Density of a unit cell is same as the density of the substance. If the density of the substance is known, number of atoms or dimensions of the unit cell can be calculated . The density of the unit cell is related to its mass(M), no. of atoms per unit cell (Z), edge length (a in cm) and Avogadro number `N_A` as :
`rho = (Z xx M)/(a^3 xx N_A)`
An element crystallizes in a structure having a fcc unit-cell an edge 100 pm. If 24 g of the element contains `24 xx 10^(23)` atoms, the density is

Options

A. `2.40 g cm^(-3)`
B. `40 g cm^(-3)`
C. `4 g cm^(-3)`
D. `24 g cm^(-3)`

Explanation

To calculate the density of the element that crystallizes in a face-centered cubic (FCC) unit cell, we can follow these steps:

### Step-by-step Solution:

1. **Identify the given values:**
- Edge length of the unit cell (a) = 100 pm = \(100 \times 10^{-10}\) cm
- Total mass of the element = 24 g
- Total number of atoms in 24 g = \(24 \times 10^{23}\) atoms

2. **Calculate the number of atoms per unit cell (Z) for FCC:**
- In a face-centered cubic (FCC) structure, there are:
- 8 corner atoms, each contributing \(\frac{1}{8}\) of an atom = \(8 \times \frac{1}{8} = 1\)
- 6 face-centered atoms, each contributing \(\frac{1}{2}\) of an atom = \(6 \times \frac{1}{2} = 3\)
- Therefore, the total number of atoms per unit cell (Z) = 1 + 3 = 4.

3. **Calculate the mass of one atom:**
- The total number of atoms in 24 g is \(24 \times 10^{23}\).
- The mass of one atom = \(\frac{24 \text{ g}}{24 \times 10^{23} \text{ atoms}} = \frac{1}{10^{23}} \text{ g} = 10^{-23} \text{ g}\).

4. **Calculate the mass of the unit cell:**
- The mass of the unit cell = \(Z \times \text{mass of one atom} = 4 \times 10^{-23} \text{ g} = 4 \times 10^{-23} \text{ g}\).

5. **Calculate the volume of the unit cell:**
- The volume of the unit cell (V) = \(a^3 = (100 \times 10^{-10} \text{ cm})^3 = 10^{-24} \text{ cm}^3\).

6. **Calculate the density (ρ) of the unit cell:**
- Using the formula \(\rho = \frac{Z \times M}{a^3 \times N_A}\), where \(M\) is the mass of the unit cell:
- \(\rho = \frac{4 \times 10^{-23} \text{ g}}{10^{-24} \text{ cm}^3} = 40 \text{ g/cm}^3\).

### Final Answer:
The density of the element is **40 g/cm³**.