You are given a small bar of an unknown metal X. You find the density of the metal to be \(10.5 \mathrm{g} / \mathrm{cm}^{3} .\) An X-ray diffraction experiment measures the edge of the face-centered cubic unit cell as \(4.09 Å\left(1 Å=10^{-10} \mathrm{m}\right) .\) Identify X.

Short Answer

Expert verified
The unknown metal X can be identified as silver (Ag) based on its density, face-centered cubic unit cell structure, and atomic mass of approximately 108 u.

Step by step solution

01

Determine the number of atoms in an FCC unit cell

In a face-centered cubic unit cell, there is 1 atom at each corner and 1 atom at the center of each face. There are 8 corners and 6 faces in an FCC unit cell. From this information, we can calculate the total number of atoms in a FCC unit cell: Number of atoms = 8 (atoms at corners) * 1/8 + 6 (atoms at faces) * 1/2 = 1 * 8 + 3 * 6 = 1 + 18 = 4 atoms per unit cell
02

Calculate the volume of the unit cell

We are given the edge length of the unit cell as 4.09 Å, and we need to convert it to meters (1 Å = 10^{-10} m): Edge length = 4.09 Å * (10^{-10} m/Å) = 4.09 * 10^{-10} m Now, we can calculate the volume of the unit cell: Volume = edge length^3 = (4.09 * 10^{-10} m)^3 = 6.84 * 10^{-29} m^3
03

Calculate the mass of the unit cell

We are given the density of the metal as 10.5 g/cm^3, and we need to convert it to kg/m^3: Density = 10.5 g/cm^3 * (1 kg/1000 g) * (100 cm/m)^3 = 10.5 * 10^3 kg/m^3 Now, calculate the mass of the unit cell using density and volume: Mass = density * volume = 10.5 * 10^3 kg/m^3 * 6.84 * 10^{-29} m^3 = 71.82 * 10^{-25} kg
04

Calculate the mass of one atom and identify the metal

We determined that there are 4 atoms in the unit cell. To find the mass of one atom, divide the mass of the unit cell by the number of atoms: Mass of one atom = (71.82 * 10^{-25} kg) / 4 = 17.96 * 10^{-25} kg Now, we need to convert the mass of one atom to atomic mass units (u) to identify the metal using the periodic table. The conversion is 1 u = 1.66054 * 10^{-27} kg. Mass of one atom in atomic mass units (u) = 17.96 * 10^{-25} kg * (1 u / 1.66054 * 10^{-27} kg) = 107.87 u With a mass of approximately 108 u, the unknown metal X is most likely silver (Ag), which has an atomic mass of about 107.87 u.

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