Ch.12 - Solids and Solid-State Materials

Problem 141c

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Chapter 12, Problem 141c

The mineral wustite is a nonstoichiometric iron oxide with the empirical formula FexO, where x is a number slightly less than 1. Wustite can be regarded as an FeO in which some of the Fe sites are vacant. It has a density of 5.75 g>cm3, a cubic unit cell with an edge length of 431 pm, and a facecentered cubic arrangement of oxygen atoms. (c) Each Fe atom in wustite is in either the +2 or the +3 oxidation state. What percent of the Fe atoms are in the +3 oxidation state?

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Determine the number of oxygen atoms per unit cell in the face-centered cubic (fcc) arrangement. In an fcc lattice, there are 4 atoms per unit cell.

Calculate the total mass of one unit cell using the density and volume of the unit cell. The volume of the unit cell can be calculated using the formula for the volume of a cube, V = a^3, where a is the edge length.

Using the empirical formula FexO and the mass of the unit cell, calculate the total number of moles of Fe and O in the unit cell. Assume the simplest case where x is close to 1 for initial calculations.

Apply charge neutrality to the system. Since each O2- ion contributes 2 negative charges, the total positive charge contributed by Fe must balance this. Use the oxidation states of Fe (+2 and +3) to set up an equation for charge balance.

Solve the charge balance equation to find the ratio of Fe(III) to total Fe. Multiply this ratio by 100 to find the percentage of Fe atoms that are in the +3 oxidation state.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Nonstoichiometric Compounds

Nonstoichiometric compounds, like wustite (FexO), do not have a fixed ratio of elements. Instead, they exhibit variations in composition due to the presence of vacancies or defects in their crystal structure. In wustite, some iron (Fe) sites are unoccupied, leading to a formula where 'x' is slightly less than 1, indicating that the actual ratio of iron to oxygen can fluctuate.

Oxidation States

Oxidation states represent the degree of oxidation of an atom in a compound, indicating how many electrons an atom has gained or lost. In wustite, iron can exist in two oxidation states: +2 and +3. Understanding these states is crucial for determining the distribution of iron atoms in the compound and calculating the percentage of iron in each state.

Density and Unit Cell Calculations

The density of a compound and its unit cell dimensions are essential for understanding its structure and composition. The density can be calculated using the formula density = mass/volume, where the mass is derived from the molar mass of the components and the volume is based on the unit cell dimensions. In wustite, knowing the edge length of the cubic unit cell helps in determining the number of formula units per cell, which is necessary for calculating the distribution of oxidation states.

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A cube-shaped crystal of an alkali metal, 1.62 mm on an edge, was vaporized in a 500.0 mL evacuated flask. The resulting vapor pressure was 12.5 mm Hg at 802 °C. The structure of the solid metal is known to be body-centered cubic. (c) What are the densities of the solid and the vapor in g>cm3?

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Assume that 1.588 g of an alkali metal undergoes complete reaction with the amount of gaseous halogen contained in a 0.500 L flask at 298 K and 755 mm Hg pressure. In the reaction, 22.83 kJ is released 1ΔH = -22.83 kJ2. The product, a binary ionic compound, crystallizes in a unit cell with anions in a face-centered cubic arrangement and with cations centered along each edge between anions. In addition, there is a cation in the center of the cube. (c) Sketch a space-filling, head-on view of the unit cell, labeling the ions. Are the anions in contact with one another?

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Europium(II) oxide is a semiconductor with a band gap of 108 kJ/mol. Below 69 K, it is also ferromagnetic, meaning all the unpaired electrons on europium are aligned in the same direction. How many f electrons are present on each europium ion in EuO? (In lanthanide ions the 4f orbitals are lower in energy than the 6s orbitals.)

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Textbook Question

The mineral wustite is a nonstoichiometric iron oxide with the empirical formula FexO, where x is a number slightly less than 1. Wustite can be regarded as an FeO in which some of the Fe sites are vacant. It has a density of 5.75 g>cm3, a cubic unit cell with an edge length of 431 pm, and a facecentered cubic arrangement of oxygen atoms. (d) Using X rays with a wavelength of 70.93 pm, at what angle would third-order diffraction be observed from the planes of atoms that coincide with the faces of the unit cells? Third-order diffraction means that the value of n in the Bragg equation is equal to 3.

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The alkali metal fulleride superconductors M3C60 have a cubic closest-packed (face-centered cubic) arrangement of nearly spherical C60 3- anions with M+ cations in the holes between the larger C603- ions. The holes are of two types: octahedral holes, which are surrounded octahedrally by six C603- ions; and tetrahedral holes, which are surrounded tetrahedrally by four C603- ions. (c) Specify fractional coordinates for all the octahedral and tetrahedral holes. (Fractional coordinates are fractions of the unit cell edge lengths. For example, a hole at the center of the cell has fractional coordinates 12, 12, 12.)

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