Crystal of Sodium form Cubic unit cell with edge length 430pm.Find the no.of atoms present per unit cell and also the type of unit cell.
P.S. No radius or density is given so its a lil bit tough dan the rest... Plz Help ASAP !!
To determine the number of atoms per unit cell and the type of unit cell for a crystal of Sodium with a cubic unit cell and an edge length of 430 pm, we can follow these steps:
Step 1: Convert the edge length from picometers (pm) to angstroms (Å). 1 pm = 0.01 Å.
Edge length (Å) = 430 pm × 0.01 Å/pm = 4.3 Å
Step 2: Compute the volume of the unit cell.
Volume (V) = Edge length^3 = (4.3 Å)^3 = 79.507 Å^3
Step 3: Determine the type of cubic unit cell based on the ratio of lengths of its edges.
In a cubic unit cell, all three edges are of equal length.
If the ratio of lengths of the three edges is:
- 1:1:1 -> Simple Cubic (SC) unit cell
- √2:√2:√2 -> Body-Centered Cubic (BCC) unit cell
- √3:√3:√3 -> Face-Centered Cubic (FCC) unit cell
Step 4: Calculate the number of atoms per unit cell for the given crystal structure.
For a simple cubic (SC) unit cell: 1 atom at each corner of the cube
Number of atoms per unit cell (SC) = 1
For a body-centered cubic (BCC) unit cell: 1 atom at each corner + 1 atom at the center of the cube
Number of atoms per unit cell (BCC) = 1 + 1 = 2
For a face-centered cubic (FCC) unit cell: 1 atom at each corner + 1 atom at the center of each face
Number of atoms per unit cell (FCC) = 1 + 6(1/2) = 4
Step 5: Compare the calculated number of atoms per unit cell with the number of atoms in Sodium (Na).
Sodium (Na) has one atom per unit cell.
Comparing the calculated values:
- Number of atoms in SC unit cell = 1 (matches Na)
- Number of atoms in BCC unit cell = 2 (does not match Na)
- Number of atoms in FCC unit cell = 4 (does not match Na)
Step 6: Deduce the type of unit cell and the number of atoms per unit cell.
Since the calculated number of atoms per unit cell in the SC unit cell matches the number of atoms in Sodium (Na), we can conclude that the crystal of Sodium has a Simple Cubic (SC) unit cell with 1 atom per unit cell.
Therefore, in a Sodium crystal with a cubic unit cell and an edge length of 430 pm, there is 1 atom present per unit cell, and the unit cell is a Simple Cubic (SC) unit cell.