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.

Verified step by step guidance

1. 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.
2. Determine the number of unit cells in the crystal by dividing the volume of the crystal by the volume of a single unit cell. The volume of a unit cell in a body-centered cubic structure can be calculated using the formula V_{cell} = (2r)^3, where r is the radius of an atom.
3. Calculate the number of atoms in the crystal. In a body-centered cubic structure, there are 2 atoms per unit cell.
4. Use the ideal gas law to find the number of moles of the metal vapor in the flask. The ideal gas law is PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature in Kelvin.
5. Compare the calculated molar mass of the metal with the molar masses of known alkali metals to identify the metal. The molar mass can be calculated by dividing the mass of the metal by the number of moles of metal vapor.

Key Concepts

Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in equilibrium with its liquid or solid phase at a given temperature. It indicates how readily a substance vaporizes; higher vapor pressures suggest a greater tendency to evaporate. In this question, the vapor pressure of the alkali metal at a specific temperature is crucial for identifying the metal based on its physical properties.

Body-Centered Cubic Structure

A 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 physical properties of the metal, including its density and atomic packing. Understanding the BCC structure helps in determining the characteristics of the alkali metal in question.

Alkali Metals

Alkali metals are a group of elements in Group 1 of the periodic table, characterized by their single valence electron, high reactivity, and low density. They include lithium, sodium, potassium, rubidium, cesium, and francium. Identifying the specific alkali metal involves comparing the calculated properties, such as molar mass and vapor pressure, with known values for these elements.