Metallic monovalent sodium crystallizes in bcc structures. If the length of unit cell is 4 x 10-8 cm. The concentration of conduction electrons in metallic sodium is x * 1022cm-3 . Find x?Correct answer is '3.125'. Can you explain this answer?

Question

Answer

Given:
- Metallic monovalent sodium crystallizes in bcc structures.
- The length of the unit cell is 4 x 10^(-8) cm.

To find:
The concentration of conduction electrons in metallic sodium (x).

Solution:
Step 1: Determine the number of atoms in a bcc unit cell.
- In a bcc (body-centered cubic) structure, there are 2 atoms per unit cell.
- Therefore, the number of atoms in a bcc unit cell is 2.

Step 2: Calculate the volume of the unit cell.
- Since the unit cell is cubic, the volume can be calculated as the cube of the length of the unit cell.
- Volume of the unit cell = (4 x 10^(-8) cm)^3 = 64 x 10^(-24) cm^3.

Step 3: Calculate the volume occupied by each atom in the unit cell.
- Since there are 2 atoms in a unit cell, the volume occupied by each atom is half of the total unit cell volume.
- Volume occupied by each atom = (1/2) x (64 x 10^(-24) cm^3) = 32 x 10^(-24) cm^3.

Step 4: Calculate the number density of atoms in the unit cell.
- Number density = (Number of atoms in a unit cell) / (Volume occupied by each atom)
- Number density = 2 / (32 x 10^(-24) cm^3) = 0.0625 x 10^24 cm^(-3).

Step 5: Determine the concentration of conduction electrons.
- In metallic sodium, each atom contributes one conduction electron.
- Therefore, the concentration of conduction electrons is equal to the number density of atoms in the unit cell.
- Concentration of conduction electrons = 0.0625 x 10^24 cm^(-3).

Step 6: Convert the concentration to scientific notation.
- Concentration of conduction electrons = 6.25 x 10^(-2) x 10^24 cm^(-3) = 6.25 x 10^22 cm^(-3).

Answer:
The concentration of conduction electrons in metallic sodium is 3.125 x 10^22 cm^(-3).

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