The number of atoms in `100 g` of an fcc crystal with density `= 10.0g cm^(-3)` and cell edge equal to `200 pm` is equal to

Options

A. `5 xx 10^(24`
B. `5 xx 10^(25)`
C. `6 xx 10^(23)`
D. `2 xx 10^(25)`

Explanation

To find the number of atoms in 100 g of an FCC crystal with a given density and cell edge, we can follow these steps:

### Step 1: Identify the number of atoms per unit cell in FCC
For a face-centered cubic (FCC) crystal structure, the number of atoms per unit cell (Z) is 4.

### Step 2: Convert the cell edge length from picometers to centimeters
The cell edge length (A) is given as 200 pm. We need to convert this to centimeters:
\[ A = 200 \, \text{pm} = 200 \times 10^{-12} \, \text{m} = 200 \times 10^{-10} \, \text{cm} = 2 \times 10^{-8} \, \text{cm} \]

### Step 3: Calculate the volume of the unit cell
The volume (V) of the unit cell can be calculated using the formula:
\[ V = A^3 \]
Substituting the value of A:
\[ V = (2 \times 10^{-8} \, \text{cm})^3 = 8 \times 10^{-24} \, \text{cm}^3 \]

### Step 4: Use the density to find the molar mass (M)
The density (ρ) is given as 10 g/cm³. The formula for density is:
\[ \rho = \frac{Z \cdot M}{V \cdot N_A} \]
Where:
- \( \rho \) = density
- \( Z \) = number of atoms per unit cell (4 for FCC)
- \( M \) = molar mass (g/mol)
- \( V \) = volume of the unit cell (cm³)
- \( N_A \) = Avogadro's number (\(6.022 \times 10^{23} \, \text{mol}^{-1}\))

Rearranging the formula to solve for M:
\[ M = \frac{\rho \cdot V \cdot N_A}{Z} \]

Substituting the known values:
\[ M = \frac{10 \, \text{g/cm}^3 \cdot 8 \times 10^{-24} \, \text{cm}^3 \cdot 6.022 \times 10^{23} \, \text{mol}^{-1}}{4} \]

Calculating this gives:
\[ M = \frac{10 \cdot 8 \cdot 6.022}{4} \times 10^{-24 + 23} = \frac{480.176}{4} = 120.044 \, \text{g/mol} \]

### Step 5: Calculate the number of moles in 100 g
To find the number of moles in 100 g of the substance:
\[ \text{Number of moles} = \frac{\text{mass}}{M} = \frac{100 \, \text{g}}{120.044 \, \text{g/mol}} \approx 0.833 \, \text{mol} \]

### Step 6: Calculate the number of atoms
Now, to find the total number of atoms, we use Avogadro's number:
\[ \text{Number of atoms} = \text{Number of moles} \times N_A = 0.833 \, \text{mol} \times 6.022 \times 10^{23} \, \text{atoms/mol} \]
Calculating this gives:
\[ \text{Number of atoms} \approx 5.01 \times 10^{23} \, \text{atoms} \]

### Final Answer
Thus, the number of atoms in 100 g of the FCC crystal is approximately \(5 \times 10^{24}\) atoms.