Al (atomic mass =27) crystallises in a cubic system with edge length (a) equal to `4Å` its density is `2.7 g//cm^(3)` The type of the unit cell is:

Options

A. Simple
B. Face centred
C. Body centred
D. None of these

Explanation

To determine the type of unit cell for aluminum (Al) given its atomic mass, edge length, and density, we can follow these steps:

### Step 1: Gather Given Information
- Atomic Mass (M) of Al = 27 g/mol
- Edge Length (a) = 4 Å = \(4 \times 10^{-8}\) cm
- Density (D) = 2.7 g/cm³

### Step 2: Convert Edge Length to Centimeters
Since the density is given in g/cm³, we need to convert the edge length from angstroms to centimeters:
\[ a = 4 \text{ Å} = 4 \times 10^{-8} \text{ cm} \]

### Step 3: Use the Density Formula
The formula for density (D) in terms of the number of atoms per unit cell (Z), atomic mass (M), edge length (a), and Avogadro's number (N_A) is given by:
\[ D = \frac{Z \times M}{a^3 \times N_A} \]
Where:
- \(N_A = 6 \times 10^{23} \text{ atoms/mol}\)

### Step 4: Rearranging the Formula to Find Z
We can rearrange the formula to solve for Z:
\[ Z = \frac{D \times a^3 \times N_A}{M} \]

### Step 5: Substitute the Values
Now we substitute the known values into the equation:
\[ Z = \frac{2.7 \text{ g/cm}^3 \times (4 \times 10^{-8} \text{ cm})^3 \times 6 \times 10^{23} \text{ atoms/mol}}{27 \text{ g/mol}} \]

### Step 6: Calculate \(a^3\)
First, calculate \(a^3\):
\[ a^3 = (4 \times 10^{-8})^3 = 64 \times 10^{-24} \text{ cm}^3 = 6.4 \times 10^{-23} \text{ cm}^3 \]

### Step 7: Substitute \(a^3\) Back into Z
Now substitute \(a^3\) back into the equation for Z:
\[ Z = \frac{2.7 \times 6.4 \times 10^{-23} \times 6 \times 10^{23}}{27} \]

### Step 8: Calculate Z
Calculating the numerator:
\[ 2.7 \times 6.4 \times 6 = 103.68 \]
Now, substituting this back:
\[ Z = \frac{103.68}{27} \approx 3.84 \]

### Step 9: Determine the Type of Unit Cell
Since Z is approximately 4, this indicates that aluminum crystallizes in a face-centered cubic (FCC) unit cell, where Z = 4.

### Conclusion
Thus, the type of unit cell for aluminum is:
**Face-Centered Cubic (FCC)**