Concept explainers
Titanium metal has a body-centered cubic unit cell. The density of titanium is 4.50 g/cm3. Calculate the edge length of the unit cell and a value for the atomic radius of titanium. (Hint: In a body-centered arrangement of spheres, the spheres touch across the body diagonal.)
Interpretation:
The edge length of the unit cell of Titanium has to be determined.
Concept introduction:
In packing of atoms in a crystal structure, the atoms are imagined as spheres. The two major types of close packing of the spheres in the crystal are – hexagonal close packing and cubic close packing.
In body-centered cubic unit cell, each of the six corners is occupied by every single atom. Center of the cube is occupied by one atom.
Each atom in the corner is shared by eight unit cells and a single atom in the center of the cube remains unshared. Thus the number of atoms per unit cell in BCC unit cell is,
Answer to Problem 57E
Answer
The edge length of the unit cell of Titanium is 328 pm.
Explanation of Solution
Explanation
Calculate the mass of unit cell of Ti.
Each unit cell contains 2 Ti atoms. Therefore two times the average mass of one Ti atom gives mass of a unit cell.
Calculate the volume of unit cell of Ti.
Density of the Titanium is given and mass of the Titanium is calculated in the previous step. Substituting these two values in the equation
Determine the edge length of unit cell of Ti.
Volume of the bcc unit cell is calculated in the previous step. Cube root of the volume of a bcc unit cell, gives the edge length of bcc unit cell.
Conclusion
The edge length of the unit cell of Titanium is determined.
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