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The density of a face-centered cubic element having 4 formula units is [tex]6.2 \, \text{g/cm}^3[/tex]. The atomic mass of the element is [tex]60.2 \, \text{amu}[/tex]. Calculate the length of the edge of the unit cell.

Answer :

To find the length of the edge of a face-centered cubic unit cell, first convert the atomic mass to grams, then calculate the mass of the unit cell, and use the density to find the volume of the unit cell. Finally, take the cube root of the volume to get the edge length, which is approximately 4.03 Å.

To calculate the length of the edge of the unit cell for a face-centered cubic (FCC) element, we can use the following data:

  • Density (ρ) = 6.2 g/cm³
  • Atomic mass (M) = 60.2 amu
  • Number of atoms per unit cell (Z) = 4

First, we convert the atomic mass (M) to grams:

M (grams) = 60.2 amu / (6.022 × 1023 atoms/mol) = 9.99 × 10-23 g/atom

Second, we calculate the mass of the unit cell by multiplying the mass of one atom by the number of atoms in the unit cell:

Mass of unit cell = 4 atoms × 9.99 × 10-23 g/atom = 3.996 × 10-22 g

Third, we use the density formula to find the volume of the unit cell:

Density (ρ) = mass of unit cell / volume of unit cell
6.2 g/cm³ = 3.996 × 10-22 g / volume of unit cell
Volume of unit cell = 3.996 × 10-22 g / 6.2 g/cm³ = 6.44 × 10-23 cm³

Finally, to find the edge length (a) of the cubic unit cell, we take the cube root of the volume:

Edge length (a) = (6.44 × 10-23 cm³)1/3
= 4.03 × 10-8 cm = 4.03 Å

Therefore, the length of the edge of the unit cell is approximately 4.03 Å.

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