Problem 90 - Chemistry The Science in Context

Question

The unit cell of an oxide of uranium consists of cubic closest-packed uranium ions with oxide ions in all the tetrahedral holes. What is the formula of the oxide?

Step-by-Step Solution

Answer

Based on this, what is the formula of the uranium oxide? Short Answer: In the ccp structure unit cell, there are 4 uranium ions and 4 oxide ions. Therefore, the formula of the uranium oxide is UO.

Step 1: Understand the atomic arrangement in the unit cell

In a cubic closest-packed (ccp) structure, the atoms are arranged in a way that maximizes atomic density. The uranium ions are in a ccp arrangement, while the oxide ions are present in all tetrahedral holes.

Step 2: Determine the number of uranium ions in a unit cell

In a ccp structure, the atoms at the vertices and face centers of the unit cell are shared with neighboring unit cells. There are 8 vertices and 6 face centers for each cube. Each vertex atom is shared by 8 unit cells, and each face center atom is shared by 2 unit cells. Therefore, the number of uranium ions in one unit cell can be calculated as follows: Number of uranium ions = (8 vertices × 1/8) + (6 face centers × 1/2) = 1 + 3 = 4

Step 3: Determine the number of tetrahedral holes in a unit cell

In a ccp structure, there is one tetrahedral hole per atom. Since there are 4 uranium ions in the unit cell, there are 4 tetrahedral holes where oxide ions are present.

Step 4: Find the formula of the oxide

Now that we know there are 4 uranium ions and 4 oxide ions in a unit cell, we can determine the empirical formula of the uranium oxide: Uranium:Oxide = 4:4 = 1:1 Thus, the formula of the uranium oxide is UO.

Key Concepts

Cubic Closest-Packed Structure

The cubic closest-packed (ccp) structure is one of the most efficient ways to pack atoms in a crystal lattice. Imagine a close arrangement where every sphere (atom) touches several others to fill space as densely as possible. In this arrangement, each atom has 12 direct neighbors, forming a repeating pattern that creates a unit cell.
The cubic closest-packed structure is often referred to interchangeably with face-centered cubic (fcc) because of how atoms are organized in a cube. Here are some key points you need to know:

• In a cubic closest-packed arrangement, atoms occupy both the corners and the centers of all cube faces.
• The atoms at the corners are shared by eight different unit cells, while the face-centered atoms are shared by two.
• This leads to having four equivalent whole atoms within the unit cell. This sharing is key to calculating quantities of elements in the lattice.

Understanding this arrangement is fundamental because it sets the stage for how we calculate the number of tetrahedral holes, which is crucial in determining the empirical formula of a compound like uranium oxide.

Tetrahedral Holes

The concept of tetrahedral holes is essential when exploring how additional atoms fit into the gaps of a crystal lattice. In a cubic closest-packed structure, these holes play a significant role:
Tetrahedral holes are formed between four atoms that form a tetrahedron, hence the name. These spaces allow smaller atoms or ions to nestle between the larger host atoms.

• In a ccp structure, there is exactly one tetrahedral hole for every atom in the lattice.
• For the cubic closest-packed arrangement of uranium in this example, with four uranium atoms per unit cell, there are thus four tetrahedral holes available.
• These holes are perfectly suited for accommodating smaller ions, such as oxide ions in our uranium oxide example.

The number and nature of these holes directly impact the stoichiometry of compounds that crystallize in such structures. For uranium oxide, filling all tetrahedral holes with oxide ions provides an essential clue in finding its empirical formula.

Empirical Formula Determination

The empirical formula is a simplified representation of a compound's constituent elements in the smallest whole number ratio. For uranium oxide, this determination involves blending our understanding of cubic closest-packed configurations and tetrahedral holes:
Based on our understanding that one unit cell of uranium oxide comprises four uranium ions in the ccp structure, and four corresponding tetrahedral holes filled with oxide ions:

• The count of uranium ions in a unit cell is straightforward—four, based on lattice composition.
• Each tetrahedral hole accommodates one oxide ion, adding up directly to four oxide ions due to a one-to-one relationship with uranium atoms.
• This gives us a total count of both uranium and oxide ions equal per unit cell.

Thus the ratio is straightforward: 4 uranium ions to 4 oxide ions. Simplifying this ratio provides the empirical formula: UO. This formula represents the basic composition per structural unit in the compound, reflecting a one-to-one ratio of elements.