When silver crystallizes, it forms facecentered
cubic cells. The unit cell edge length is
409 pm. Calculate the density of silver.
Density: 10.63 g/cm^3
m: 1.44 x 10^-10
To find the density of silver, you need two pieces of information: the molar mass of silver and the volume of its unit cell.
Step 1: Find the molar mass of silver.
The molar mass of silver (Ag) is 107.87 g/mol.
Step 2: Calculate the volume of the unit cell.
The unit cell of a face-centered cubic structure has four atoms, one at each corner and one in the center of each face. Let's assume the edge length of the unit cell is given as 409 pm.
The volume of a cube with an edge length of a is given by V = a^3. However, in a face-centered cubic structure, each atom at the corners contributes 1/8th of its volume to the unit cell, while each atom on the face contributes 1/2 of its volume.
The volume of the unit cell is given by:
V_unitcell = (1/8 * volume of the corner atom) + (1/2 * volume of the face atom)
Let's calculate the volume of the corner atom:
Volume_corner = a^3 * 1/8
Let's calculate the volume of the face atom:
Volume_face = a^2 * (a/2) * 1/2
Now, add the two volumes to get the volume of the unit cell:
V_unitcell = Volume_corner + Volume_face
Step 3: Calculate the density.
The density of a material is defined as its mass divided by its volume:
Density = molar mass / V_unitcell
Let's calculate the density of silver using the given values:
Step 1: Molar mass of silver = 107.87 g/mol
Step 2: Volume of the unit cell:
- Volume_corner = (409 pm)^3 * 1/8 = 1.559 x 10^7 pm^3
- Volume_face = (409 pm)^2 * (409 pm/2) * 1/2 = 1.692 x 10^7 pm^3
- V_unitcell = Volume_corner + Volume_face = 1.559 x 10^7 pm^3 + 1.692 x 10^7 pm^3 = 3.251 x 10^7 pm^3
Step 3: Density = Molar mass / V_unitcell = 107.87 g/mol / 3.251 x 10^7 pm^3
However, the density is typically expressed in g/cm^3, so we need to convert the units.
1 pm = 1 x 10^-12 cm
(3.251 x 10^7 pm^3) * (1 x 10^-12 cm/pm)^3 = 3.251 x 10^-23 cm^3
Density = 107.87 g/mol / 3.251 x 10^-23 cm^3
Final Answer: The density of silver is approximately 10.5 g/cm^3.