The density of solid Cr is 7.15 g/cm3. How many atoms are present per cubic centimeter of Cr?
As a solid, Cr adopts a body-centered cubic unit cell. How many unit cells are present per cubic centimeter of Cr?
The density is 8.28e22
To determine the number of atoms present per cubic centimeter of Cr, we need to consider the unit cell arrangement in a body-centered cubic (BCC) crystal structure.
1. Determining the number of atoms per unit cell:
In a BCC crystal structure, there is one atom at each corner of the unit cell and one atom at the center of the unit cell. Hence, the total number of atoms per unit cell (N_uc) is 2.
2. Calculating the volume of the unit cell:
The volume of a BCC unit cell can be calculated using the formula:
V_uc = (4 * r^3 * π) / 3
where r is the radius of the atom. In the case of a BCC structure, the body diagonal of the unit cell is equal to 4 * r.
3. Converting density to mass per unit cell:
The density (ρ) of Cr given is 7.15 g/cm³. Since the volume of the unit cell (V_uc) is known, we can calculate the mass (m_uc) of the unit cell using the equation:
m_uc = ρ * V_uc
4. Determining the number of unit cells per cubic centimeter:
To find the number of unit cells per cubic centimeter, we need to calculate the volume of one unit cell and then divide the total volume (in cm³) by the volume of one unit cell.
Once we know the number of unit cells per cubic centimeter, we can multiply that by the number of atoms per unit cell to find the number of atoms present per cubic centimeter of Cr.
Overall, the steps involved are as follows:
- Calculate the volume of the unit cell (V_uc) using the formula.
- Calculate the mass (m_uc) of the unit cell using the density (ρ) and the volume (V_uc).
- Convert the mass of the unit cell to the number of atoms using Avogadro's number.
- Calculate the volume of one unit cell.
- Determine the number of unit cells per cubic centimeter.
- Multiply the number of unit cells by the number of atoms in each unit cell to find the number of atoms per cubic centimeter.
By following these steps, we can find the solutions to both questions.
To find the number of atoms present per cubic centimeter of Cr, we need to know the number of atoms in the unit cell.
In a body-centered cubic (BCC) unit cell, there is one atom at the center of the cube and eight atoms at the corners. So, the total number of atoms per unit cell is 1 atom (center) + 8 atoms (corners) = 9 atoms.
To calculate the number of unit cells per cubic centimeter, we need to determine the volume of a unit cell.
The volume of a BCC unit cell can be calculated using the formula:
Volume of unit cell = (side length of unit cell)³
However, we need to convert the density of Cr to the side length of the unit cell in order to use this formula.
Density (ρ) = mass (m) / volume (V)
ρ = 7.15 g/cm³
Since we know the density of Cr and want the side length of the unit cell, we can rearrange the formula to solve for the side length of the unit cell:
Side length of unit cell (a) = (mass of unit cell) / (density of Cr)
To find the mass of the unit cell, we need to know the atomic mass of one Cr atom. The atomic mass of Cr is approximately 52 g/mol.
Using Avogadro's number (6.022 x 10^23 atoms/mol), we can calculate the mass of one Cr atom:
Mass of one Cr atom = (Atomic mass of Cr) / (Avogadro's number)
= 52 g/mol / (6.022 x 10^23 atoms/mol)
Now, we can calculate the mass of the unit cell:
Mass of unit cell = (9 atoms per unit cell) x (Mass of one Cr atom)
Finally, we can substitute the mass of the unit cell and the density of Cr into the equation for the side length of the unit cell. Then, we can calculate the volume of the unit cell and the number of unit cells per cubic centimeter:
Side length of unit cell (a) = (Mass of unit cell) / (Density of Cr)
Volume of unit cell = (Side length of unit cell)³
Number of unit cells per cubic centimeter = 1 cm³ / Volume of unit cell
Let's calculate the values step-by-step.