Copper has density of 8.94 and crystallizes with the face-centered cubic unit cell. Calculate the radius of a copper atom.
There are 4 atoms to the unit cell so the mass of a unit cell is
4*atomic mass Cu/6.02E23 = ?
density = mass/volume
Substitute mass and density and solve for volume = ?
V1/3 = edge length = a
4r = a*21/2
Solve for r. My periodic table says Cu is 1.28 angstroms.
127.3
To calculate the radius of a copper atom, we can use the formula:
Radius = (3 * Volume / (4 * π * density))^1/3
Given information:
Density of copper = 8.94 g/cm³
To calculate the volume of the unit cell, we need to know the edge length (a) of the unit cell.
In a face-centered cubic (FCC) structure, there are four atoms per unit cell.
The volume of the unit cell can be calculated as:
Volume = (a)^3 * (number of atoms per unit cell / Avogadro's number)
Avogadro's number = 6.022 × 10^23 mol⁻¹
For an FCC unit cell, there are four atoms per unit cell.
Substituting the given values:
Density = 8.94 g/cm³
Number of atoms per unit cell = 4
Avogadro's number = 6.022 × 10^23 mol⁻¹
Let's calculate the radius of a copper atom step-by-step:
Step 1: Calculate the volume of the unit cell
Volume = (a^3 * 4) / (6.022 × 10^23)
Step 2: Rearrange the formula to solve for a (edge length):
a^3 = (Volume * 6.022 × 10^23) / 4
Step 3: Calculate the edge length (a):
a = (Volume * 6.022 × 10^23) / 4 ^(1/3)
Step 4: Calculate the radius
Radius = (3 * Volume / (4 * π * density)) ^(1/3)
Now, we can substitute the value of a (edge length) obtained from Step 3 into the formula to calculate the radius.
To calculate the radius of a copper atom, we need to make use of the given information about its density and crystal structure.
Step 1: Determine the coordination number of a face-centered cubic (FCC) unit cell.
In a FCC unit cell, each atom is surrounded by 12 nearest neighbors. This implies that the coordination number (CN) is 12.
Step 2: Calculate the volume of the unit cell.
The volume (V) of a FCC unit cell can be calculated using the formula:
V = (a^3) / 4, where 'a' is the length of the edge of the unit cell.
Step 3: Calculate the volume of a copper atom.
Since there are 4 atoms per unit cell in FCC structure, the volume occupied by a copper atom (V_atom) can be calculated by dividing the volume of the unit cell by the number of atoms per unit cell:
V_atom = V / 4
Step 4: Calculate the mass of a copper atom.
The density (d) of copper is given as 8.94 g/cm³. Using the formula:
density = mass / volume, we can rearrange to find the mass (m) of a copper atom:
m = density * V_atom
Step 5: Calculate the radius of a copper atom.
The volume of a sphere (V_sphere) can be calculated using:
V_sphere = (4/3) * π * (radius^3). Rearranging the formula, we can solve for the radius of an atom:
radius = cube root [(3 * mass) / (4 * π * density)]
Now, let's calculate the radius of a copper atom using the given data:
Step 1: CN = 12
Step 2: V = (a^3) / 4
Step 3: V_atom = V / 4
Step 4: m = density * V_atom
Step 5: radius = cube root [(3 * m) / (4 * π * density)]
After performing these calculations, we can find the radius of a copper atom.