Unit cell of the iron crystal has edge length of 288 pm and density of 7.86 g `cm^(-3)`. Determinte the type of crystal lattice. Atomic mass of Fe= 56 g `mol^(-1)`

Explanation

To determine the type of crystal lattice for iron given the edge length and density, we can follow these steps:

### Step 1: Gather the Given Data
- Edge length of the unit cell (a) = 288 pm = 288 × 10^(-12) m = 2.88 × 10^(-8) cm
- Density (d) = 7.86 g/cm³
- Atomic mass of Fe (m) = 56 g/mol

### Step 2: Write the Formula for Density
The formula for density (d) in terms of the number of atoms in the unit cell (z), molar mass (m), Avogadro's number (N_A), and volume (V) is given by:
\[ d = \frac{z \cdot m}{N_A \cdot V} \]
For a cubic unit cell, the volume (V) can be expressed as:
\[ V = a^3 \]
Thus, the density formula becomes:
\[ d = \frac{z \cdot m}{N_A \cdot a^3} \]

### Step 3: Rearrange the Formula to Solve for z
Rearranging the formula to solve for z gives us:
\[ z = \frac{d \cdot N_A \cdot a^3}{m} \]

### Step 4: Substitute the Known Values
Now, we can substitute the known values into the equation. We need to use Avogadro's number:
\[ N_A = 6.022 \times 10^{23} \text{ mol}^{-1} \]

Substituting the values:
- d = 7.86 g/cm³
- N_A = 6.022 × 10^{23} mol⁻¹
- a = 2.88 × 10^(-8) cm
- m = 56 g/mol

Calculating \( a^3 \):
\[ a^3 = (2.88 \times 10^{-8})^3 = 2.39 \times 10^{-23} \text{ cm}^3 \]

Now substituting into the equation for z:
\[ z = \frac{7.86 \cdot (6.022 \times 10^{23}) \cdot (2.39 \times 10^{-23})}{56} \]

### Step 5: Calculate z
Calculating the numerator:
\[ 7.86 \cdot 6.022 \times 10^{23} \cdot 2.39 \times 10^{-23} \approx 7.86 \cdot 6.022 \cdot 2.39 \approx 111.4 \]

Now divide by 56:
\[ z \approx \frac{111.4}{56} \approx 1.99 \]

### Step 6: Determine the Type of Crystal Lattice
The value of z indicates the number of atoms per unit cell:
- If z = 1, it is Simple Cubic (SC).
- If z = 2, it is Body-Centered Cubic (BCC).
- If z = 4, it is Face-Centered Cubic (FCC).

Since z ≈ 2, we conclude that the crystal lattice of iron is **Body-Centered Cubic (BCC)**.

### Summary
The type of crystal lattice for iron is BCC.

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