Question

A certain mineral has a cubic unit cell with calcium at each corner, oxygen at the center of each face, and titanium at its body center. What is the formula of the mineral? An alternate way of drawing the unit cell has calcium at the center of each cubic unit cell. What are the positions of titanium and oxygen in such a representation of the unit cell? How many

Short answer

The formula of the mineral is CaTiO3. In the alternate representation, titanium atoms would be at the corners and oxygen atoms at the face centers of the unit cell.

Step-by-step solution

Determine the Cell Content Ratios

Each corner of the cubic cell is shared by eight cubes, so effectively, the contribution of each corner to one cube is 1/8. As there are eight corners, the amount of calcium in one cube becomes 1. Each face is shared by two cubes, so the contribution of oxygen in each cube becomes 1/2 * 6 = 3, as there are six faces in a cube. The cube body center is not shared with any other cube, so there is one titanium atom in each cube.

Formula for Unit Cell

The formula of the mineral is determined based on the calculated amounts in Step 1. There is 1 calcium, 3 oxygen, and 1 titanium atom in one cube, which gives the formula CaTiO3.

Determine Alternate Positions

When calcium is at the center of the unit cell, titanium and oxygen have to find a place at the corners and face centers (that were previously occupied by calcium and oxygen respectively), based on the 'corner-face' swap principle in a symmetrical cubic unit cell. So, the titanium atoms would now be at the corners and the oxygen atoms at the face centers of this alternate unit cell.

Key concepts

Understanding the Cubic Unit Cell

When beginning to understand the structure of crystalline solids, the concept of a cubic unit cell is fundamental. Picture a unit cell as the smallest repeating unit in the crystal lattice structure, much like a single square on a checkerboard that, when repeated, creates the entire board field.

In a cubic unit cell, all edges have equal lengths, and all angles are right angles, making the cube symmetrical and simple to visualize. There are various types of cubic unit cells, including simple cubic, body-centered cubic (BCC), and face-centered cubic (FCC), each with a distinct arrangement of atoms.

As we delve into chemistry exercises, it's essential to comprehend how atoms are positioned within these cubes. Atoms can be located at the corners, on the edges, at the face centers, or right in the center of the cube, known as the body center. The arrangement of these atoms determines the physical and chemical properties of the material.

Chemical Formula Determination from Unit Cell

Having established the structure of a unit cell, the next logical step is determining the chemical formula of a compound based on this structure. Determination of the chemical formula from a unit cell involves counting the atoms within the cell and considering the way they are shared with adjacent cells.

In the exercise provided, we have calcium at each corner, oxygen in the center of each face, and titanium at the body center. To determine the chemical formula, we count each type of atom, adjusting for their positions: a corner atom is shared by eight different unit cells, a face-centered atom is shared by two, and a body-centered atom belongs entirely to its unit cell.

Understanding these principles, we can derive the chemical formula. Specifically, the counts of atoms adjust to one calcium atom, three oxygen atoms, and one titanium atom per unit cell, yielding the formula \( \text{CaTiO}_3 \). This counting method is crucial for solving many problems in solid-state chemistry and materials science.

Deciphering the Crystal Lattice Structure

The crystal lattice structure is at the heart of many properties in materials science and crystalline chemistry. It embodies a three-dimensional arrangement of points denoting the periodic array in which atoms, ions, or molecules are situated in a crystalline material.

Each point in a crystal lattice structure represents the position of a repeating element of the structure, much like how multiple units of a simple pattern, when seamlessly combined, create a complex design in tiling. This structural pattern defines the substance's properties, such as its melting point, density, and how it interacts with light and electricity.

In the exercise, we see a shift from one representation of a unit cell to another, implying that the arrangement of atoms within a unit cell can vary while still maintaining the overall stoichiometry and symmetry of the crystal lattice. If calcium moves to the center, titanium and oxygen reposition to the corners and face centers, respectively, still retaining the cubic symmetry while altering our spatial perspective of the unit cell. This interchangeability within the structure reflects the versatile nature of crystalline arrangements and their significant role in understanding material properties.