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 128 g of the crystal is

Options

A. `2xx10^(24)`
B. `4xx10^(24)`
C. `1xx10^(24)`
D. `3xx10^(24)`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the face-centered cubic (FCC) structure
In a face-centered cubic (FCC) unit cell, there are:
- 8 atoms at the corners (each corner atom contributes 1/8 of an atom to the unit cell)
- 6 atoms on the faces (each face atom contributes 1/2 of an atom to the unit cell)

Calculating the total number of atoms (Z) in the FCC unit cell:
\[ Z = 8 \times \frac{1}{8} + 6 \times \frac{1}{2} = 1 + 3 = 4 \]

### Step 2: Use the density formula
The formula for density (D) is given by:
\[ D = \frac{Z \cdot M}{N_A \cdot A^3} \]
Where:
- \(D\) = density (8 g/cm³)
- \(Z\) = number of atoms per unit cell (4 for FCC)
- \(M\) = molar mass (g/mol)
- \(N_A\) = Avogadro's number (\(6.022 \times 10^{23}\) mol⁻¹)
- \(A\) = edge length of the unit cell in cm

### Step 3: Convert the edge length to cm
Given the edge length \(A = 400 \text{ pm}\):
\[ A = 400 \text{ pm} = 400 \times 10^{-10} \text{ cm} \]

### Step 4: Calculate the volume of the unit cell
\[ A^3 = (400 \times 10^{-10})^3 = 64 \times 10^{-30} \text{ cm}^3 \]

### Step 5: Substitute values into the density formula
Now substituting the known values into the density formula:
\[ 8 = \frac{4 \cdot M}{6.022 \times 10^{23} \cdot 64 \times 10^{-30}} \]

### Step 6: Solve for the molar mass (M)
Rearranging the equation to solve for \(M\):
\[ M = \frac{8 \cdot 6.022 \times 10^{23} \cdot 64 \times 10^{-30}}{4} \]
Calculating \(M\):
\[ M = \frac{8 \cdot 6.022 \cdot 64}{4} \times 10^{-7} \approx 128 \text{ g/mol} \]

### Step 7: Calculate the number of moles in 128 g
Using the molar mass to find the number of moles:
\[ \text{Number of moles} = \frac{\text{mass}}{\text{molar mass}} = \frac{128 \text{ g}}{128 \text{ g/mol}} = 1 \text{ mol} \]

### Step 8: Calculate the number of atoms
Using Avogadro's number to find the total number of atoms:
\[ \text{Number of atoms} = \text{Number of moles} \times N_A = 1 \text{ mol} \times 6.022 \times 10^{23} \text{ atoms/mol} = 6.022 \times 10^{23} \text{ atoms} \]

### Final Answer
The number of atoms present in 128 g of the crystal is \(6.022 \times 10^{23}\) atoms.

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