If the density of solid iron is 7.87 g/cm3 , what is the packing efficiency of Fe if it adopts a body‑centered cubic unit cell? The molar mass of Fe is 55.847 g/mol .
Nevermind I got it was 68%
Good work. 68% is correct.
Well, before I start clowning around, let me give you a straight answer.
For a body-centered cubic (BCC) structure, the packing efficiency can be calculated by dividing the volume of the atoms in a unit cell by the total volume of the unit cell.
In a BCC unit cell, there are two atoms: one atom at each corner of the cube, and an additional atom in the center. The volume of atoms in the unit cell can be calculated as (number of atoms) * (volume of each atom) = 2 * (4/3 * π * (r^3)) + 1 * (4/3 * π * (r^3)).
Now, the volume of the unit cell is simply the length of one side cubed, since it's a cube.
Packing efficiency is given by (volume of atoms) / (volume of unit cell).
Got it? Or should I throw in a few jokes to keep things lively?
To find the packing efficiency of a body-centered cubic (BCC) unit cell, we need to consider the arrangement of atoms in the cell.
In a BCC unit cell, there are two atoms: one atom at each corner of the cube and another atom in the center of the cube.
The formula for packing efficiency is given by:
Packing efficiency = (Number of atoms in the unit cell * Atomic mass of the element) / (Volume of the unit cell * Density of the element)
Since we have 2 atoms in a BCC unit cell, the number of atoms in the unit cell is 2.
The atomic mass of Fe is given as 55.847 g/mol.
The density of solid iron is given as 7.87 g/cm3.
Now, let's calculate the volume of the BCC unit cell:
The edge length of the cube can be calculated using the formula:
Edge length = (4 * (Density / Atomic mass))^0.5
Substituting the given values, we get:
Edge length = (4 * (7.87 g/cm3 / 55.847 g/mol))^0.5
≈ (0.2813 cm)^0.5
≈ 0.530 cm
Now, let's calculate the volume of the unit cell:
Volume of the unit cell = (Edge length)^3
= (0.530 cm)^3
≈ 0.149 cm3
Finally, let's calculate the packing efficiency:
Packing efficiency = (2 * 55.847 g/mol) / (0.149 cm3 * 7.87 g/cm3)
= 111.694 g/mol / 1.17463 cm3
≈ 94.98 %
Therefore, the packing efficiency of Fe in a body-centered cubic unit cell is approximately 94.98%.
To calculate the packing efficiency of iron (Fe) in a body-centered cubic (bcc) unit cell, we need to determine the volume occupied by each iron atom in the unit cell and compare it to the total volume of the unit cell.
First, let's find the volume of the bcc unit cell. In a bcc structure, there are 2 atoms per unit cell. Therefore, the volume of the unit cell (V) can be determined using the following formula:
V = (4 * r^3) / 3
Where "r" is the distance from the center of the unit cell to the atom at the corners.
Since the Fe atom is at the corners of the unit cell and the body-center of the unit cell, the distance from the center to the corner atom (r) can be calculated as follows:
a = 4r / sqrt(3)
Where "a" is the length of one edge of the unit cell.
Next, let's calculate the volume of one Fe atom in the unit cell. Since there are two Fe atoms per unit cell, we can calculate the volume of one Fe atom (V_atom) as:
V_atom = V / 2
Now, we need to divide the molar mass of Fe by Avogadro's number to determine the mass of one Fe atom (m_atom):
m_atom = molar mass / Avogadro's number
Finally, we can calculate the density (rho) of iron using the equation:
rho = m_atom / V_atom
To find the packing efficiency, we can compare the density of iron to the density of solid iron (given in the question) using the formula:
packing efficiency = (rho / density of solid Fe) * 100%
Now, let's substitute the given values into the formulas and calculate the packing efficiency:
Given:
Density of solid iron (Fe) = 7.87 g/cm^3
Molar mass of Fe = 55.847 g/mol
Solution:
Step 1: Calculate the volume of the bcc unit cell:
V = (4 * r^3) / 3
Step 2: Calculate the distance from the center to the corner atom:
a = 4r / sqrt(3)
Step 3: Calculate the volume of one Fe atom in the unit cell:
V_atom = V / 2
Step 4: Calculate the mass of one Fe atom:
m_atom = molar mass / Avogadro's number
Step 5: Calculate the density of Fe:
rho = m_atom / V_atom
Step 6: Calculate the packing efficiency:
packing efficiency = (rho / density of solid Fe) * 100%