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Chapter 5: Problem 56

When an oxide of uranium crystallizes the uranium cations form an expanded cubic close-packed array with an oxide ion in each tetrahedral hole. (a) Determine the coordination numbers of the two ions. (b) Write the formula of the oxide.

Short Answer

Expert verified
The coordination number for uranium is 12 and for oxide is 4. The formula of the oxide is UO.

Step by step solution

01

Determine uranium cation coordination number

In a cubic close-packed (ccp) or face-centered cubic (fcc) array, each sphere (cation) is surrounded by 12 other spheres. Thus, the coordination number for the uranium cation in this arrangement will be 12.
02

Determine oxide ion coordination number

Each oxide ion fills a tetrahedral hole in the ccp structure. In a tetrahedral hole, an ion is surrounded by 4 cations. Therefore, the coordination number for the oxide ion is 4.
03

Calculate the formula of the oxide

Given that each oxide ion is in a tetrahedral hole and considering that in a ccp structure there is one tetrahedral hole per sphere (cation), there will be an equal number of oxide ions to uranium ions. Hence, the simplest whole number ratio of uranium to oxygen is 1:1. The formula for the oxide is UO.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cubic Close-packed Structure
In crystallography, a cubic close-packed structure is a highly ordered arrangement of atoms within a crystal. This structure is also known as a face-centered cubic (fcc) lattice, and it’s characterized by layers of atoms stacked in a sequence where each atom is surrounded by 12 others.

The cubic close-packed structure is one of the most efficient ways to fill space with identical spheres, achieving a packing efficiency of about 74%. This structure plays a critical role in determining the properties of the material, including its density and how it interacts with other substances.
Tetrahedral Holes in Crystals
Crystals often have interstices or 'holes' where smaller atoms can reside. One common type of interstitial site is called a tetrahedral hole. These holes are spaces where four atoms in the structure form a tetrahedron.

In a cubic close-packed structure, each atom touches four atoms in the layer above and four in the layer below, creating a void in the middle. These tetrahedral holes are essential for understanding how different atoms can integrate into metal structures, forming compounds, and are crucial for determining the stoichiometry of the resulting compounds.
Uranium Oxide Formula
The formula for a compound gives the proportion of each element present. For uranium oxide, where uranium cations form an expanded cubic close-packed array with each oxide ion occupying a tetrahedral hole, we can deduce the formula using the coordination numbers.

Since the uranium cation has a coordination number of 12 and the oxide ion has a coordination number of 4 within this structure, and considering that there's one oxide ion per uranium ion, the simplest ratio is 1:1. Thus, the chemical formula for uranium oxide is represented as UO, indicating an equal number of uranium and oxygen atoms.
Face-centered Cubic Lattice
A face-centered cubic lattice is a type of crystal structure where each face of the cube has an atom at its center, in addition to the atoms at the cube's corners. This configuration leads to each atom being surrounded by 12 equidistant neighbors, hence a coordination number of 12.

The face-centered cubic lattice is identical to the cubic close-packed structure in terms of the arranged layers of atoms. It is pervasive in metals and their alloys, with materials structured in this way often displaying high density and stability.

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Most popular questions from this chapter

(a) If a pure element crystallizes with a primitive cubic lattice, what percentage of the unit cell is empty space? (b) How does this percentage compare with that of empty space in an fcc unit cell?

Classify each of the following solids as ionic, network, metallic, or molecular: (a) iron pyrite (fool's gold), \(\mathrm{FeS}_{2}\); (b) octane (a component of gasoline), \(\mathrm{C}_{8} \mathrm{H}_{18}\); (c) cubic boron nitride (a compound with a structure similar to that of diamond, but with alternating boron and nitrogen atoms), BN; (d) calcium sulfate (gypsum), \(\mathrm{CaSO}_{4}\); (e) the chromium plating on a motorcycle.

All the alkali metals crystallize with bec structures. (a) Find a general equation relating the metallic radius to the density of a bcc solid of an element in terms of its molar mass and use it to deduce the atomic radius of each of the following elements, given the density of each \(\left(\right.\) in \(\left.\mathrm{g} \cdot \mathrm{cm}^{-3}\right): \mathrm{Li}, 0.53 ; \mathrm{Na}, 0.97 ; \mathrm{K}, 0.86 ; \mathrm{Rb}, 1.53\); Cs, 1.87. (b) Find a factor for converting the density of a bcc element into the density that it would have if it crystallized in a ccp structure. (c) Calculate what the densities of the alkali metals would be if they were ccp. (d) Which, if any, would float on water?

An oxide of rhenium crystallizes with a cubic unit cell that has a rhenium cation at each corner and an oxide ion at the center of each edge of the crystal. (a) Determine the coordination numbers of the two ions. (b) Write the formula of the oxide.

For which of the following molecules will dipole-dipole interactions be important: (a) \(\mathrm{O}_{2} ;\) (b) \(\mathrm{O}_{3} ;\) (c) \(\mathrm{CO}_{2}\); (d) \(\mathrm{SO}_{2}\) ?
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