Question

The fraction of volume occupied by atoms in a face centered cubic unit cell is: (a) \(0.32\) (b) \(0.48\) (c) \(0.68\) (d) \(0.74\)

Short answer

The fraction of volume occupied by atoms in a face centered cubic unit cell is approximately 0.74.

Step-by-step solution

Understanding the face-centered cubic (FCC) structure

A face-centered cubic unit cell consists of atoms at each of the eight corners and in the center of each face. Each corner atom is shared by eight adjacent unit cells and each face-centered atom is shared by two unit cells, making an effective number of atoms inside a single unit cell equal to 4.

Calculating the volume of one atom

In a FCC unit cell, the atoms are in contact along the face diagonal. The face diagonal length is four times the radius of an atom (r), thus the face diagonal is given by \(4r\). The edge length (a) of the unit cell can then be found using the diagonal of a square: \(a\sqrt{2}=4r\), which gives us \(a=\frac{4r}{\sqrt{2}}=2\sqrt{2}r\).

Calculating the volume of the unit cell

The volume of the cube (V_{cell}) is given by the cube of the edge length \(a\), so we get \(V_{cell}=a^3=(2\sqrt{2}r)^3=16\sqrt{2}r^3\).

Calculating the volume occupied by atoms

Since there are effectively 4 atoms in the unit cell and the volume of one atom (considered as a sphere) is \(\frac{4}{3}\pi r^3\), the total volume occupied by atoms in the unit cell is \(4\times\frac{4}{3}\pi r^3=\frac{16}{3}\pi r^3\).

Calculating the fraction of volume occupied

The fraction of volume occupied by atoms is the ratio of the volume occupied by atoms to the volume of the unit cell. This is given by\[\frac{\text{Volume occupied by atoms}}{\text{Volume of unit cell}} = \frac{\frac{16}{3}\pi r^3}{16\sqrt{2}r^3}\].

Simplifying the expression

Upon simplifying the expression, we get \[\frac{16\pi}{3\times16\sqrt{2}}\], which further simplifies to \[\frac{\pi}{3\sqrt{2}}\].

Calculating the numerical value

Using a calculator, we can find the decimal value for \(\frac{\pi}{3\sqrt{2}}\) which is approximately 0.7405...

Comparing with given options

The calculated decimal value (approximately 0.74) corresponds to option (d), so this is the fraction of volume occupied by atoms in a face centered cubic unit cell.

Key concepts

Face-Centered Cubic Structure

The face-centered cubic (FCC) structure is one of the most prevalent crystal structures found in nature, important for both its high packing efficiency and its properties in solid state chemistry. Imagine a cube where each corner and the center of each face features an atom. These corner atoms aren't solely part of one unit cell—they're shared between eight unit cells. Face-centered atoms, however, are only shared between two. This unique arrangement means each unit cell effectively contains 4 atoms.

In the FCC, atoms are arranged in a way such that they touch each other along the face diagonals. Visualize this by thinking of a dice where, instead of spots, each face has a bulging atom right at its center. The interplay of these atoms in the FCC structure endows the material with characteristics that are exploited in various industrial applications, including metallurgy and electronics.

Unit Cell Calculation

To grasp unit cell calculation, it's key to understand the relationship between the atoms' sizes and the cell's edge length. In an FCC structure, atoms touch along the diagonal spanning the face, which is four times the atomic radius (\r\text{r}). From that, we can tease out the unit cell's edge length using the face diagonal and a bit of Pythagorean magic. This gives us the formula to link the atomic radius and the cube's edge: \(a = 2\sqrt{2}r\). With the edge length in hand, calculating the unit cell volume is straightforward: just raise the edge length to the third power.

The beauty of this process lies in its simplicity. Once we have the volume of the unit cell and the volume an atom occupies—envisioned as a sphere—we can measure how much of the cell is actually 'filled' by atoms. This is vital for understanding the material's density and, hence, predicting its properties in differing contexts, such as strength, ductility, or thermal conductivity.

Solid State Chemistry

Solid state chemistry dives into the study of the synthesis, structure, and properties of solid phase materials. The face-centered cubic unit cell helps us understand the fundamental properties of metals and ceramics. The packing of atoms within the cell informs its properties: the closer the packing, the higher the melting point and the greater the density.

The concept of the fraction of volume occupied by atoms—a central topic in our exercise—is crucial in this field. It affects electrical conductivity, malleability, and a variety of other physical characteristics. By calculating the occupied volume fraction, as shown in Steps 4 through 8, we develop a deeper comprehension of why materials behave the way they do when they are stretched, bent, heated, or electrified.