Calculate the theoretical density of Al from the following data: Crystal Structure = FCC Atomic radius = 1.86 Å Atomic weight = 26.98 g/mo
The Correct Answer and Explanation is :
To calculate the theoretical density of aluminum (Al) with a face-centered cubic (FCC) crystal structure, an atomic radius of 1.86 Å, and an atomic weight of 26.98 g/mol, we can follow these steps:
1. Determine the Number of Atoms per Unit Cell (n):
In an FCC structure, there are 4 atoms per unit cell.
2. Calculate the Edge Length (a) of the Unit Cell:
The relationship between the atomic radius (R) and the edge length (a) in an FCC structure is given by:
[ a = 2\sqrt{2} \times R ]
Given that the atomic radius R = 1.86 Å = 1.86 × 10⁻¹⁰ m, we can calculate:
[ a = 2\sqrt{2} \times 1.86 \times 10^{-10} \, \text{m} ]
[ a \approx 4.24 \times 10^{-10} \, \text{m} ]
3. Calculate the Volume of the Unit Cell (V₀):
The volume of the unit cell is the cube of the edge length:
[ V_0 = a^3 ]
[ V_0 = (4.24 \times 10^{-10} \, \text{m})^3 ]
[ V_0 \approx 7.64 \times 10^{-29} \, \text{m}^3 ]
4. Calculate the Mass of the Unit Cell (m₀):
The mass of the unit cell is the product of the number of atoms per unit cell (n), the atomic weight (M), and Avogadro’s number (Nₐ):
[ m_0 = n \times M / N_a ]
[ m_0 = 4 \times 26.98 \, \text{g/mol} / 6.022 \times 10^{23} \, \text{atoms/mol} ]
[ m_0 \approx 1.79 \times 10^{-22} \, \text{g} ]
5. Calculate the Theoretical Density (ρ):
Density is mass per unit volume:
[ \rho = m_0 / V_0 ]
[ \rho = 1.79 \times 10^{-22} \, \text{g} / 7.64 \times 10^{-29} \, \text{m}^3 ]
[ \rho \approx 2.34 \times 10^{3} \, \text{g/m}^3 ]
[ \rho \approx 2.34 \, \text{g/cm}^3 ]
Therefore, the theoretical density of aluminum is approximately 2.34 g/cm³.
Explanation:
The theoretical density of a crystalline material can be calculated using the formula:
[ \rho = \frac{n \times M}{V_0 \times N_a} ]
Where:
- ( n ) = number of atoms per unit cell
- ( M ) = atomic weight
- ( V_0 ) = volume of the unit cell
- ( N_a ) = Avogadro’s number
For aluminum with an FCC structure, there are 4 atoms per unit cell. The atomic weight is 26.98 g/mol, and Avogadro’s number is 6.022 × 10²³ atoms/mol. The volume of the unit cell is calculated from the edge length, which is derived from the atomic radius using the relationship ( a = 2\sqrt{2} \times R ).
By substituting these values into the density formula, we obtain the theoretical density of aluminum as approximately 2.34 g/cm³.
This theoretical value is close to the experimentally measured density of aluminum, which is around 2.70 g/cm³. The slight difference can be attributed to factors such as atomic packing imperfections, temperature variations, and measurement uncertainties.