An element with atomic mass 93 g `mol^(-1)` has density 11.5 g `cm^(-3)`. If the edge length of its unit cell is 300 pm, identify the type of unit cell.

Explanation

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

### Step 1: Gather the given data
- Atomic mass (M) = 93 g/mol
- Density (D) = 11.5 g/cm³
- Edge length (a) = 300 pm = 300 x 10⁻¹² m = 3 x 10⁻⁸ cm

### Step 2: Use the formula for density
The formula for density in terms of the number of atoms per unit cell (Z), atomic mass (M), Avogadro's number (Nₐ), and the volume of the unit cell (a³) is given by:

\[ D = \frac{Z \cdot M}{a^3 \cdot N_a} \]

### Step 3: Rearrange the formula to find Z
We can rearrange the formula to solve for Z:

\[ Z = \frac{D \cdot a^3 \cdot N_a}{M} \]

### Step 4: Substitute the values into the equation
Now, we will substitute the known values into the equation. First, we need to calculate \(a^3\):

\[ a^3 = (3 \times 10^{-8} \text{ cm})^3 = 27 \times 10^{-24} \text{ cm}^3 = 2.7 \times 10^{-23} \text{ cm}^3 \]

Now substituting the values into the equation for Z:

- Density (D) = 11.5 g/cm³
- Avogadro's number (Nₐ) = 6.022 x 10²³ mol⁻¹
- Atomic mass (M) = 93 g/mol

\[ Z = \frac{11.5 \, \text{g/cm}^3 \cdot 2.7 \times 10^{-23} \, \text{cm}^3 \cdot 6.022 \times 10^{23} \, \text{mol}^{-1}}{93 \, \text{g/mol}} \]

### Step 5: Calculate Z
Now we will calculate Z:

\[ Z = \frac{11.5 \cdot 2.7 \times 10^{-23} \cdot 6.022 \times 10^{23}}{93} \]

Calculating the numerator:

\[ 11.5 \cdot 2.7 \approx 31.05 \]
\[ 31.05 \times 6.022 \approx 186.57 \]

Now, substituting back into the equation for Z:

\[ Z = \frac{186.57 \times 10^{0}}{93} \approx 2 \]

### Step 6: Identify the type of unit cell
Since Z = 2, we can identify the type of unit cell. In solid state chemistry:
- Z = 1 corresponds to Simple Cubic (SC)
- Z = 2 corresponds to Body-Centered Cubic (BCC)
- Z = 4 corresponds to Face-Centered Cubic (FCC)

Thus, the unit cell type is **Body-Centered Cubic (BCC)**.

### Final Answer
The type of unit cell for the element is **Body-Centered Cubic (BCC)**.

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