Question

Silicon tetrafluoride molecules are arranged in a body-centered cubic unit cell. How many silicon atoms are in the unit cell?

Short answer

There are 2 silicon atoms in the body-centered cubic unit cell.

Step-by-step solution

Understand body-centered cubic unit cell

In a body-centered cubic (bcc) unit cell, one atom is located at the center, and one atom is located at each of the eight corners of the cube. So, in total there are 9 positions where atoms can occupy in a complete unit cube.

Consider corner occupancy

The atoms located at the corners of the cube are shared with adjacent cubes. As a consequence, only 1/8 of each of these eight corner atoms is within the specific cube. So, the contribution of corner atoms to one unit cell is \( \frac{1}{8} \times 8 = 1 \) atom.

Consider center occupancy

The atom located in the center of the cube is not shared with any other cubes. That is, this entire atom lies within the specific cube. So, the contribution of the center atom to one unit cell is \( 1 \) atom entirely inside the unit cell.

Calculate total atoms in unit cell

To get the total number of atoms in the unit cell, add the contributions from the corner atoms and the center atom. This is \( 1 (from corner atoms) + 1 (from center atom) = 2 \) atoms in the unit cell.

Key concepts

Silicon Tetrafluoride

Silicon tetrafluoride (SiF₄) is a compound consisting of one silicon atom and four fluorine atoms. It's important to note that silicon, like carbon, can form four bonds, leading to tetrahedral-shaped molecules, such as SiF₄. In a solid state, these molecules arrange themselves in a specific pattern to form a crystal. Understanding the arrangement of SiF₄ in its crystal form is crucial for determining properties such as melting point, hardness, and its role in technological applications.

When discussing SiF₄ in the context of crystal structures, the molecular geometry is also an essential consideration. The tetrahedral molecular arrangement results in a symmetrical and uniform structure, which influences how these molecules interact in the solid state. This molecular symmetry plays a role in how the crystal structure of SiF₄ can be modeled in terms of unit cells.

Crystal Structure

A crystal structure represents the orderly arrangement of atoms, molecules, or ions in a solid. The smallest repeating unit of this structure is known as the 'unit cell.' When repeated in three-dimensional space, unit cells stack together to form the crystal lattice and ultimately, the entire crystal.

One of the simplest and most common types of unit cells is the body-centered cubic (bcc) structure. This structure features atoms at all eight corners of a cube, with an additional atom at the very center. When visualizing this, imagine stacking building blocks neatly together, with some blocks only partially within one specific boundary; this is reminiscent of how atoms are shared at corner positions in the bcc structure.

In various substances, including metals and some inorganic compounds, the bcc unit cell is one of several arrangements determining the physical properties of the material. For silicon tetrafluoride, if we consider it in a bcc structure, this dramatically impacts the way we calculate the number of atoms per unit cell, as we'll see in the next section.

Unit Cell Atom Counting

Unit cell atom counting is the process used to determine the number of atoms within a single unit cell of a crystal. This is not always straightforward, as atoms can be shared between multiple unit cells. For example, in a body-centered cubic (bcc) unit cell, atoms located at the corners are shared among eight adjacent cells, and thus each contributes only 1/8th of an atom to the unit cell.

To perform the count accurately, one must account for the atom at the center of the bcc structure, which belongs entirely to the unit cell, as well as the contributions from the atoms at the corners. This results in a mathematical approach: summing the fractional parts of the corner atoms and adding the whole center atom. The beauty of this counting method is that it tells us not only how many atoms are present but also provides insights into material properties such as density and packing efficiency.

In our case with silicon tetrafluoride arranged in a bcc unit cell, we find two silicon atoms per unit cell: one whole atom at the center and the sum of one eighth from each of the eight corner positions. This counting method is essential for understanding material compositions in chemistry, materials science, and several other related fields.