A crystalline solid substance has a density of `10 g//cm^(3)` and the length of the edge of the unit cell (FCC) is `2.0A^(0)`. How many number of atoms are present in 200 grams of the solid? 

Options

A. `2xx10^(23)`
B. `1xx10^(26)`
C. `1xx10^(25)`
D. `5xx10^(27)`

Answer

To solve the problem step-by-step, we need to find the number of atoms present in 200 grams of a crystalline solid with a given density and unit cell edge length. Here’s how we can do it:

### Step 1: Understand the parameters given
- Density (ρ) = 10 g/cm³
- Edge length of the unit cell (a) = 2 Å = 2 × 10⁻⁸ cm (since 1 Å = 10⁻¹⁰ m and 1 cm = 10⁻² m)
- For a face-centered cubic (FCC) structure, the number of atoms per unit cell (z) = 4
- Mass of the substance (m) = 200 g

### Step 2: Use the formula for density
The density of a crystalline solid can be expressed using the formula:
\[ \rho = \frac{z \cdot M}{N_A \cdot a^3} \]
Where:
- ρ = density
- z = number of atoms per unit cell
- M = molar mass (in grams per mole)
- \(N_A\) = Avogadro's number (approximately \(6.022 \times 10^{23}\) atoms/mole)
- a = edge length of the unit cell (in cm)

### Step 3: Rearranging the formula to find the number of atoms (n)
To find the number of atoms in a given mass, we can rearrange the density formula:
\[ n = \frac{z \cdot m}{\rho \cdot a^3} \]
Where:
- n = number of atoms in the mass m

### Step 4: Calculate \(a^3\)
First, we need to calculate \(a^3\):
\[ a = 2 \times 10^{-8} \text{ cm} \]
\[ a^3 = (2 \times 10^{-8})^3 = 8 \times 10^{-24} \text{ cm}^3 \]

### Step 5: Substitute the values into the formula
Now we can substitute the values into the rearranged formula:
\[ n = \frac{4 \cdot 200 \text{ g}}{10 \text{ g/cm}^3 \cdot 8 \times 10^{-24} \text{ cm}^3} \]

### Step 6: Calculate n
Calculating the denominator:
\[ 10 \cdot 8 \times 10^{-24} = 8 \times 10^{-23} \text{ g} \]
Now substituting back:
\[ n = \frac{800 \text{ g}}{8 \times 10^{-23} \text{ g}} = 1 \times 10^{25} \]

### Conclusion
Thus, the number of atoms present in 200 grams of the solid is:
\[ n = 1 \times 10^{25} \]