An element crystallizes in a structure having fee unit cell of an edge 200 pm. Calculate the density if 200 g of this element contains `24 xx 10^(23)` atoms.

Explanation

To solve the problem, we will calculate the density of the element that crystallizes in a face-centered cubic (FCC) structure with a given edge length and mass of the element containing a specific number of atoms.

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Edge length of the FCC unit cell, \( a = 200 \) pm = \( 200 \times 10^{-12} \) m
- Mass of the element, \( m = 200 \) g
- Number of atoms in the given mass, \( N = 24 \times 10^{23} \) atoms

2. **Calculate the Mass of One Atom:**
The mass of one atom can be calculated using the formula:
\[ \text{Mass of one atom} = \frac{\text{Total mass}}{\text{Number of atoms}} = \frac{200 \text{ g}}{24 \times 10^{23} \text{ atoms}} \]
\[ = \frac{200}{24 \times 10^{23}} \text{ g/atom} \]
\[ = \frac{200}{24} \times 10^{-23} \text{ g/atom} \approx 8.33 \times 10^{-23} \text{ g/atom} \]

3. **Calculate the Mass of the Unit Cell:**
In an FCC structure, there are 4 atoms per unit cell. Therefore, the mass of the unit cell can be calculated as:
\[ \text{Mass of unit cell} = \text{Mass of one atom} \times \text{Number of atoms in unit cell} = 8.33 \times 10^{-23} \text{ g/atom} \times 4 \]
\[ = 33.32 \times 10^{-23} \text{ g} \]

4. **Calculate the Volume of the Unit Cell:**
The volume of the unit cell can be calculated using the formula for the volume of a cube:
\[ \text{Volume} = a^3 = (200 \times 10^{-12} \text{ m})^3 \]
\[ = 8 \times 10^{-30} \text{ m}^3 \]
To convert this to cm³:
\[ 1 \text{ m}^3 = 10^6 \text{ cm}^3 \implies \text{Volume} = 8 \times 10^{-30} \text{ m}^3 \times 10^6 \text{ cm}^3/\text{m}^3 = 8 \times 10^{-24} \text{ cm}^3 \]

5. **Calculate the Density:**
Density (\( \rho \)) is calculated using the formula:
\[ \rho = \frac{\text{Mass}}{\text{Volume}} = \frac{33.32 \times 10^{-23} \text{ g}}{8 \times 10^{-24} \text{ cm}^3} \]
\[ = \frac{33.32}{8} \text{ g/cm}^3 \approx 4.165 \text{ g/cm}^3 \]

6. **Final Result:**
The density of the element is approximately \( 4.165 \text{ g/cm}^3 \).