Ch.12 - Solids and Solid-State Materials

Problem 137c

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Chapter 12, Problem 137c

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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Calculate the volume of the cube-shaped crystal using the formula for the volume of a cube, V = s^3, where s is the length of an edge of the cube.

Determine the number of unit cells in the crystal by considering the body-centered cubic (bcc) structure. In a bcc lattice, there are 2 atoms per unit cell. Calculate the number of unit cells by dividing the total volume of the cube by the volume of a single unit cell.

Calculate the molar mass of the alkali metal using the Avogadro's number and the number of atoms obtained from the number of unit cells.

Use the ideal gas law, PV = nRT, to find the number of moles of the alkali metal in the vapor phase. Use the temperature in Kelvin and the vapor pressure in atmospheres.

Calculate the density of the solid and the vapor. For the solid, use the mass (from molar mass and number of moles) and the volume of the cube. For the vapor, use the mass and the volume of the flask.

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Key Concepts

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

Density

Density is defined as mass per unit volume, typically expressed in grams per cubic centimeter (g/cm³). It is a crucial property that helps in identifying substances and understanding their behavior in different states. For solids, density can be calculated using the formula density = mass/volume, while for gases, it can be derived from the ideal gas law under specific conditions.

Body-Centered Cubic (BCC) Structure

The body-centered cubic (BCC) structure is a type of crystal lattice where atoms are located at each corner of a cube and a single atom is positioned at the center of the cube. This arrangement affects the packing efficiency and density of the solid. Understanding the BCC structure is essential for calculating the mass and volume of the alkali metal crystal, which are needed to determine its density.

Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in equilibrium with its liquid or solid phase at a given temperature. It is an important concept in thermodynamics and helps in understanding the behavior of substances in the gaseous state. In this question, the vapor pressure of the alkali metal at 802 °C is used to calculate the density of the vapor, which is essential for comparing it to the density of the solid.

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Open Question

The mineral magnetite is an iron oxide ore that has a density of 5.20 g/cm³. At high temperature, magnetite reacts with carbon monoxide to yield iron metal and carbon dioxide. When 2.660 g of magnetite is allowed to react with sufficient carbon monoxide, the CO₂ product is found to have a volume of 1.136 L at 298 K and 751 mm Hg pressure. (a) What mass of iron in grams is formed in the reaction? (b) What is the formula of magnetite?

Textbook Question

A group 3A metal has a density of 2.70 g/cm³ and a cubic unit cell with an edge length of 404 pm. Reaction of A 1.07 cm³ chunk of the metal with an excess of hydrochloric acid gives a colorless gas that occupies 4.00 L at 23.0 °C and a pressure of 740 mm Hg. (a) Identify the metal.

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Textbook Question

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. (b) Use the data in Figure 5.19 to identify the alkali metal.

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Textbook Question

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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Textbook Question

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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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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