Silicon has a diamond (Diamond) crystal structure and has a lattice constant a = 0.543 nm. Atomic mass of silicon Considering that silicon is 28.08 g / mol, (a) the atomic density per unit volume (b) of silicon density g / cm3 Find in terms of.
The Correct Answer and Explanation is:
Correct Answer
Given Data:
- Silicon has a diamond cubic structure
- Lattice constant, a = 0.543 nm = 0.543 × 10⁻⁷ cm
- Atomic mass of silicon = 28.08 g/mol
- Avogadro’s number = 6.022 × 10²³ atoms/mol
- Number of atoms per unit cell in diamond cubic structure = 8 atoms
(a) Atomic Density per Unit Volume
Atomic density = (Number of atoms per unit cell) ÷ (Volume of unit cell)
Volume of unit cell = a³a=0.543×10−7 cma = 0.543 \times 10^{-7} \, \text{cm} a=0.543×10−7cmVolume=a3=(0.543×10−7)3=1.602×10−22 cm3\text{Volume} = a^3 = (0.543 \times 10^{-7})^3 = 1.602 \times 10^{-22} \, \text{cm}^3 Volume=a3=(0.543×10−7)3=1.602×10−22cm3
Total atoms per unit cell = 8
Atomic density:Atomic Density=81.602×10−22=4.995×1022 atoms/cm3\text{Atomic Density} = \frac{8}{1.602 \times 10^{-22}} = 4.995 \times 10^{22} \, \text{atoms/cm}^3 Atomic Density=1.602×10−228=4.995×1022atoms/cm3
(b) Density of Silicon in g/cm³
Density formula:Density=Mass of atoms in one unit cellVolume of unit cell\text{Density} = \frac{\text{Mass of atoms in one unit cell}}{\text{Volume of unit cell}} Density=Volume of unit cellMass of atoms in one unit cell
Mass of one silicon atom:Mass per atom=28.08 g/mol6.022×1023 atoms/mol=4.662×10−23 g\text{Mass per atom} = \frac{28.08 \, \text{g/mol}}{6.022 \times 10^{23} \, \text{atoms/mol}} = 4.662 \times 10^{-23} \, \text{g} Mass per atom=6.022×1023atoms/mol28.08g/mol=4.662×10−23g
Total mass in one unit cell:Total mass=8×4.662×10−23=3.7296×10−22 g\text{Total mass} = 8 \times 4.662 \times 10^{-23} = 3.7296 \times 10^{-22} \, \text{g} Total mass=8×4.662×10−23=3.7296×10−22g
Density:Density=3.7296×10−221.602×10−22=2.33 g/cm3\text{Density} = \frac{3.7296 \times 10^{-22}}{1.602 \times 10^{-22}} = 2.33 \, \text{g/cm}^3 Density=1.602×10−223.7296×10−22=2.33g/cm3
Final Answers
(a) Atomic density = 4.995 × 10²² atoms/cm³
(b) Density of silicon = 2.33 g/cm³
Explanation
Silicon crystallizes in a diamond cubic structure which is a common structure for group IV elements like carbon, silicon, and germanium. In this arrangement, each silicon atom is covalently bonded to four neighboring silicon atoms forming a tetrahedral geometry. The unit cell of this structure contains eight atoms.
To calculate atomic density, we first need to determine the volume of the cubic unit cell. The lattice constant a is the edge length of the cube, given as 0.543 nanometers which converts to 0.543 × 10⁻⁷ centimeters. The volume of the cube is obtained by raising this length to the power of three.
Once the volume is known, the atomic density is found by dividing the number of atoms per unit cell by the unit cell’s volume. This gives the number of atoms present per cubic centimeter of silicon crystal.
To calculate the density of silicon in grams per cubic centimeter, the mass of one silicon atom is required. The atomic mass is given as 28.08 grams per mole, and dividing this by Avogadro’s number provides the mass of one silicon atom. Multiplying the mass of one atom by the total number of atoms per unit cell gives the mass of the unit cell. Finally, dividing this mass by the unit cell’s volume provides the density.
The final calculated density of silicon matches the known experimental value of approximately 2.33 g/cm³, which confirms the consistency of this method. This information is essential in semiconductor physics, materials science, and engineering applications where silicon is widely used.
