The cubic unit cell of a metal (molar mass= 63.55 g`"mol"^(-1)`) has an edge length of 362pm. Its density is `8.92gcm^(-3)`. The type of unit cell is

Options

A. Primitive
B. Face centred
C. Body centred
D. End centred

Answer

To determine the type of unit cell for the given metal, we will follow these steps:

### Step 1: Convert the edge length from picometers to centimeters
Given:
- Edge length \( a = 362 \, \text{pm} \)

We know that:
\[ 1 \, \text{pm} = 10^{-10} \, \text{cm} \]

So,
\[ a = 362 \, \text{pm} = 362 \times 10^{-10} \, \text{cm} = 3.62 \times 10^{-8} \, \text{cm} \]

### Step 2: Use the density formula to find the number of atoms per unit cell (Z)
The formula for density \( \rho \) is given by:
\[ \rho = \frac{Z \cdot M}{a^3 \cdot N_A} \]
Where:
- \( \rho \) = density = \( 8.92 \, \text{g/cm}^3 \)
- \( M \) = molar mass = \( 63.55 \, \text{g/mol} \)
- \( a \) = edge length in cm = \( 3.62 \times 10^{-8} \, \text{cm} \)
- \( N_A \) = Avogadro's number = \( 6.023 \times 10^{23} \, \text{mol}^{-1} \)

### Step 3: Rearranging the formula to solve for Z
Rearranging the density formula gives:
\[ Z = \frac{\rho \cdot a^3 \cdot N_A}{M} \]

### Step 4: Calculate \( a^3 \)
Calculating \( a^3 \):
\[ a^3 = (3.62 \times 10^{-8} \, \text{cm})^3 = 4.74 \times 10^{-23} \, \text{cm}^3 \]

### Step 5: Substitute values into the equation for Z
Substituting the values:
\[ Z = \frac{8.92 \, \text{g/cm}^3 \cdot 4.74 \times 10^{-23} \, \text{cm}^3 \cdot 6.023 \times 10^{23} \, \text{mol}^{-1}}{63.55 \, \text{g/mol}} \]

### Step 6: Perform the calculation
Calculating \( Z \):
\[ Z = \frac{8.92 \cdot 4.74 \times 10^{-23} \cdot 6.023 \times 10^{23}}{63.55} \]
\[ Z \approx 4.00 \]

### Step 7: Determine the type of unit cell
From the value of \( Z \):
- If \( Z = 1 \), it is a simple cubic unit cell.
- If \( Z = 2 \), it is a body-centered cubic (BCC) unit cell.
- If \( Z = 4 \), it is a face-centered cubic (FCC) unit cell.

Since we found \( Z = 4 \), the type of unit cell is **Face-Centered Cubic (FCC)**.

### Final Answer:
The type of unit cell is **Face-Centered Cubic (FCC)**.

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