A crystalline solid of pure substance has a face-centred cubic structure with a cell edge of 400 pm. If the density of the substance in the crystal is `8 g cm^(-3)` then the number of atoms present in 256 g of the crystal is `N xx 10^24`. The value of N is...............

Answer

To solve the problem step by step, we need to find the number of atoms present in 256 g of a crystalline solid with a face-centered cubic (FCC) structure, given its density and cell edge length.

### Step 1: Understand the given data
- **Cell edge (A)** = 400 pm = \(400 \times 10^{-12}\) m
- **Density (d)** = 8 g/cm³
- **Mass of the crystal (m)** = 256 g

### Step 2: Convert cell edge to centimeters
Since the density is given in g/cm³, we should convert the cell edge from picometers to centimeters:
\[ A = 400 \text{ pm} = 400 \times 10^{-12} \text{ m} = 400 \times 10^{-12} \times 100 \text{ cm} = 4 \times 10^{-8} \text{ cm} \]

### Step 3: Calculate the volume of the unit cell
The volume (V) of the cubic unit cell can be calculated using the formula:
\[ V = A^3 \]
Substituting the value of A:
\[ V = (4 \times 10^{-8} \text{ cm})^3 = 64 \times 10^{-24} \text{ cm}^3 = 6.4 \times 10^{-23} \text{ cm}^3 \]

### Step 4: Use the density formula to find the number of atoms (Z)
The formula for density (d) is:
\[ d = \frac{Z \cdot m}{V} \]
Where:
- \(Z\) = number of atoms per unit cell
- \(m\) = molar mass of the substance (in grams)
- \(V\) = volume of the unit cell (in cm³)

Rearranging the formula to find Z:
\[ Z = \frac{d \cdot V}{m} \]

### Step 5: Calculate the mass of the unit cell
To find the mass of the unit cell, we can use the density:
\[ \text{Mass of unit cell} = d \cdot V = 8 \text{ g/cm}^3 \cdot 6.4 \times 10^{-23} \text{ cm}^3 = 5.12 \times 10^{-22} \text{ g} \]

### Step 6: Find the number of moles in 256 g of the substance
Using the molar mass (M), we can find the number of moles (n):
\[ n = \frac{\text{mass}}{\text{molar mass}} = \frac{256 \text{ g}}{M} \]

### Step 7: Calculate the total number of atoms in 256 g
The total number of atoms (N) can be calculated as:
\[ N = n \cdot Z \]
Substituting the values:
\[ N = \frac{256 \text{ g}}{M} \cdot Z \]

### Step 8: Calculate the value of N
Since we know that in FCC, Z = 4 (there are 4 atoms per unit cell), we can substitute Z into the equation:
\[ N = \frac{256 \text{ g}}{M} \cdot 4 \]

### Step 9: Find the value of N in the required format
We know that \(M\) can be calculated from the mass of the unit cell:
\[ M = \frac{5.12 \times 10^{-22} \text{ g}}{Z} = \frac{5.12 \times 10^{-22}}{4} = 1.28 \times 10^{-22} \text{ g} \]
Now substituting back into the N equation:
\[ N = \frac{256 \text{ g}}{1.28 \times 10^{-22}} \cdot 4 = 8 \times 10^{24} \]

### Final Answer
The value of \(N\) is 2, thus:
\[ N = 2 \times 10^{24} \]