Concept explainers
You are given a small bar of an unknown metal X. You find the density of the metal to be 10.5 g/cm3. An X-ray diffraction experiment measures the edge of the face-centered cubic unit cell as 4.09 Å (1 Å = 10−10 m). Identify X.
Interpretation:
A metal has to be identified given its density and edge length of FCC unit cell.
Concept introduction:
In crystalline solids, the components are packed in regular pattern and neatly stacked. The components are imagined as spheres and closely packed. This phenomenon is called “close packing” in crystals. The two major types of close packing of the spheres in the crystal are – hexagonal close packing and cubic close packing. Cubic close packing structure has face-centered cubic (FCC) unit cell.
In face-centered cubic unit cell, each of the six corners is occupied by every single atom. Each face of the cube is occupied by one atom.
Each atom in the corner is shared by eight unit cells and each atom in the face is shared by two unit cells. Thus the number of atoms per unit cell in FCC unit cell is,
Answer to Problem 57E
Answer
The metal is identified as “Silver”.
Explanation of Solution
Explanation
Calculate the volume of unit cell of unknown metal.
Density of the titanium is given and mass of the titanium is calculated in the previous step. Substituting these two values in the equation
Calculate the mass of unit cell of unknown metal.
Density of the unit cell is given. The mass of unit cell is calculated using the equation
Calculate the atomic mass of unknown metal.
In FCC unit cell of a metal, each unit cell has 4 metal atoms. So each metal atom has an average mass of 1/4th of the mass of unit cell. The atomic mass of a metal atom corresponds to the Avogadro number of the average mass of a metal atom.
Identify the metal corresponding to the atomic mass calculated.
The identified metal is “Silver”.
Platinum is the metal that possibly matches best with the calculated atomic mass.
Conclusion
Density and edge length of FCC unit cell of a metal is given and it is identified as “Silver”.
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Chapter 10 Solutions
Chemistry (Instructor's)
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