12.3.4 Calculating Atomic Mass and Radius from a Unit Cell

Question 1

Calculating Atomic Mass and Radius from a Unit Cell

Answer 1

• Atomic mass and atomic radius can be calculated based on unit cell data.

Question 2

note

Answer 2

• In the face-centered cubic (fcc) lattice shown, each corner represents one eighth of a sphere. There is a half sphere in the middle of each face. Thus there are four atoms in one unit cell.
• If the unit cell type, the length of an edge, and the density of the crystal are known, then the volume, mass and radius of individual atoms may be calculated.
• Problem: Silver crystallizes in a face-centered cubic crystal lattice. The length of one edge of the unit cell is 408.6 picometers. The density of silver metal is 10.50 g/cm 3 . What is the mass of a silver atom?
• Use volume in the density formula to calculate the mass of a unit cell.
• Dividing the mass of a unit cell by the number of atoms in the unit cell allows calculation of the mass of a single atom.
• Problem: What is the radius of a silver atom?
• Allow the length of the unit cell edge (d) to be the base of a right triangle. The value for the hypotenuse of this right triangle, which contains four atomic radii, can be calculated using the Pythagorean theorem.
• Dividing the resulting value by 4 yields the atomic radius of a silver atom.
• A knowledge of unit cell arrangement permits calculation of values for mass and radius for other standard crystal arrangements.
• The simple cubic unit cell has a single atom in the unit cell. The cell edge (d) / atomic radius (r) relationship is d = 2 · r.
• The body-centered unit cell contains two atoms.The cell edge / atomic radius relationship is · d = 4 · r.
• The face-centered unit cell contains four atoms.The cell edge / atomic radius relationship is · d = 4 · r.

Question 3

What must be known about a cubic unit cell in a metal crystal in order to be able calculate the mass and radius of an atom of that metal?

Answer 3

Size of unit cell, type of unit cell, and density of the metal

Question 4

In a face centered cubic crystal lattice, each corner atom is shared between how many cells?

Answer 4

eight

Question 5

The mass of a silver atom can be determined if we know the density of silver and the length of a(n) ____ .

Answer 5

unit cell

Question 6

The radius of a silver atom is 144 pm. Silver forms a face-centered cubic crystal. What is the length of the unit cell in a silver atom?

Answer 6

407 pm

Question 7

The length of the edge of a face-centered cubic cell of iridium is 383 pm. What is the radius of an atom of iridium?

Answer 7

135 pm

Question 8

What is the total number of atoms contained in a simple cubic cell?

Answer 8

one

Question 9

Examine the diagram of a face centered cubic unit cell. What is the length of the cell, d, in terms of the radius of an atom, r?

Answer 9

d = 2 sqrt 2r

Question 10

For convenience in doing calculations with crystals and unit cells, chemists like to think of atoms as what?

Answer 10

Hard spheres

Question 11

What is the difference between a simple cubic, a face-centered cubic, and a body-centered cubic unit cell?

Answer 11

They differ in the number, arrangement, and spacing of atoms

Question 12

The length of the edge of a face-centered cubic cell of iridium is 383 pm. The density of iridium is 22.42 g / cm3. What is the mass of an atom of iridium?

Answer 12

3.15 × 10^−22 g