The element platinum has ccp packing with a face-centered cubic unit cell. The density of platinum is 21.5 g/cm3. Calculate the volume (cm3) of the unit cell of platinum.
You should be able to do this from the data and the procedures in the previous problems.
To calculate the volume of the face-centered cubic (FCC) unit cell of platinum, we need to know the atomic weight (mass) of platinum (Pt). The atomic weight of platinum is approximately 195.08 g/mol.
Step 1: Determine the number of platinum atoms in the unit cell:
In a face-centered cubic (FCC) unit cell, there are 4 atoms located at each of the corners and 1 atom located at the center of each face. Thus, each unit cell contains 4 corner atoms and 1 face-centered atom, totaling 5 platinum atoms.
Step 2: Determine the mass of platinum atoms in the unit cell:
To determine the mass of platinum atoms in the unit cell, we need to calculate the mass of one platinum atom. This can be done by dividing the atomic weight of platinum (195.08 g/mol) by Avogadro's number (6.022 x 10^23 atoms/mol).
Mass of one platinum atom = Atomic weight / Avogadro's number
= 195.08 g/mol / (6.022 x 10^23 atoms/mol)
≈ 3.245 x 10^-22 g
Step 3: Calculate the mass of platinum in the unit cell:
To calculate the mass of platinum in the unit cell, we multiply the mass of one platinum atom (from step 2) by the number of platinum atoms in the unit cell (from step 1).
Mass of platinum in the unit cell = Mass of one platinum atom x Number of platinum atoms
= 3.245 x 10^-22 g x 5 atoms
≈ 1.62 x 10^-21 g
Step 4: Calculate the volume of the unit cell:
To calculate the volume of the unit cell, we need to use the density of platinum (21.5 g/cm^3) and the mass of platinum in the unit cell (from step 3).
Density = Mass / Volume
Volume = Mass / Density
Volume of the unit cell = Mass of platinum in the unit cell / Density
≈ (1.62 x 10^-21 g) / (21.5 g/cm^3)
≈ 7.53 x 10^-23 cm^3
Therefore, the volume of the unit cell of platinum is approximately 7.53 x 10^-23 cm^3.
To calculate the volume of a unit cell in a face-centered cubic (FCC) structure, we need to know the edge length of the unit cell, denoted as "a."
In an FCC structure, there are atoms at each of the eight corners of the unit cell and additional atoms at each of the face centers. Each atom at the corner contributes 1/8th of its volume to the unit cell, while each atom at the face center contributes its full volume.
The formula to calculate the volume of a unit cell in an FCC structure is given by:
Volume = (Number of atoms at the corner * 1/8th volume per atom) + (Number of atoms at the face center * volume per atom)
In the case of platinum, since it has FCC packing, there are 4 atoms in each unit cell. This means that there are 8 atoms at the corners and 6 atoms at the face centers (since each face has one atom).
Now, the density of platinum is given as 21.5 g/cm^3. We can use the density to calculate the volume of each platinum atom.
The molar mass of platinum is approximately 195.1 g/mol. Since there are 4 platinum atoms in each unit cell, the mass of each platinum atom in the unit cell would be (195.1 g/mol) / (4 atoms) = 48.775 g.
Now, we can use the density and mass to calculate the volume per platinum atom:
Density = Mass / Volume
Volume = Mass / Density
Volume = 48.775 g / 21.5 g/cm^3
Volume = 2.272 cm^3
Now, we can substitute the values into the formula for the volume of the unit cell:
Volume = (8 atoms * 1/8th volume per atom) + (6 atoms * volume per atom)
Volume = (8 atoms * 1/8 * 2.272 cm^3/atom) + (6 atoms * 2.272 cm^3/atom)
Volume = (8 cm^3) + (13.632 cm^3)
Volume = 21.632 cm^3
Therefore, the volume of the unit cell of platinum in its face-centered cubic (FCC) packing is 21.632 cm^3.