Density of a unit cell is same as the density of the substance. If the density of the substance is known, number of atoms or dimensions of the unit cell can be calculated . The density of the unit cell is related to its mass(M), no. of atoms per unit cell (Z), edge length (a in cm) and Avogadro number `N_A` as :
`rho = (Z xx M)/(a^3 xx N_A)`
A metal X (at. mass = 60) has a body centred cubic crystal structure. The density of the metal is `4.2 g cm^(-3)`. The volume of unit cell is

Options

A. `8.2 xx 10^(-23) cm^(3)`
B. `4.75 xx 10^(-23) cm^(3)`
C. `3.86 xx 10^(23) cm^(3)`
D. `3.86 xx 10^(-23) cm^(3)`

Explanation

To find the volume of the unit cell for the metal X with a body-centered cubic (BCC) structure, we can use the formula for density:

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

Where:
- \(\rho\) = density of the substance (4.2 g/cm³)
- \(Z\) = number of atoms per unit cell (for BCC, \(Z = 2\))
- \(M\) = atomic mass of the substance (60 g/mol)
- \(a\) = edge length of the unit cell (in cm)
- \(N_A\) = Avogadro's number (\(6.022 \times 10^{23}\) mol⁻¹)

### Step 1: Rearranging the formula
We want to find \(a^3\), so we rearrange the formula to solve for \(a^3\):

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

### Step 2: Substitute the known values
Now, we can substitute the known values into the equation:

\[ a^3 = \frac{2 \cdot 60 \, \text{g/mol}}{4.2 \, \text{g/cm}^3 \cdot 6.022 \times 10^{23} \, \text{mol}^{-1}} \]

### Step 3: Calculate the numerator
Calculating the numerator:

\[ 2 \cdot 60 = 120 \, \text{g/mol} \]

### Step 4: Calculate the denominator
Calculating the denominator:

\[ 4.2 \, \text{g/cm}^3 \cdot 6.022 \times 10^{23} \, \text{mol}^{-1} = 2.52924 \times 10^{24} \, \text{g/cm}^3 \cdot \text{mol}^{-1} \]

### Step 5: Calculate \(a^3\)
Now substituting these values into the equation for \(a^3\):

\[ a^3 = \frac{120}{2.52924 \times 10^{24}} \approx 4.75 \times 10^{-23} \, \text{cm}^3 \]

### Step 6: Conclusion
Thus, the volume of the unit cell is:

\[ \boxed{4.75 \times 10^{-23} \, \text{cm}^3} \]