The formula for determination of density of unit cell is

Options

A. `(a^(3)xxN_(A))/(ZxxM)g//cm^(3)`
B. `(ZxxM)/(a^(3)xxN_(A))g//cm^(3)`
C. `(a^(3)xxM)/(ZxxN_(A))g//cm^(3)`
D. `(N_(A)xxM)/(Zxxa^(3))g//cm^(3)`

Answer

To derive the formula for the density of a unit cell in a cubic crystal, we can follow these steps:

### Step 1: Understand the Definitions
- **Density (ρ)** is defined as mass per unit volume.
- For a unit cell, we need to determine both the mass of the unit cell and its volume.

### Step 2: Determine the Volume of the Unit Cell
- For a cubic unit cell, the volume (V) can be calculated using the formula:
\[ V = a^3 \]
where \( a \) is the length of the edge of the cube.

### Step 3: Determine the Mass of the Unit Cell
- The mass of the unit cell (m_cell) is determined by the number of atoms in the unit cell (Z) and the mass of each atom (m):
\[ \text{Mass of unit cell} = Z \times m \]
- The mass of an atom can be expressed in terms of the molar mass (M) and Avogadro's number (\( N_A \)):
\[ m = \frac{M}{N_A} \]
- Therefore, the mass of the unit cell can be rewritten as:
\[ m_{\text{cell}} = Z \times \frac{M}{N_A} \]

### Step 4: Combine Mass and Volume to Find Density
- Now, we can substitute the expressions for mass and volume into the density formula:
\[ \rho = \frac{\text{Mass of unit cell}}{\text{Volume of unit cell}} = \frac{Z \times \frac{M}{N_A}}{a^3} \]
- This simplifies to:
\[ \rho = \frac{Z \times M}{N_A \times a^3} \]

### Final Formula
- The final formula for the density of a unit cell in a cubic crystal is:
\[ \rho = \frac{Z \times M}{N_A \times a^3} \]