An element with molar mass `2.7 xx 10^(2) kg mol^(-1)` forms a `2.7 xx 10^(3) kg^(-3)`, what is the nature of the cubic unit cell?

Explanation

To determine the nature of the cubic unit cell based on the given molar mass and density, we can follow these steps:

### Step 1: Understand the given values
- Molar mass (M) = \(2.7 \times 10^2 \, \text{kg/mol}\)
- Density (d) = \(2.7 \times 10^3 \, \text{kg/m}^3\)

### Step 2: Use the formula for density of a cubic unit cell
The formula for the density of a cubic unit cell is given by:
\[ d = \frac{Z \cdot M}{N_A \cdot a^3} \]
where:
- \(d\) = density
- \(Z\) = number of formula units per unit cell
- \(M\) = molar mass
- \(N_A\) = Avogadro's number (\(6.022 \times 10^{23} \, \text{mol}^{-1}\))
- \(a\) = edge length of the unit cell

### Step 3: Rearrange the formula to find Z
We can rearrange the formula to solve for \(Z\):
\[ Z = \frac{d \cdot N_A \cdot a^3}{M} \]

### Step 4: Calculate the edge length in meters
Given that the edge length \(a\) is \(405 \, \text{pm}\) (picometers), we convert it to meters:
\[ a = 405 \, \text{pm} = 405 \times 10^{-12} \, \text{m} \]

### Step 5: Calculate \(a^3\)
Now, we calculate \(a^3\):
\[ a^3 = (405 \times 10^{-12})^3 = 6.63 \times 10^{-29} \, \text{m}^3 \]

### Step 6: Substitute values into the equation for Z
Now, substitute the values into the equation for \(Z\):
\[ Z = \frac{(2.7 \times 10^3) \cdot (6.022 \times 10^{23}) \cdot (6.63 \times 10^{-29})}{(2.7 \times 10^2)} \]

### Step 7: Calculate Z
Calculating \(Z\):
\[ Z = \frac{(2.7 \times 10^3) \cdot (6.022 \times 10^{23}) \cdot (6.63 \times 10^{-29})}{(2.7 \times 10^2)} \approx 6 \]

### Step 8: Determine the nature of the cubic unit cell
The value of \(Z\) indicates the number of atoms per unit cell.
- 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.
- If \(Z = 6\), it indicates a more complex structure, possibly a close-packed structure.

Since we calculated \(Z \approx 6\), the nature of the cubic unit cell is likely a close-packed structure.

### Final Answer
The nature of the cubic unit cell is a close-packed structure.