a crystal lattice and that, by repetition in three dimensions, can build up the lattice. Think of it as a fundamental building block, much like a Lego brick, from which the complete crystal can be constructed.

In a simple cubic lattice, the unit cell takes the shape of a perfect cube with atoms at each of its eight corners. Since each atom at a corner is shared by eight adjacent unit cells, each unit cell effectively contains one atom. The properties of the entire crystal can be understood by studying this single unit cell, which simplifies complex three-dimensional structures into more manageable representations.

Recognizing the geometry and spatial dimensions of a unit cell allows students to better visualize the arrangement of atoms in a solid and provides a basis for understanding more complex crystal structures.