Iron crystallizes in body centered cubic system with edge length `2.86Å`. The density of iron is nearly `X` g/ml. What is the value of `X` here?
Report your answer by rounding it upto nearest whole number.

Answer

To find the density of iron crystallizing in a body-centered cubic (BCC) structure with an edge length of 2.86 Å, we can follow these steps:

### Step 1: Determine the number of atoms per unit cell (Z)
In a body-centered cubic (BCC) structure:
- There are 8 corner atoms, each contributing \( \frac{1}{8} \) of an atom to the unit cell.
- There is 1 atom at the center contributing 1 full atom.

Thus, the total number of atoms \( Z \) in a BCC unit cell is:
\[ Z = 8 \times \frac{1}{8} + 1 = 1 + 1 = 2 \]

### Step 2: Convert the edge length from Ångströms to centimeters
The edge length \( A \) is given as 2.86 Å. To convert this to centimeters:
\[ A = 2.86 \, \text{Å} = 2.86 \times 10^{-8} \, \text{cm} \]

### Step 3: Use the density formula
The density \( D \) of a crystal can be calculated using the formula:
\[ D = \frac{Z \cdot M}{N_A \cdot A^3} \]
Where:
- \( D \) is the density,
- \( Z \) is the number of atoms per unit cell (which we found to be 2),
- \( M \) is the molar mass of iron (approximately 56 g/mol),
- \( N_A \) is Avogadro's number (\( 6.02 \times 10^{23} \, \text{mol}^{-1} \)),
- \( A \) is the edge length in cm.

### Step 4: Calculate \( A^3 \)
First, calculate \( A^3 \):
\[ A^3 = (2.86 \times 10^{-8} \, \text{cm})^3 = 2.33 \times 10^{-23} \, \text{cm}^3 \]

### Step 5: Substitute the values into the density formula
Now we can substitute the values into the density formula:
\[ D = \frac{2 \cdot 56 \, \text{g/mol}}{6.02 \times 10^{23} \, \text{mol}^{-1} \cdot 2.33 \times 10^{-23} \, \text{cm}^3} \]

### Step 6: Calculate the density
Calculating the numerator:
\[ 2 \cdot 56 = 112 \, \text{g} \]
Calculating the denominator:
\[ 6.02 \times 10^{23} \cdot 2.33 \times 10^{-23} \approx 1.40 \, \text{g/cm}^3 \]
Now, substituting these values:
\[ D = \frac{112}{1.40} \approx 80 \, \text{g/cm}^3 \]

### Step 7: Convert to g/ml
Since \( 1 \, \text{g/cm}^3 = 1 \, \text{g/ml} \), we can say:
\[ D \approx 7.94 \, \text{g/ml} \]

### Step 8: Round to the nearest whole number
Thus, rounding 7.94 to the nearest whole number gives us:
\[ X \approx 8 \, \text{g/ml} \]

### Final Answer
The density of iron is approximately \( 8 \, \text{g/ml} \).

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