An element (with atomic mass=250 g) crystallises in a simple cube. If the density of unit cell is 7.2 g `cm^(-3)` , what is the radius of the element ?

Options

A. `1.93xx10^(-6)` cm
B. `1.93xx10^(-8)` cm
C. `1.93xx10^(-8) Ã…`
D. `1.93 xx 10^(-8)`m

Explanation

To find the radius of the element that crystallizes in a simple cubic structure, we can follow these steps:

### Step 1: Understand the given data
- Atomic mass (m) = 250 g
- Density (ρ) = 7.2 g/cm³
- For a simple cubic structure, the number of atoms per unit cell (Z) = 1.

### Step 2: Use the formula for density
The formula for density in terms of the unit cell parameters is given by:

\[ \rho = \frac{Z \cdot m}{a^3 \cdot N_A} \]

Where:
- \( \rho \) = density
- \( Z \) = number of atoms per unit cell
- \( m \) = atomic mass
- \( a \) = edge length of the unit cell
- \( N_A \) = Avogadro's number (approximately \( 6.022 \times 10^{23} \) mol⁻¹)

### Step 3: Rearrange the formula to solve for \( a \)
Rearranging the formula to find \( a \):

\[ a^3 = \frac{Z \cdot m}{\rho \cdot N_A} \]

Substituting the known values:

\[ a^3 = \frac{1 \cdot 250 \, \text{g}}{7.2 \, \text{g/cm}^3 \cdot 6.022 \times 10^{23} \, \text{mol}^{-1}} \]

### Step 4: Calculate \( a^3 \)
Calculating the denominator:

\[ 7.2 \, \text{g/cm}^3 \cdot 6.022 \times 10^{23} \, \text{mol}^{-1} \approx 4.334 \times 10^{24} \, \text{g/cm}^3 \]

Now substituting back into the equation:

\[ a^3 = \frac{250}{4.334 \times 10^{24}} \approx 5.77 \times 10^{-23} \, \text{cm}^3 \]

### Step 5: Calculate \( a \)
Taking the cube root to find \( a \):

\[ a \approx (5.77 \times 10^{-23})^{1/3} \approx 3.86 \times 10^{-8} \, \text{cm} \]

### Step 6: Relate \( a \) to \( r \)
For a simple cubic structure, the relationship between the edge length \( a \) and the radius \( r \) of the atom is given by:

\[ a = 2r \]

Thus, we can find \( r \):

\[ r = \frac{a}{2} = \frac{3.86 \times 10^{-8}}{2} \approx 1.93 \times 10^{-8} \, \text{cm} \]

### Final Answer
The radius of the element is approximately \( 1.93 \times 10^{-8} \, \text{cm} \).

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